Progress in Approximation Theory: An International Perspective

Springer Series in Computational

Mathematics

19

Editorial Board

R.L. Graham, Murray Hill (NJ) J. Stoer, Warzburg

R. Varga, Kent (Ohio)

A.A. Gonchar E.B. Saff Editors

Progress in Approximation Theory An International Perspective

Springer-Verlag New York Berlin Heidelberg London Paris Toyko Hong Kong Barcelona Budapest

AA Gonchar Steklov Mathematics Institute 117966 Moscow GSP-1 Russia

E.B. Saff Institute for Constructive Mathematics University of South Florida Tampa, FL 33620 USA

Mathematics Subject Classification (1991): 30-06, 33-06, 41-06

With 9 figures.

Library of Congress Cataloging-in-Publication Data Progress in approximation theory: an international perspectivel

edited by A.A. Gonchar and E.B. Saff. p. cm. - (Springer series in computational mathematics;

19.) "Proceedings of an international conference on approximation

theory that was held March 19-22, 1990, at the University of South Florida, Tampa"-Pref.

Includes bibliographical references (p. 33-35) and index. ISBN-13:978-1-4612-7737-8 1. Approximation theory-Congresses. I. Gonchar, A.A. (Andrei

A.) II. Saff, E.B., 1944- . III. Series. QA221.P78 1992 511 '.4-dc20 92-24316

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© 1992 Springer-Verlag New York, Inc. Softcover reprint of the hardcover 1st edition 1992

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Preface

This volume is the proceedings of an international conference on approximation theory that was held March 19 to 22, 1990, at the University of South Florida, Thmpa. The conference was a historic event in the sense that it brought together for the first time a large number of approximation theorists from the United States and from Russia, the Ukraine, and several other former Soviet countries. In addition, 69 other delegates from European, North American, and Asian countries were present.

The conference was hosted by the Institute for Constructive Mathematics, and the organizing committee consisted of A.A. Gonchar, E.B. Saff, S. Khrushchev, R.A. DeVore, and P. Nevai. Both plenary and shorter invited research announcements were presented.

This proceedings differs in several important respects from typical conference publications. First, all of the contributions that appear were by invitation of the editors. Second, much latitude was granted to authors in preparing their manuscripts; the typical page limitations were relaxed in order to encourage contributions that present not only new results but include detailed perspectives on the subject area.

This volume is designed to give an overview of current research activities in approximation theory and special functions that truly reflects the international nature of these subject areas. For example, 8 of the 19 chapters are authored or coauthored by mathematicians from former Soviet countries. The contents include a wide range of topics, which we now briefly describe.

Of interest to mathematicians in special functions are new results dealing with q-hypergeometric functions, difference hypergeometric functions, and basic hypergeometric series with Schur function argument. Several chapters concern the theory of orthogonal polynomials and expansions, including generalizations of Szego type asymptotics and connections with Jacobi matrices. The convergence theory for Pade and Hermite-Pade approximants is explored in three chapters in which techniques from potential theory are emphasized. The relatively new topics of wavelets and fractals are featured in chapters dealing with invariant measures and nonlinear approximation. Applications of results concerning approximation by entire functions and the problem of analytic continuation are also included. In addition, generalizations of de Brange's inequality for univalent functions are presented from an operator point of view in a quasi-orthogonal Hilbert space setting.

v

vi Preface

Further topics include rearrangements of functions, harmonic analysis, numerical estimates for the de Bruijn-Newman constant, and approximation by polynomials with varying weights.

It is the sincere hope of the organizers that the conference along with this proceedings will act as a catalyst for future joint research and extensive cooperation among approximation theorists of all nations.

The organizers are particularly grateful for the support of the National Science Foundation, the former Soviet Academy of Sciences, and the University of South Florida.

We are also indebted to the local organizing committee consisting of M.E.H. Ismail, M. Parrott, B. Shekhtman, V. Thtik, and C. Williams for their help in planning activities and hosting our foreign guests. Special notes of appreciation are further extended to Maria Carvalho, who served efficiently and energetically as the conference secretary, and to Rafael Munoz, who carefully typed many of the manuscripts.

Moscow, Russia Thmpa, Florida

A.A. Gonchar E.B. Saf!

Contents

Preface..................................................... v List of Participants. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix

Difference Hypergeometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 N.M. Atakishiyev and S.K. Suslov

Pad~ Approximants for Some q-Hypergeometric Functions. . . . . . . . . 37 M.E.H. Ismail, R. Pertine, and J. Wimp

Summation Theorems for Basic Hypergeometric Series of Schur Function Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 S.C. Milne

Orthogonal Polynomials, Recurrences, Jacobi Matrices, and Measures. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 P. Nevai

Szego lYPe Asymptotics for Minimal Blaschke Products. . . . . . . . . . . . 105 A.L. Levin and E.B. Sa/I

Asymptotics of Hermite-Pad~ Polynomials. . . . . . . . . . . . . . . . . . . . . . . 127 A.I. Aptekarev and H. Stahl

On the Rate of Convergence of Pad~ Approximants of Orthogonal Expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 A.A. Gonchar, E.A. Rakhmanov, and S.p. Suetin

Spurious Poles in Diagonal Rational Approximation. . . . . . . . . . . . . . . 191 D.S. Lubinsky

Expansions for Integrals Relative to Invariant Measures Determined by Contractive Affine Maps......................... 215 C.A. Micchelli

Approximation of Measures by Fractal Generation Thchniques ...... 241 S.Demko

Nonlinear Wavelet Approximation in the Space C (Rd) ••••••••••••• 261 R.A. DeVore, P. Petrushev, and X.M. Yu

vii

viii Contents

Completeness of Systems of Translates and Uniqueness Theorems for Asymptotically Holomorphic Functions . . . . . . . . . . . . . . . . . . . . .. 285 A.A. Borichev

Approximation by Entire Functions and Analytic Continuation ..... 295 N. U Arakelyan

Quasi-Orthogonal Hilbert Space Decompositions and Estimates of Univalent Functions. II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 315 N.K. Nikolskii and V.I. Vasyunin

On the Differential Properties of the Rearrangements of Functions. .. 333 v.l. Kolyada

A Class of I.M. Vinogradov's Series and Its Applications in Harmonic Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 353 K.l. Oskolkov

A Lower Bound for the de Bruijn-Newman Constant A. II.......... 403 T.S. Norfolk, A. Ruttan, and R.S. Varga

On the Denseness of Weighted Incomplete Approximations . . . . . . . .. 419 P. Borwein and E.B. Saff

Asymptotics of Weighted Polynomials. . . . . . . . . . . . . . . . . . . . . . . . . .. 431 M. v. Golitschek, G. G. Lorentz, and Y. Makovoz

List of Participants

Dan Amir, Faculty of Exact Sciences, leI-Aviv University, Ramat Aviv 69978, Israel.

G.A. Anastassiou, Department of Mathematical Sciences, Memphis State University, Memphis, TN 38152, USA.

Milne Anderson, Department of Mathematics, University College, London WC16 EBT, UK.

A.I. Aptekarev, Keldysh Institute of Applied Mathematics, Miusskaya Sq. 4, 125047 Moscow A-47, Russia.

Norair U Arakelyan, Institute of Mathematics, Armenian Academy of Sciences, M~shal Bagramian Ave. 24 - B, 375019 - Yerevan, Armenia.

Remi Arcangeli, Laboratoire de Mathematiques Appliquees, Universite de Pau, Avenue de l'Universite, 64000 Pau, France.

Richard A. Askey, Department of Mathematics, University of Wisconsin, Madison, WI 53706, USA.

Natig Mamed ogly Atakishiyev, Physics Institute, Narimanov pro 33, Baku 370143, Azerbaijan.

R V. Atkinson, Department of Mathematics, University of Toronto, Toronto, Canada M5S IA1.

Bogdan M. Baishanski, Department of Mathematics, Ohio State University, Columbus, OH 43210-1174, USA.

George A. Baker, Jr., T-I1, MS-B262, Los Alamos National Laboratory, Los Alamos, NM 87545, USA.

Laurent Baratchart, Res. des Hartes, 700 ch. des Combes, 06600 Antibe, France.

V.I. Belyi, Institute for Applied Mathematics and Mechanics, Ukrainian Academy of Science, ul. Roze Luxemburg 74, Ukraine.

Hubert Berens, Mathematisches Institut, UniversiHit Erlangen - Niirnberg, Bismarckstrasse 1112, D-8520 Erlangen, Germany.

ix

x Participants

Hans-Peter Blatt, Lehrstuhl fUr Mathematik-Angewandte Mathematik, Katholische Univ. Eischstatt, Ostenstrasse 18, D-8078 Eichstatt, Germany.

Andre Boivin, Department of Mathematics, University of Western Ontario, London, Canada N6A 5B7.

Ranko Bojanic, Department of Mathematics, Ohio State University, Columbus, OH 43210-1174, USA.

A.A. Borichev, Steklov Mathematical Institute, St. Petersburg Branch, 27 Fontanka, 191011 St. Petersburg, Russia.

Peter Borwein, Department of Mathematics, Dalhousie University, Halifax, Canada B3H 4H8.

Dietrich Braess, Mathematisches Institut, Ruhr- Universitat, 4630 Bochum, Germany.

Bruno Brosowski, Universitat Frankfurt, Fachbereich Mathematik, D-6000 Frankfurt a.M., Germany.

M.D. Buhmann, DAMTP, University of Cambridge, Silver Street, Cambridge CB3 9EW, UK.

Paul L. Butzer, Lehrstuhl A. fUr Mathematik, Thchnishe Hochschule Aachen, D-5100 Aachen, Germany.

Alfred Cavaretta, Jr., Mathematics Department, Kent State University, Kent, OH 44242, USA.

Bruce L. Chalmers, Department of Mathematics, University of California, Riverside, CA 92521, USA.

Jairo A. Charris, Department of Mathematics and Statistics, OF. 315, National University of Colombia, Bogota, Colombia.

Weiyu Chen, Department of Mathematics, University of Alberta, Edmonton, Canada T6G 2Gl.

Sandra Cooper, Department of Pure and Applied Mathematics, Washington State University, Pullman, WA 99163-2930, USA.

Carl de Boor, Department of Mathematics, University of Wisconsin, Madison, WI 53705, USA.

Marcel G. de Bruin, Delft University of Thchnology, Faculty of Thchnical Mathematics and Informatics, 2600 A J Delft, The Netherlands.

Jesus S. Dehesa, Departamento de Fisica Moderna, Facultad de Ciencias, Universidad de Granada, 18071 Granada, Spain.

Participants xi

Biancamaria della Vecchia, Istituto per Applicazioni della MatematicaCNR, 80131 Napoli, Italy.

Stephen Demko, School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA.

Baiquio Deng, Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA.

Catherin Detail/e, Department of Mathematics, Facultes Univ. ND de la Paix, Rempart de la Vierge, 8, B-5000 Namur, Belgium.

Ron A. De Yore, Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA.

z. Ditzian, Department of Mathematics, University of Alberta, Edmonton, Alberta, Canada T6G 2G 1.

Andre Draux, Laboratoire d'Analyse Numerique, Universite de Lille, 59655 Villeneuve d'Ascq Cedex, France.

Michael Eiermann, Institiit fUr Praktische Mathematik, Universitiit Karlsruhe, D-7500 Karlsruhe 1, Germany.

Tamas Erdelyi, Department of Mathematics, Ohio State University, Columbus, OH 43210-1174, USA.

Mosche Feder, Department of Mathematics, St. Lawrence University, Canton, NY 13617, USA.

Bernd Fischer, Institute of Applied Mathematics, University of Hamburg, D-2000 Hamburg 13, Germany.

Thomas Fischer, Fachbereich Mathematik, Wolfgang Goethe-Universtitiit, 6000 Frankfurt 1, Germany.

Wolfgang H. Fuchs, Department of Mathematics, Cornell University, Ithaca, NY 14853, USA.

Dieter Gaier, Department of Mathematics, University of Giessen, Arndtstrasse 2, 63 Giessen, Germany.

Jeffrey Geronimo, School of Mathematics, Georgia Institute of Technology, Atlanta, GA 30332, USA.

Manfred v. Golitschek, Institiit fUr Angewandte Mathematik, Universitiit Wiirzburg, Am Hubland, 8700 Wiirzburg, Germany.

Andrei A. Gonchar, Steklov Mathematics Institute, Vavilova 42, 117966 Moscow GSP-l, Russia.

xii Participants

Laura Gori, Dipartimento di Metodi e Modelli, "La Sapienza," via Scarpa 10, 00161 Rome, Italy.

William B. Gragg, Department of Mathematics, Naval Post-Graduate School, Monterey, CA 93943, USA.

P.R. Graves-Morris, Department of Mathematics, University of Bradford, Bradford, West Yorks BD7 lDP, UK.

Matthew He, Department of Mathematics, Science, and Technology, Nova University, Ft. Lauderdale, FL 33314, USA.

Weighu Hong, Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA.

Gary Howell, Department of Applied Mathematics, Florida Institute of Technology, Melbourne, FL 32901, USA.

Chai-Chang Hsiao, Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA.

Yingkang Hu, Department of Mathematics and Computer Science, Georgia Southern College, Statesboro, GA 30460-8093, USA.

Mourad E.H. Ismail, Department of Mathematics, University of South Florida, Thmpa, FL 33620, USA.

Marie-Paule Istace, Department of Mathematics, Facultes Universite ND de la Paix, Rempart de la Vierge, 8, B-5000 Namur, Belgium.

K.G. Ivanov, Institute of Mathematics, Bulgarian Academy of Sciences, Sofia 1090, Bulgaria.

W.B. Jones, Department of Mathematics, University of Colorado, Boulder, CO 80309-0426, USA.

Henry Kallioniemi, Hagelvagen 302, S-951 48 Lulea, Sweden.

Valery Kalyagin, Gorky Institut Polytechnic, Minina 24, Russia.

Boris S. Kashin, Steklov Mathematics Institute, Vavilova 42, 117966, Moscow GSP-l, Russia.

Sergei V. Khrushchev, Steklov Mathematics Institute, Vavilov 42, 117966, Moscow GSP-l, Russia.

S. Kiss, Department of Mathematics, Ohio State University, Columbus, OH 43210-1174, USA.

Fadimba Kolli, Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA.

Participants xiii

Victor I. Kolyada, Department of Mathematics, Mathematics Institute, Odessa, Ukraine.

Ibm H. Koornwinder, Centre for Mathematics and Computer Science, 1009 AB Amsterdam, The Netherlands.

Rolitza Kovacheva, Institute of Mathematics, Bulgarian Academy of Sciences, Sofia 1090, Bulgaria.

George Kyriazis, Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA.

Michael Lachance, Department of Mathematics, University of MichiganDearborn, Dearborn, MI 48128, USA.

Andrea Lajorgia, Facolta d'Ingegneria, Monteluco di Roio, 67040-I'Aquila, Italy.

David 1. Leeming, Department of Mathematics and Statistics, University of Victoria, Victoria, Canada V8W 2Y2.

F. David Lesley, Department of Mathematics, San Diego State University, San Diego, CA 92182, USA.

Dany Leviatan, School of Mathematics, Tel Aviv University, 69978 Thl Aviv, Israel.

A.L. Levin, Department of Mathematics, Open University of Israel, Max Rowe Educational Center, 16 Klausner Street, P.O. Box 39328, 61392 ThlAviv, Israel.

Wu Li, Department of Mathematics, Pennsylvania State University, University Park, PA 16802, USA.

X. Li, Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA.

Glenn M. Lilly, 3482 Landsdowne Drive, Apt. 137, Lexington, KY 40517, USA.

Xiaoyan Liu, Department of Mathematics, University of South Florida, Thmpa, FL 33620, USA.

Maria Laura Lo Cascio, Dipartimento di Metodi e Modelli Matematici per Ie Scienze Appticate, via Scarpa, 10-00161 Rome, Italy.

P.D. Loach, School of Mathematics, University of Bristol, Bristol BS6 6LG, UK.

G. Lopez-Lagomasino, Fac. Mat. y Cib., University Habana, Habana 4, Cuba.

xiv Participants

G. G. Lorentz, Department of Mathematics, University of Texas, Austin, TX 78712, USA.

Rudolf A. Lorentz, Postfach 1240, 5205 St. Augustin 2, Germany.

Lisa (Jacobsen) Lorentzen, Division of Mathematical Sciences, University of Trondheim - NTH, N-7034 Trondheim, Norway.

D.S. Lubinsky, Department of Mathematics, Witwatersrand University, WITS 2050, Republic of South Africa.

Wolfgang Luh, Fachbereich 4/Mathematik, Universitat Trier, Postfach 3825, D-5500 Trier, Germany.

Francisco Marcellan, Departamento de Matematica Aplicada, E. T.S. Ingenieros Industriales, c/o Jose Gutierrez Abascal2, 28006 Madrid, Spain.

David R. Masson, Department of Mathematics, University of Toronto, Toronto, Canada M5S 1A1.

Giuseppe Mastroianni, I.A.M.-C.N.R., via P. Castellino 111, 80131 Napoli, Italy.

Syed M. Mazhar, Department of Mathematics, Ohio State University, Columbus, OH 43210-1174, USA.

Scott Metcalf, Wallace 402, Eastern Kentucky University, Richmond, KY 40475-3133, USA.

H.N. Mhaskar, 931 East Lemon Avenue, Glendora, CA 91740-3614, USA.

Charles A. Micchelli, IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA.

Stephen C. Milne, Department of Mathematics, Ohio State University, Columbus, OH 43210-1174, USA.

Ram Mohapatra, Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA.

Lee Mong-Shu, Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA.

J. Myjak, Facolta di Ingegnena, Unive. dell'Aquila, 67100 L'Aquila, Italy.

Paul Nevai, Department of Mathematics, Ohio State University, Columbus, OH 43210-1174, USA.

Nikolai K. Nikolskii, Steklov Mathematical Institute, St. Petersburg Branch, Fontanka 27, St. Petersburg, 191011, Russia.

Participants xv

Olav Njastad, Department of Mathematics, University of 1tondheim, N-7034 1tondheim - NTH, Norway.

Martine Olivi, 1655 Ave. St. Lambert, 06100 Nice, France.

Konstantin I. Oskolkov, Steklov Mathematics Institute, Vavilova 42, 117966 Moscow GSP-l, Russia.

Judith Palagal/o Price, Department of Mathematics, University of Akron, Akron, OH 44325-4002, USA.

K. Pan, Department of Mathematics, University of California, Riverside, CA 92521, USA.

Mary Parrott, Department of Mathematics, University of South Florida, Thmpa, FL 33620, USA.

Pencho P. Petrushev, Institute of Mathematics, Bulgarian Academy of Sciences, 1090 Sofia, Bulgaria.

A. Pinkus, Department of Mathematics, Thchnion, Haifa, Israel.

Vasil Popov, Department of Mathematics, Thmple University, Philadelphia, PA 19122, USA.

Marc Prevost, USTL Flandres Antois, Laboratoire d'Analyse Numerique et d'Optimisation, B4t M3, 59655 Villeneuve d'Ascq Cedex, France.

T. Price, Department of Mathematics, University of Akron, Akron, OH 44325, USA.

H. Qiao, Department of Mathematics, University of South Florida, Thmpa, FL 33620, USA.

E.A. Rakhmanov, Steklov Mathematics Institute, Vavilov 42, 117966 Moscow GSP-l, Russia.

Lothar Reichel, Department of Mathematics, Kent State University, Kent, OH 44242, USA.

Paolo Emilio Ricci, Dipartimento di Metodi e Modelli Matematici, Universita degli Studi di Roma, 10-00161 Rome, Italy.

Ted J. Rivlin, IBM Thomas J. Watson Research Center, P.O. Box 218, Yorktown Heights, NY 10598, USA.

Rene Rodriguez, Department of Mathematics, University of Central Florida, Orlando, FL 32816, USA.

Frode Ronning, Department of Mathematics and Statistics, University of 1tondheim, AVH, N-7055 Dragvoll, Norway.

xvi Participants

Andre Ronveaux, Department of Physics, Facultes N.D. de la Paix, B-5000 Namur, Belgium.

David Ross, Embry-Riddle University, Daytona Beach, FL 32114, USA.

John A. Roulier, Department of Computer Science and Engineering, UISS, University of Connecticut, Storrs, CT 06269-3155, USA.

Stephan Ruscheweyh, Mathematisches Institut, Universitat Wiirzburg, D-8700 Wiirzburg, Germany.

E.B. Saff, Institute for Constructive Mathematics, Department of Mathematics, University of South Florida, Thmpa, FL 33620, USA.

Elisabetta Santi, Dipartimento di Energetica, Universita di l'Aquila, 67040 Roio Poggio -l'Aquila, Italy.

Darrell Schmidt, Department of Mathematical Sciences, Oakland University, Rochester, MI 48309-4401, USA.

A. Sharma, Department of Mathematics, University of Alberta, Edmonton, Canada T6G 2Gl.

Robert Sharpley, Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA.

Boris Shekhtman, Department of Mathematics, University of South Florida, Thmpa, FL 33620, USA.

Nikolai Shirokov, LOMI, St. Petersburg 193231, Russia.

Jamil A. Siddiqi, Department of Mathematics and Statistics, Laval University, Quebec, Canada GIK 7P4.

Rafat Nabi Siddiqi, Department of Mathematics, Kuwait University, Safat, Kuwait 13060.

Mehrdad Simkani, Department of Mathematics, University of MichiganFlint, Flint, MI 48502-2186, USA.

Herbert Stahl, TFH/FB2, Luxemburger Str. 10, 0-1000 Berlin 65, Germany.

Sergei P. Suetin, Steklov Mathematics Institute, Vavilova 42, 117966 Moscow GSP-l, Russia.

Sergei K. Suslov, Kurchatov Institute of Atomic Energy, Moscow 123182, Russia.

G.D. Taylor, Department of Mathematics, Colorado State University, Fort Collins, CO 80523, USA.

Participants xvii

J.P. Thiran, Department of Mathematics, Facultes Universite ND de la Paix, Rempart de la Vierge, 8, B-5000 Namur, Belgium.

Suzanne Thiry, Department of Mathematics, Facultes Universite ND de la Paix, Rempart de la Vierge, 8, B-5000 Namur, Belgium.

M. Vittoria Tirone, Dipartimento Metodi e Modelli Matematici, 16, via Antonio Scarpa, 00161 Napoli, Italy.

Vilmos Totik, Department of Mathematics, University of South Florida, Thmpa, FL 33620, USA.

George Vairaktarakus, Department of Mathematics, University of South Carolina, Columbia, SC 29208, USA.

Walter van Assche, Department of Mathematics, Katholieke Universiteit, Celestijnenlaan 200 B, B-3030 Leuven, Belgium.

V.C. Varadachari, 651 South McKnight Road, St. Paul, MN 55119, USA.

Richard S. Varga, Institute for Computational Mathematics, Kent State University, Kent, OH 44242, USA.

A.K. Varma, Department of Mathematics, University of Florida, Gainesville, FL 32611, USA.

Valeri V. Vavilov, Moscow State University, Department of Mathematics and Mechanics, 119899 Moscow, B-234, Russia.

A.L. Volberg, Laboratory of Mathematical Analysis, V.A.-Styeklov Mathematical Institute, Fontanka 27, 191011 St. Petersburg, Russia.

Jiasong Wang, Department of Mathematics, Nanjing University, Nanjing, 210008, China.

Franck Wielonsky, Chemin du Verde, 06570 St. Paul de Vence, France.

Carol Williams, Department of Mathematics, University of South Florida, Thmpa, FL 33620, USA.

Jet Wimp, Department of Mathematics and Computer Science, Drexel University, Philadelphia, PA 19104, USA.

Nancy Wyshinski, Department of Mathematics, University of Colorado, Boulder, CO 80309-0426, USA.

Xiang Ming Yu, Department of Mathematics, Southwest Missouri State University, Springfield, MO 65804, USA.

John Zhang, Department of Mathematics, Ohio State University, Columbus, OH 43210-1174, USA.

xviii Participants

R. Zhang, Department of Mathematics, University of South Florida, Tampa, FL 33620, USA.

Zvi Ziegler, Department of Mathematics, Technion, Haifa 32000, Israel.

Difference Hypergeometric Functions

N .M. Atakishiyev S.K. Suslov

ABSTRACT The particular solutions for hypergeometric-type difference equations on non-uniform lattices are constructed by using the method of undetermined coefficients. Recently there has been revived interest in the classical theory of special functions. In particular, this theory has been further developed through studying their difference analogs [AWl], [AW2], [NUl], [NSU], [ASI], [AS2] , and [S]. Quantum algebras [D], [VS], [KR], [K], which are being developed nowadays, provide a natural basis for the group-theoretic interpretation of these difference special functions. In the present paper we discuss a method of constructing the solutions for difference equations of hypergeometric type on non-uniform lattices [AS2].

1 Classical Special Functions of Hypergeometric Type

Special functions of mathematical physics, i.e. classical orthogonal polynomials, hypergeometric functions and Bessel functions, are particular solutions of the differential equation, [EMOT] and (NU2] ,

u(x)y" + r(x)y' + >.y = 0, (1.1)

where u( x) and r( x) are polynomials of respective degrees at most two and one, and>' is a constant.

As is well known, equation (1.1) can be rewritten in the self-adjoint form

(upy')' + >.py = 0, (up)' = rp. (1.2)

It is convenient to construct particular solutions of equation (1.1) by using the method of undetermined coefficients (see, for example, the classical work (B2]).

Theorem 1. If a is a root of the equation u(x) = 0, then equation (1.1) has a particular solution of the form

00

y(x) = E CI'I(x - at, 1'1=0

PROGRESS IN APPROXIMATION THEORY (A.A. Gonchar and E.B. Saff, eds.), ©Springer-Verlag (1992) 1.~35.

(1.3)

1

2 N.M. Ata.kishiyev, S.K. Suslov

where Cr.+! --= A + n[r' + l(n - 1)0""]

(n + l)[T(a) + nOO'(a)] . (1.4)

In the case when the equation oo(z) = 0 does not have solutions, series (1.3) will satisfy equation (1.1) if

Cn +2 = _ A + nT' Cn (n + l)(n + 2)00

(1.5)

and a is a root of the equation :r(z) = O.

The proof of Theorem 1 follows from the identity

(1.6)

which can be easily verified. Here Tm(e) = T(e) + moo' (e) and An = -nT' -(1/2)n(n - 1)00". •

With the aid of linear transformations of independent variable equation (1.1) for r' f. 0 may be reduced to one of the following canonical forms [NU2]

z(l - z)y" + h - (a + f3 + l)z]y' - af3y = 0,

zy" + ('Y - z)y' - ay = 0,

y" - 2zy' + 2vy = o. According to (1.3) - (1.5) the appropriate particular solutions are the hypergeometric function, the confluent hypergeometric function and the Hermite function, respectively. Generally speaking, these solutions arise under definite restrictions on variable and parameters. The solutions can be extended to wider regions by anaJytic continuation.

2 The General Series Expansions for Difference Analogs of Special Functions

As is well known (see, for example, [NUl], [NSU], [AS1], [AS2], and [S]), the theory of classical special functions admits a further generalization, if one replaces (1.1) by a difference equation of hypergeometric type on a lattice z = z(z) with the non-uniform step .6.z(z) = z(z + 1) - z(z) [NUl]:

_ .6. [VY(Z)] _ 1 [.6.Y(Z) VY(Z)] oo[z(z)] Vz1(z) Vz(z) + T[Z(Z)] '2 .6.z(z) + Vz(z) + AY(Z) = O. (2.1)

Difference Hypergeometric Functions 3

Here Vy(z) = .6.y(z-l) = y(z) - y(z -1), ZI(Z) = z(z+ 1/2); u(z) and fez) are polynomials ofrespective degrees at most two and one, and ~ is a constant.

For the following_ classes of non-uniform lattices

{ CIZ2 + C2Z + C3 ,

z(z) = CIQ-II + C2q" + C3

(2.2)

(2.3)

(Q,C1 , C2 and C3 are constants) equation (2.1) has the simple property: the difference differentiation of (2.1) yields an equation of the same type [NUl}.

By analogy with equation (1.1), we shall look for solutions of equation (2.1) in the form of a "power" expansion. To this end it is convenient to rewrite this equation in the self-adjoint form (see, for example, [NSU] and [NU2])

.6. [ Vy(Z)] Vz1(z) o-(z)p(z)Vz(z) + ~p(z)y(z) = 0,

(2.4) .6.

V Z 1(Z) [o-(z)p(z)] = T(Z)p(Z),

where o-(z) = u[z(z)] - !f[z(Z)]VZ1(Z), T(Z) = f[z(z)]. (2.5)

Lemma. For the lattices (2.2) ·and (2~3) the equality

p_l(Z)VZ~(Z) {o-(z)p(z)V:CZ) [z(z) - z(e)](n)}

=1(n)-y(n -l)o-(e - n + l)[z(z) - z(e _ 1)](n-2) +

+ 1(n)Tn-l(e - n + l)[z(z) - z(e - l)](n-l) - ~n[z(z) - z(e)](n) (2.6)

is valid. Here the definitions

Tv (s)VZV+l(S) = o-(s + II) + T(S + II)VZ1(S + II) - o-(s),

~p -1(1') [(t(1' -1);:' + i1(1' - 1)0"] ,

m-l [z(z) - z«()}(m) = II [z(z) - z«( - k)], (2.7)

1:=0

zp(s) = z(s + 1'/2)

4 N.M. Atakishiyev, S.K. Suslov

and the notations

a(p) = { 1, ~(p) = {

1', q/J/2 _ q-/J/2

ql/2 _ q-l/2

have been used for the lattices (2.2) and (2.3), respectively.

(2.8)

Formula (2.6) follows from the main identity, obtained in [AS3] (see also [SD for studying the properties of the moments of the classical orthogonal polynomials of a discrete variable.

This lemma is also valid for arbitrary exponents n, if one introduces the "generalized power" according to [ASl] and [S]. For the proof, see Appendix.

Theorem 2. The series of the form

00 n-l

y = y[z(z)] = E Cn II [z(z) - z(a + k)], n=O k=O

where a is a finite root of the equation

and

Cn+l = Cn

u(z) = u[z(z)] - !7'[z(Z)]VZl(Z) = 0

p + -y(n)[a(n - 1)7" + (1/2)-y(n - I)O"]}Vzn +1(a) -y(n + 1)[u(a + n) + r(a + n)Vz1(a + n)]

(2.9)

(2.10)

(2.11)

satisfies a non-homogeneous equation of the type (2.1) on the non-uniform lattices (2.2) and (2.3) with the right-hand side

N-l

G(z) = lim (A - AN)CN II [z(z) - z(a + k)]. (2.12) N_oo

k=O

Proof. Let us write the difference equation of hypergeometric type (2.1) in the self-adjoint form (2.4) and look for its solutions in the form of the following expansion in "generalized powers"

N

y = y(z) = lim ~ cn[z(z) - z(~')](n), N_oo L..J

n=O

(2.13)

where e is a constant. Substituting (2.13) into (2.4), with the aid of identity (2.6) with e - n + 1 = a and u(a) = 0, we obtain

[p-l~ (up~) + A] y = VZl Vz


= lim {~cn'Y(n)Tn_l(a) [z(z) - z(a + n _ 2)](n-l) + N-+oo L..J

n=O

+ t. en(.\ - .\n) [z(z) - z(a + n - 1)](n)} = G(z),

provided that en+! .\n -.\ 7n"" = 'Y(n + I)Tn(a)·

From (2.14), in view of (2.7), (2.11) follows. The theorem is proved .

(2.14)

• It is of interest to compare Theorem 2 and its proof to the result in [AS 2] . The solution (2.9) and the function (2.12) can be rewritten in the fol

lowing explicit form

~ nrr-l (.\ - .\J:)[z(a + k) - z(z)] y(z) = Co t:o J:=O Tj,(ah(k + 1) ,

(2.15) G(z) = Co lim (.\ - .\N)SCN(.\,Z).

N-+oo

Here Co is an arbitrary periodic function with unit period; SCm (.\, z) is the m-th term in the sum in (2.15). (Similar notations will be used in all subsequent relations of the type (2.15).)

When .\ = .\V and v = m = 0,1,2, ... , solution (2.15) is a polynomial of degree m and G(z) == o. The polynomial solutions of equation (2.1) can also be obtained by the Rodrigues formula [NSU]. Comparison of the coefficients of the highest powers for these two cases yields

m-l

Co = Bm rr Tn(a). n=O

Remark 1. Using the expansion

y(z) = LCn[z(z) - z(e)](a+n), n

it is also not difficult to find solutions of the more general form

( ) _ [() _ ( _1)](a) ~ nrr-l (.\ - .\a+J:)[z(a + a + k) - z(z)] u z - Co z z z a + a L..J ( ) ( L 1) ,

n=O J:=O Ta+l: a 'Y a + a; + (2.16)


where u(a) = 0 and -y(a)ra_l(a) = 0 (in particular, putting a = 0 we recover (2.15».

Remark 2. With the aid of Theorem 2, we have constructed the particular solutions of non-homogeneous equation of the type (2.1) with the righthand side (2.12) for the general case, i.e. when the equation u(z) = 0 has at least one finite root. In those cases when equation (2.10) has an infinite root, solutions can be obtained on the basis of the same identity (2.6).

If the equation u(z) = 0 does not have a solution, then one can choose a root of the equation u(z) + r(z)Vzl(Z) = 0 as a point e = b. In this case the right-hand side of identity (2.6) is equal to

u(b-n+1) ( 1) --y(n) b.zn(b _ n) [z(z) - z(b)] n- - An[Z(Z) - z(b)]

and an appropriate expansion has the form

00

y(z) = L: cn[z(z) - z(b)](n) , n=O

Cn+l (A - An)VZn+l(b - n) ~ - -y(n + 1)u(b - n)

Therefore

y(Z) = ~ nrr-l (A - Ak)[Z(b - k) - z(z)]

Co ~ k=O rk(b - kh(k + 1) ,

(2.17)

In the general case we have

u(Z) = Co[z(z) _ z(b)](a) fIT (A - Aa+k)[Z(b - a - k) - z(z)] n=O k=O ra+k(b - a - kh(a + k + 1)

(2.18)

provided that u(b) + r(b)Vzl(b) = 0 and -y(a)ra_l(b - a + 1) = o. Remark 3. Using the identity [AS1] and [S]

p-l(z)vz~(z) [~(Z)p(Z)v~z) Cz(z) _1z(e)]<I'»)] =

= -Y(J.'h(J.' + 1)u(e + 1) [z(z) - z(e + 1)](1'+2)

-Y(J.')r-I'-I(e + 1) [z(z) - z(e)](I'+I)

(2.19)


(cf. (2.6», it is also possible to construct solutions of the non-homogeneous equation of the t!!pe (2.1) in the form of an expansion in inverse generalized powers:

00

y(z) = ~ [z(z) _ z(:n_ 1)](a+n)' cn+1 i(a + n)T_a_n_l(a) -- = ..:....:..~--':---.....:..-'-

Cn A - A-a-n-l

(u(a) = 0, A = A-a);

G( ) Ii i(a + N)r_a_N_l(a) z = - N!f1oo CN [z(z) _ z(a _ 1)](a+N+1)'

Hence it follows that

v(z) = Co ~~-~-~~x [z(z) - z(a - 1)](a)

~ nrr-l i(a + k)r_a_k_l(a) (220) ~k=O (A - A_a_k_l)[Z(Z) - z(a - a - k - 1)]' .

Co' •

G(z) = - [z(z) _ z(a _ 1)](a) J~oo <PN+1(A, z).

If the equation u(z) = 0 has an infinite root or has no solution at all, then for constructing solutions of a non-homogeneous equation of the type (2.1) in the form of the expansion in inverse "powers" one can use the same reasonings as in Remark 2. If u(b) + r(b)'\7z1(b) = 0 and A = A-a, then

v(z) = Co -;-[ z"7"( z"7") -_-z7.( b~+-a---:)-:;] (a~) X

f: IT i(a + k)La_k_l(b + a + k + 1) (2.21) n=O k=O (A - A_a_k_l)[Z(Z) - z(b + a + k + 1)]'

G(z)

3 The Form of Solutions

In constructing solutions of difference equations of the hypergeometric type (2.1) on non-uniform lattices (2.2) - (2.3) by the scheme expounded above, we encounter four cases I-IV. The corresponding coefficients of this equation are listed in the table at the end of this paper.

I. Lattice z(z) = C1Z2 + C2z + C3 • According to (2.5) the function u(z) is an arbitrary polynomial of degree four in the variable z. For C l "lOwe


have O'(z) + r(z)Vzl(Z) = O'(-z - JJ), where JJ = C2C11 • The following cases are possible.

1. In the case when O'(z) = A n!=1 (z - ZI:) we choose a = ZI and -' = -'". The solutions of non-homogeneous equations (2.16) and (2.20) are [ILVW]

() r(a+zl-Z)r(a+zl+z+JJ) u Z = X

r(ZI - Z)r(ZI + Z + p)

(3.1)

(a=O; a= I-JJ-ZI-Zp, p=2,3,4);

v(Z) = r(1 - ZI + z)r(l- ZI - Z - JJ) x r(l- ZI + Z + a)r(l- ZI - Z - JJ + a)

00

I:'l'T'O":':""~~~~~~~~~~~~ n=O

(a = -II, a = ZI + Z2 + Z3 + Z4 + 2p + 11- 1)

respectively; the functions G( z) are equal to

(3.2)

(3.3)

respectively. Additional solutions arise as a result of the successive interchanges ZI +-+ Z2, Z3, Z4.

Appropriate linear combinations of the functions (3.1) and (3.2) satisfy the homogeneous equation (2.1). The Bailey transformation ([Ba], p. 29,or [GR]) relates these solutions with analogs of integral representations, considered in [R], [AS1], and [S].

2. When O'(z) = B n!=I(z - ZI:), a = Z1. the solutions (2.16) and (2.20) are

u(z) r( a + ZI - Z )r( a + ZI + Z + p) = x r(ZI - Z)r(ZI + z + JJ)

t (a -1I)n(a + ZI - z)n(a + ZI + Z + JJ)n (3.5) n=O (a + ZI + Z2 + P)n(a + ZI + Z3 + P)n(a + l)n

(a = 0, a = 1 - JJ - ZI - Z2, a = 1 - JJ - ZI - Z3);


() r(1-ZI+Z)r(1-I'-ZI-Z) V Z = X

r(1- v - ZI + z)r(1- v-I' - z)

(-v, 1 -I' - v - Zl - Z2,

3 F2

1-v- ZI + Z,

(3.6)

1 - I' - v - Zl - Z3 ) ,

1 - I' - v - ZI - Z

respectively. In all these cases G(z) == 0 for Re(Z2 + Z3 + I' + v) > o. The successive interchanges ZI +-+ Z2, Z3 lead to other solutions.

3. For u(z) = C(z - ZI)(Z - Z2) according to (2.16) we find the following solutions

(3.7)

which correspond to G(z) == 0 when IACrC-II < 1. Two more solutions arise after substitution ZI +-+ Z2.

4. When u(z) = D(z - zd, the following formal solution

(3.8)

arises from (2.16). This formal series does not converge unless it terminates. 5. In the case when 17 = E = const. the formal solution has the form

( ) _,,(_A/u)n/2( c)n(a-n+1 ) (a-n+1 ) u Z - ~ r(n + 1) - 1 2 - Z n 2 + Z + I' n·

(3.9)

II. Lattice z(z) = C2z + C3. The functions u(z) and u(z) + r(z)VTzl(Z) are arbitrary polynomials of second degree with coinciding coefficients of z2(see table). Let us discuss the following solutions.

1. In the general case, when

(3.10)

10 N .M. Atakishiyev, S.K. Suslov

u(z) + r(z)V'Xl(Z) = A(z + Z3)(Z + Z4) solutions (2.16) and (2.20) with a = ZI and A = A/I have the form ([Bl] and [T2])

u(Z) =

(3.11)

(a = 0, a = 1- ZI - Z3, a= 1- ZI - Z4);

v(Z) = (3.12)

(a = -v, a = ZI + Z2 + Z3 + Z4 + v-I)

respectively, and G(z) == 0 when Re(z - Z2) > O. An interchange ZI +-+ Z2 leads to additional solutions.

Formulas (2.18) and (2.21) give solutions, which can be obtained from (3.11) and (3.12) as a result of the substitutions z -+ -z, ZI +-+ Z3, and Z2 +-+ Z4·

2. In the case of

u(z) = B(z - zI), u(z) + r(z)V'x1(Z) = D(z + Z2) (3.13)

according to (2.16) and (2.20) we find

u(z) = rea + Z1 - z) f: (a - v)n(a + ZI - z)n (1- B)n r(ZI - z) n=O (a + ZI + Z2)n(a + l)n D

(3.14)

(a = 0, a = 1 - Z1 - Z2);

v(z) r(1 - Z1 + z) ( -v, 1 - v - Z1 - Z2

=-:-~-....;;...-~ 2F1 r(l- v - Z1 + z)

1- v - Z1 + z ~1 ). I-BID

(3.15)

Here G(z) == 0 when 11 - BD-11 < 1. Formulas (2.18) and (2.21) lead to solutions, originating from (3.14) - (3.15) as a result of the substitutions z -+ -z, Z1 +-+ Z2, and B +-+ D.

3. If B = D in (3.13), then the corresponding solutions have the form

u(z) = rea + ZI - z) f: (a + ZI - z)n (A C?)n r(Z1 - z) n=O (a + Z1 + Z2)n(a + l)n B '

(3.16)


4. When o'(z) = B(z - Z1), but o'(z) + r(z)Vz1(z) = E, formulas (2.16) and (2.20) give

u(z) = 2FO(-V,Zl-Z,-B/E), (3.17)

f(l- Z1 + z) v(z) = f(1 ) 1F1(-V, 1- v - Z1 + z,E/B).

- v - Z1 + z

Analogously, when o'(z) = C and O'(z)+r(z)Vz1(Z) = D(Z+Z2) we come to solutions, originating from (3.17) as a result of substitutions z ~ -z, Z1 ~ Z2, B ~ -D and E ~ C.

5. For 0' = B(z - zt) and 0' + rVz1 = 0 we have

u(z) = [z(z) - Z(Z1 + V - 1)](11) = f(~(I- Z1 + z) ). (3.18) - v - Z1 + z

Solution for the case 0' = 0 and 0' + rVz1 = D(z + Z2) is obtained if we substitute z ~ -Z,Z1 ~ Z2 and B ~ -D.

6. For 0' = C and 0' + rV Zl = E we have

u(z) = qZ, E2q2 + (AC~ - E - C)q + C = O. (3.19)

7. To pass to the simplest cases when 0' - C = 0' + rVz1 = 0 and 0' = 0' + rVz1 - E = 0 it suffices to put E = 0 or C = 0 in (3.19).

8. When 0' = 0' + rVz1 = constant, we have the following solutions

( ) = '"' (-A/O')n/2(_C )n (a - n + 1 _ ) u z L.J f( ) 2 2 z, . n+l

n n

(3.20)

( I-a l+a ) --+z -- -z AC2

u+(z) = 2F1 2 ' 2 __ 2,

1/2 40'

(3.21)

( A)1/2 ( a) (l-a/2+Z,I+a/2_Z u_(z) = -- C2 z- - 2F1

0' 2· 3/2

ACi) . 40'

In. Lattice z(z) = C1q-Z + C2qz + Ca. According to (2.5) in the most general case we have

o'(z) = Aq2z + BqZ + C + Dq-Z + Eq-2z (3.22)

(A, B, C, D and E are arbitrary numbers). For C1C2 f:. 0 the equality o'(z) + r(z)Vz1(z) = 0'( -z - F) holds with F = lnl'/lnq and I' = C2C11.


The corresponding coefficients of equation (2.1) are given in the table at the end of this paper. Let us now consider all the possible cases.

1. For O'(z), = Aq-2z n;=I(qZ - zp), zp i- 0, Iql < 1, we choose qa = ZI (all other cases can be obtained by substituting ZI +-+ Z2, Z3, Z4 in final formulas) and ~ = ~II' Solutions of non-homogeneous equation (2.1) are

u(Z) =

V(Z) =

( Q = OjQ = In (-q-) /Inq, p= 2,3,4), JJZIZp

(qa+z+l / ZI, qa-z+1 / JJZl j q)oo x (ql+z / ZI, ql-z / JJZl j q)oo

(Q = -II, Q = In(JJ2z1z2z3z4qll-l)/ln q),

(3.24)

which correspond to (2.16) and (2.20), respectivelYj and functions G(z) are equal to

G(z) =

In formulas (3.23) - (3.26) the notations (Vj q)o = 1, (Vj q)n = (1 - v)(1 -vq) ... (1- vqn-l) and (Vj q)oo = limn_oo(vj q)n have been used. The functions in (3.23) and (3.24) admit representations as basic hypergeometric series

q,t ) =

(3.27)


with r = 3 and s = O. We followed the notations in [GR]. Appropriate linear combinations of functions (3.23) and (3.24) satisfy the

homogeneous equation (2.1). The Bailey transformation ([Ba], p. 69j[GR)) relates these solutions with analogs of integral representations, considered in [NR], [ASl], and [S].

2. In the case when O'(z) = Aq-Z n!=l(qZ - zp), zp # 0 and Iql < 1, solutions (2.16) and (2.20) for qll = Zl and A = All have the form

u(Z) =

v(Z)

(Zlq-Z,jjZlqZjq)oo x (Zl qa-z ,jjZl qa+z j q)oo

( a = OJ a = In (-q ) /Inq, p= 2,3), jjZlZ2

(ql-II+Z / Zl, ql-II-Z / jjZl j q)oo X

(q1+Z / Z, ql-z / IlZl j q)oo

I-II -II q q ,--,

jjZlZ2

respectively. The right-hand side is nonzero only for (3.28):

G( ) _ 3/2-a A(tl'-", zlq-Z, jjZlqZ j q)oo Z -q Ci(l-q)2(jjZlZ2qa,jjZlz3qa,qa+ljq)00.

(3.28)

(3.29)

(3.30)

It is not hard to write down a combination of solutions (3.28) that satisfies the homogeneous equation (2.1).

3. For O'(z) = A(qZ - Zl)(qZ - Z2), Zl, Z2 # 0 and Iql < 1, according to (2.16) and (2.20) we have

u(Z) = (Zlq-Z,jjZlqZjq)oo X

(Zlqa-z, jjZl qa+z j q)oo

00 (qa-lI,zlqa-z,jjZlqa+z jq)n n

?; (jjZlZ2qa, qa+lj q)n q,

(3.31)


v(z)

( a = 0, a = In (-q ) lIn q) ; PZ1Z2

q-II, q1-11

PZ1Z2

2<P2 Z2

q,q-Zl

q1-II+Z q1-II-z

Zl PZ1

An interchange Zl +-+ Z2 yields more solutions. 4. For O'(z) = A(qZ - Zl)qZ, Zl I- 0 and Iql < 1 we find

u(z) = '0'> ( q-', z~-: pz,q' q,q),

G(z)

v(z)

q3/2 A( q-II , Zl q-Z , PZ1 qZ ; q)oo . q(1 - q)2(q; q)oo '

q-II

q1-II+z q1-II-z 2-11 )

q, :z~ , Zl PZ1

(3.32)

G(z) == o.

(3.33)

(3.34)

G(z) == O.

5. To construct solutions of equation (2.1) for the case O'(z) = Aq2z it is necessary to go back to the initial identity (2.6). Let us set there e -n+ 1 = a and Tn _1(a) = 0, which give a = !(1- n - G) with G = F + i1l'In- 1q, F = Inp/lnq, and P = C2C11 • The required expansion has the form

[ ( G n 1)] (n) u(z) = Len x(z) ~ X -'2 + T '

n


( l+n+G)' -y(n + Ih(n + 2)0' - 2 (3.35)

It is necessary to consider expansions in two different but related generalized powers, i.e.

(3.36)

A(ql-II IIq2+2z ,,-lq2-2Z. q2) G ( ) (C q-Z + G qZ)ql/2 , -,. , -,. , 00

- z = 1 2 C~(I_q)2(q3,q2jq2)00

Analogous reasonings in the case of expansion in inverse generalized powers lead to the solution of the homogeneous equation:

v(z) = qz2+Fz(_pql-II+2z,_p-lql-II-2Zjq2)00 x

(3.38)

6. H A = E = 0, then in (3.22) we have ~II == 0. For O'(z) = Bq-Z(qZ -Zl)(qZ - Z2), qG = Z1 ::f:. 0, and Z1,2 = (-C ± JC2 - 4BD)/2B, according to (2.16) we find the following solutions of the homogeneous equation:

(3.39)

16 N.M. Ata.kishiyev, S.K. Suslov

...:.( Z:-,l:..,;:q_-~%;..:., I'-;Z""i-l~q%~j~q.<..,;)oo~ X

(ql-% ql+%. ) --,--,q I'Z2 z2 00

(3.40)

)-Two more solutions are obtained as a result of the interchange Zl ~ Z2.

7. For u(z) = B(q% - zt}, qa = Zl i- 0 there arises the following solution of the homogeneous equation

(3.41)

8. We have been unable as yet to construct solutions of the initial equation when u(z) = Bq%.

9. For u = C = constant, in complete analogy with the Case 5 we find the following solutions of the homogeneous equation:

q2, _ ACl C2 (1 _ q)2 ) , (3.42) uql/2

IV. Lattice z(z) = Clq-% + C3 . According to (2.5) we have

u(z) = C + Dq-% + Eq-2%,

u(z) + r(z)V'zl(Z)

(A, B, C, D and E are arbitrary constants).

Let us discuss possible cases.

(3.44)


1. For nonzero A, E and C it is convenient to represent (3.44) as

(3.45)

with -1 -D±~D2_4EC -B±~B2_4AC

Z - Z3,4 = 1,2 - 2E' 2A

In this case the solutions (2.16) with qG = Z1 and (2.21) with q-b = Z3

have the form (cf. [H2] and [H3J)

u(Z) =

(3.46)

( or = 0, or = In(-q-)/lnq, or = In(-q-)/lnq) , Z1Z3 Z1 Z4

v(Z) =

(3.47)

respectively, and

3/2-a A(qa-II, Z1z2z3z4qa+II-1, Z1q-.J j q)oo

q C1(1- q)2(Z1z3qa, Z1 zd a , qa+1; q)oo ' (3.48)

G(Z) =

(3.49)

Appropriate linear combinations of the functions (3.46) and (3.47) satisfy the homogeneous equation.

Formulas (2.18) with q-b = Z3 and (2.20) with qG = Z1 yield

u(z) = q-a.J (Z3q.J; q)oo X (Z3qa+.Jj q)oo

(3.50)


( a = 0, a = In (-q ) /lnq, a·= In (-q ) /lnq) , Z1 Z3 Z2Z3

v(Z)

(3.51)

(a = -II, a = In (Z1Z2z3z4qU-1)/lnq)

and G(z) == 0 when Iq-" z4"11 < 1 and Iq" z2"11 < 1, respectively (Iql < 1). 2. In the case when C = 0 in (3.44) with a, b = -00 and Iql < 1, the

solutions of the homogeneous equation are [He1], [He2] , [He3] , [1'1], [H1], and [GR]

u(z) -() -a" ~ q , 'Aq ,q n 1-" 00 ( a-u E a+u-l.) (A ) n

= U Z = q D --q n=O (lrqa,qa+1;q)n B

(3.52) (a = 0, a = In(qBD- 1)/lnq),

v(z)

(3.53) (a = -II, a = In(EA- 1qU-1)/lnq).

Formula (2.20) with q-a = -DE-1 and Iql < 1 also gives solutions of the homogeneous equation

(_DE-1qa+,,+1. q) ( ) a" , 00 X

V Z = q (-DE-1q"+1;q)oo

(a = -1I,a = In(EA-1qU-1)/lnq).

According to (2.16) with q-a = -DE-1 and to (2.21) with q-b -AB-1(lql < 1), we find that

(-ED-1q-";q)00 u(z) = x

(_ED-1 qa-"j q)oo

Difference Hypergeometric Functions

(a = 0, a = In(ADB-1 E-1q)/lnq) ,

v(z) = (_AB-lqa-z+ljq)oo X

(-AB-lql-zjq)oo

respectively, and the functions G(z) are equal to

A(qa-v EA-lqa+v-l _ED-lq-Z. q) 3/2-a' , , 00

q G1(I-q)2(Wqa,qa+1jq)00' G(z) =

E(qa ADB-l E-1qa+l. q) _ql/2-a , , 00

G1(1 - q)2 (qa+V+1, iqa-v+2, _~ql-Z j q) 00

19

(3.55)

(3.57) The corresponding linear combinations of functions (3.55) and(3.56) are solutions of the homogeneous equation.

For B = -qA, D = _q2 A, E = q2aA and 11 = -I from (3.52) with a = 0, we may sum the series by the q-binomial theorem

The right-hand side also arises from a general analog of the integral representation for particular solutions of homogeneous equation [ASI] and [S].

Taking the linear combination ula=o + (b - q)/(a - q)Vla=l of solutions (3.52) and (3.53) with B = -qA, D = -qbA, E = q2aA and 11 = -I, we obtain the Rarnanujan's ItPl function, which is the left-hand side of Ramanujan's sum

The right-hand side of the abo.ve identity can be readily obtained from a general analog of integral representation. For proof and references to known proofs of Rarnanujan's sum see [GR].


3. In the case when A = 0 in (3.44), from (2.16) and (2.20) with qtJ = ZI we find

(3.58)

( a = 0, a = In ( -g:J /Inqj I ~ z2qv I < 1, Iql < 1) ,

v(z) (Z-lql-V+iI. q) ( q-V, _ CB ql-v = -Vii 1 , co ZI

q (1 l+iI) 2</'1 ZI q jq co -1 I-v+iI zl q

(IZ;lqill < 1, Iql < 1),

.,.,'.1+. ) (3.59)

respectively. Here G(z) == 0 and zl~ = [-D± ..jD2 - 4ECj/2E. The interchange ZI +-+ Z2 leads to three mo~e solutions.

Formula (2.18) with q-" = -CB-1 also gives solutions of the homogeneous equation:

u(z) = (_BC-l ql-a-il jq)co X

(_BC-l ql-ilj q)co

(a=o, a=In(-g~)/Inq, a=ln(-g!)/Inq). Formula (2.21) yields

(_BC-lql-v-iI j q)co X v(z) =

(_BC-l ql-;-ilj q)OQ

(-V B I-v B I-v q ,--q ,--q

3</'2 CZ1 CZ2

0, _BC-lql-V-iI

G(z)

.,. ) , (3.61)


4. When E = 0 in (3.44), it is convenient to use (3.45) with Z1 = -DO-1 and Z2 = O. Solution (2.16) has the form

u(Z) =

G(Z) = (3.62)

( a = 0, a = In (-q ) /Inq, a = In (-q ) /Inq) . Z1Z3 Z1 Z4

An appropriate linear combination of any two functions (3.62) is a solution of the homogeneous equation.

From (2.18) and (2.21) with q-b = Z3 we have

(3.63)

(a=o, a = In (-q ) /Inq) , Z1 Z3

v(Z) (

q1-V 1 -v

(z- q1-V-Z q) q, --3 , Z1Z3

-1 1 Z 2<P1 (Z3 q - ,q) -1 1-v-z

Z3 q .,ZlZ4." ) . (3.64)

In these cases G(z) == 0 if Iql < 1. The interchange Z3 +-+ Z4 leads to additional solutions.

5. Let O'(z) = Dq-Z + Eq-2z and O'(z) + r(z)Vz1(Z) = Aq-2z. For a = -00 and Iql < 1 formula (2.16) gives the following formal solution

(3.65)

which has meaning only if the series terminates. If q-a = -DE-1, then

(3.66) 3/2A(q-V, EA-1qv-1, _ED-1q-z j q)oo

= q 01(1 - q)2(qj q)oo


According to (2.20) and (2.21) when a,b = -00 and Iql < 1 there arise the following solutions of the homogeneous equation:

_ 00 (qa j q)n(_DE-1ql+z)n v(z) v(z) - qaz ~ ~~~"........,....---=-~.,....-= - L...- (qa+lI+1 AE-lqa-II+2. q)

n=O' , n

(3.67) (a = -v, qa = EA-1qll-l).

For q-a = -DE-1 we find that G(z) == 0 and

(_DE-1qa+z+l. q) qaz , 00 X (_DE-lqz+1j q)oo v(z) =

(a = -v, qa = EA-1qll-l).

6. Let u(z) = Eq-2z and u(z) + r(z)Vz1(z) = Aq-2Z + Bq-z. For a, b = -00 and Iql < 1, according to (2.16) and (2.18), we have G(z) == 0 and

(3.69)

Similarly formulas (2.20) and (2.21) for a, b = -00 and Iql < 1 yield the following solutions of the homogeneous equation:

az 00 (qa j q)nqn(n-l)/2(BE-1qa+z+2)n v( z) = q ~ =-.....:~-=-:-:----,-.,,~--:=--...,--~ L...- (qa+II+1 AE-lqa-lI+z. q)

n=O ' , n

(a = -v, qa = EA-1qll-l).

Formula (2.21) with q-b = -BA-l gives

(3.70)

00 (a) n V Z - (_AB- 1 q<>-S+l j )90 ~ q jq nq ( ) - (-AB l ql Sjqfoo L...- (qa+II+1 AE-lqa-II+2 _AB-lqa-z+1. q) ,

n=O' , , n

(3.71) G(z) = _ql/2-a E(qa j q)oo

Cr(l- q)2(qa+II+1, AE-lqa-II+2, _AB-lql-zj q)oo

(a = _v,qa = EA-1qll-l).

A linear combination satisfies the homogeneous equation. (In what follows all solutions will be solutions of the homogeneous equation.)


7. Let u(z) = Dq-Z + Eq-2z and u(z)+r(z)'Vx1(Z) = Bq-z. In the case of a, b = -00 and Iql < 1 formulas (2.16) and (2.18) lead to the following solutions of the homogeneous equation [H2]

For q-a = -DE- 1 from (2.16).we find

(3.72)

q. -ED-'q'+"-' ).

(3.73)

According to (2.20) for q-a = 0 and q-a = -DE- 1(1ql < 1) one may construct two more solutions

(3.74)

q. - ~q'-"+' ).

8. In the case of u(z) = Dq-Z and u(z) + r(z)'Vx1(Z) = Aq-2z + Bq-Z formula (2.16) with a = -00 and Iql < 1 leads to the following solutions

(3.75)

AB-1 1-z ) . q, - q ,


and formula (2.21) for q-b = -BA-1 gives

(3.76)

9. For u(z) = C + Dq-Z + Eq-2z and u(z) + r(z)Vz1(Z) = C according to (2.16) we come to the following formal solution

u(z) =

-1 zl,2

( _II -z I Z2 II) 2tpO q ,Zlq q, Zl q ; (3.77)

-D±VD2 -4EC 2E

which has meaning only if the series terminates. In the case of (2.20) the solution of the homogeneous equation is

Further solutions arise as a result of the interchange Zl +-+ Z2. 10. Let u(z) = C and u(z)+r(z)Vz1(Z) = Aq-2Z +Bq-Z +C. According

to (2.18) and (2.21) we find

u( z) = ,"', ( .~', :.q. q,zi"'~')' (3.79)

v(z)

Here Z3,4 = (-B ± VB2 - 4AC)/2A. 11. For u(z) = C + Dq-Z and u(z) + r(z)Vz1(z) = Bq-Z + C formulas

(2.16) and (2.18) give

u(z) = (_DC-1q-z;q)00 X

(_DC-1 qOt-z; q)oo (3.80)

- q ,qn A 1 1 2 00 ( DC-1 Ot-z.) [CC2 ]n ;(DB-1qOt,qOt+1;q)n - BDq1/2( -q) ,


u(z) (_BC-1q1-a-Z;q)00 X

(_BC-1q1-Z; q)oo

00 (_CB-1qa+z. q) q1n (n-1) [C2qa-Z ] n ""' ' n -A 1 (1 _ )2 (3 81) ~ (DB-1qa,qa+1;q)n B q1/2 q .

(a = 0, qa = qBD-1).

12. In the case of (1' = Eq-2z and (1' + TVZ1 = Aq-2z we have

1 { _ [ _ ] 1/2} u(Z)=q-IIZ; qr,2=2E E+Aq-A± (E+Aq-A)2_4EAq ,

where .x = ACr(1 _ q)2q-1/2. (3.82)

13. Let (1' = Dq-Z + Eq-2z and (1' + TV Zl = O. According to (2.16) for q-a = 0 and Iql < 1 we find

q , q n II-Z - q, q 00 00 ( -II.) (E )n (ED-1 -z. ) u(z) =?; (q;q)n - D q = (-ED-1q"-Z;q)00 (3.83)

(we have made use of the q-binomial theorem). 14. In the case of (1' = 0 and (1' + TVZ1 = Aq-2z + Bq-Z according to

(2.18) with q-b = 0 one finds

-( ) ( _II I A 1-Z) (_AB-1q1-II-Z; q)oo u z = l<PO q q,- B q = (-AB-1q1-z;q)00 . (3.84)

15. Let (1' = Dq-Z and (1' + TVZ1 = Aq-2Z. Formulas (2.16) and (2.18) lead to following formal solution

u(z) =2 <Po (q-II,O I q,- ~q1-Z);

and formulas (2.20) and (2.21) yield

(3.85)

v(z) = q-IIZ 2<PO (q-II,O I q,-~ q211+Z-1) , (3.86)

if the series terminate. 16. When (1' = Eq-2z and (1' + TVZ1 = Bq-Z according to (2.16) and

(2.18) we find

(3.87)


and in view of (2.20) and (2.21) we have

(3.88)

When II = -1 and E = B q3/2 we come to the well-known series for the theta-function:

00

= L: qn2 /2tn = (_q1/2t, _q1/2r1 , qj q)oo, t = q-Z n=-oo

(we have made use of the Jacobi triple product identity). The right-hand side also arises from a general analog of an integral representation (cf. [S]).

17. For u = a + Dq-Z and u + rV Zl = a we find the following formal solution:

( ) ( D -z I aal ( )2) U Z = 2<;'0 -aq ,0 q,->"D2q1/21-q . (3.89)

18. For u = a and u + rVz1 = Bq-Z + a we have

ii(z) = '1'" ( -C:-'O> 0, ~ c;.~;: (1 - 0)' ) . (3.90)

19. In the case of u = Dq-Z and u + rVz1 = Bq-Z we find

U1(Z) = 1<;'1 ( 0 q,>"C;~~:(1-q)2)' DB-1 q

(3.91)

C:q1-z 2 ) q,>.. D q1/2 (1- q) .

Function U1 arise in calculating matrix elements of the quantum group of plane motions [VK].

20. For u - Eq-2z = U + rVz1 = 0 the solution has the form

f{ = 1 _ >.. aUl - q)2 Eq1/2 (3.92)


21. Analogously, for u = u + TVX1 - Aq-2z = 0 we have

u(z) = q-IIZ, -II _ 1 _ A e?(1- q)2 q - Aq3/2 (3.93)

22. In the case when u - Dq-Z = u + TVX1 = 0 we find

00 1 [e?(1- q)2 _z]n (e?(I- q)2 -z )-1 u(z) = L: -( .) A D 1/2 q = A D 1/2 q ,q (3.94)

n=O q, q n q q 00

(we have used Euler's identity). 23. For u = u + TVX1 - Bq-Z = 0 we obtain

_ ( ) _ ~ qn(n-1) /2 [-A e?(1 - q)2 -z] n _ (A e?(1- q)2 -z ) U Z - L.J (.) B 1/2 q - B 1/2 q , q

n=O q, q n q q 00

(3.95) The series is summed using another identity of Euler.

24. Let u = u + TVX1 = e 'I O. The solution of the homogeneous equation has the form

00 qn(n-1)/4 ( A)n/2 ( ) "en -nz

U Z = L.J 1 r (n + 1) -;;: q = n=O q

(3.96)

The expansions in even and odd powers only are also solutions of the initial equation:

(3.97)

g', C1(1 - g)'(~/u)i'I'-" ).

Most of the solutions of equation (2.1), which we have considered here, are particular or limiting cases of the preceding ones. Polynomial solutions arise in the following cases: 1-1,2; II-I to 4; 111-1 to 5; IV-1 to 10, 15 and 16. Some of these solutions arise in studying matrix elements, Clebsch-Gordan


and Racah coefficients for the quantum SU(2) group [D], [VS], [KR], and [K] (cf. [NSU)).

We have discussed the construction of solutions for canonical types of the equation (2.1) by the method of undetermined coefficients. By now it is clear that the theory of solutions to homogeneous and non-homogeneous forms of equation (2.1) is very rich. We have additional results concerning integral representation for solutions of (2.1), and functional relationships among different types of solutions. This work will appear elsewhere.

We thank P.P. Kulish, M.V. Savel'yev, V.N. Tolstoi and M.E.H. Ismail for discussions. The final version of this paper was prepared while the authors visited the University of South Florida in Tampa and we gratefully acknowledge their hospitality. We are grateful to Jodi Anderson for her careful preparation of this manuscript.

4 Appendix. A Proof of the Lemma

To prove identity (2.6) we consider first the two following statements.

Proposition 1. Under the hypothesis of main lemma the equality

is valid, where

D.8 {U(S)p" (S)[Z"+l(S - 1) - Z"+l(e)](I')} =

= P(s)[z,,(s) - z,,(e)](I'-l)p,,(s)V"z,,+1(s) (4.1)

P(S) = -Y(Jl)u(s) + T,,(S)[Z,,_I'(s + Jl) - z"-I'(e + 1)]. (4.2)

Proof. Appyling the product rule D.[J(s - l)g(s)] = f(s)D.g(s) +g(s)V"f(s) with f(s) = [z"+1(s) - Z"+1(e)](I') and g(s) = u(s)p,,(s) and then using relations (11), (20), and (16) from Ref. [ASl] (or (2.3), (2.22) and (2.8) from [S]), we come to (A.l). •

Proposition 2. Under the hypotheses of the lemma, the function in (A.2) has the form [AS3]

P(s) = Do + D1[z,,(s) - z,,(e - Jl + 1)] + (4.3)

+D2[z,,(s) - z,,(e - Jl + l)][z,,(s) - z,,(e + 1)],

where

Do -Y(Jl)u(e - Jl + 1), Dl = TI'+,,(e - Jl + 1), (4.4)


D () -' l()-" 2 = a JJ Til + 21 JJ tr II .

Proof. Let us find out how function (A.2) depends on variable s. According to Lemma 1 from [ASl] or (2.2) from [S] we have

1(JJ)tr(S) = 1(JJ)O"II (s) - ir,,(sh(JJ)VzlI+1(s),

where

1(JJ)VZII+1(s) =z (s+ JJ~V) -z (s+ V;JJ).

Substituting in (A.2) and taking into account equation (5) from Ref. [ASl] (or (1.9) from [S]) gives

pes) = 1(JJ) 0"11 [ZII(S)] + a(JJ)r,,[zlI(s)]zlI(s) + (4.5)

Hence it follows that the function P( s) is a polynomial of degree at most two in the variable ZII(S). Thus it can be written in the form (A.3), where Do, D1 , and D2 do not depend on s. The coefficient Do in (A.3) can be easily found, if one puts s = e - JJ + 1, i.e.

Do = pee - JJ + 1) = 1(JJ)tr(e - JJ + 1).

According to (A.2) and (A.3), when s = e + 1 we have

pee + 1) = 1(JJ)tr(e + 1) + TII(e + 1)[zlI_,.(e + JJ + 1) - ZII_,.(e + 1)] =

= 1(JJ)tr(e - JJ + 1) + DdzlI(e + 1) - zlI(e - JJ + 1)].

Since (see Eqs. (6) or (1.10) from Refs. [ASl] and [S], respectively)

ZII-,.(e + JJ + 1) - ZII-,.(e + 1) = 1(JJ)VZII+1(e + 1),

zlI(e + 1) - zlI(e - JJ + 1) = 1(JJ)~ZII-"+1(e), then with the aid of the relationship (2.7) one finds that

Dl = tr(e + v + 1) + T(e + v + I)Vz1(e + v+ 1) - tr(e - JJ + 1) = ~ZII-"+l(e)

= T,.+II(e - JJ + 1).

Comparing the coefficients in (A.3) and (A.5) yields

D2 = a(JJ)r,,' + i1(JJ)O"/'.

This proves the proposition. •


Corollary. Substituting (A.3) into (A.l), with the aid of the relations

[x,,(s) - x,,(e - p + l)][x,,(s) - x,,(e)](I'-l) = [x,,(s) - x,,(e)](I'),

[x,,(s) - x,,(e)](I')[x,,(s) - x,,(e + 1)] = [x,,(s) - x,,(e + 1)](1'+1),

we come to the identity

l:!.. {u(s)p,,(s)[x"+1(s - 1) - X"+l(e)](I')} =

p,,(S)VX,,+l(S) {Do[x,,(s) - x,,(e)](I'-l)+

+Dl[X,,(S) - x,,(e)](I') + D2[x,,(s) - x,,(e + 1)](I'+1)}.

Substituting

[x"+1(s - 1) - X"+l(e)](I') =

= -y-l(p + I)V:(S) [x,,(s) - x,,(e + 1)](1'+1)

(4.6)

into (A.6) and setting II = O,p = n-l,s = z,e -+ e -1 we come to formula (2.6). •

Tab

le.

Mai

n c

har

acte

rist

ics

of

equ

atio

n (

2.1

)

I. L

atti

ce x

(z)

= C

1Z2 +

C2z

+ C

3 II

I. L

atti

ce x

(z)

= C

lq-Z

+ C

2qz +

C3

-----

-----------

U

AZ4

+ B

Z3 +

Cz2

+ D

z +

E

Aq2

z +

BqZ

+ C

+ D

q-Z

+ E

q-2

z

iT A

CI

2 (x

-C

3)2 +

C11

(2p2

A -

~pB +

C)(

x -

C3)

+

HA

C;2

+ E

CI

2 )(x

-C

3?+

+ !

(p3

A _

p2

B +

pC

-D

) +

E,

p =

C 2C

11

+ l

(BC

-l +

DC

-l)(

x -

C)

+ C

-A

S.

-E

£2

2

2 1

3 C

2

(;'1

C11

[(2A

(p +

m)

-B

)z(z

+ p

+ m

)+

~

Tm

(f- q

)C1

X

+ A

(p +

m)3

-B

(p +

m)2

+ C

(p +

m)

-D

] x[

(Ap

-2q-

2m _

E)(

q-Z

+ pq

z+m

) +

Bp

-lq-

m _

D]

I

>."

/lC

I2 [

B -

A(2

p +

/1-

1)]

q!(l-

q)-

2(1

_ q

-")(

A _

Ep2

q"-1

)/C~

II.

Lat

tice

x(z

) =

C2z

+ C3

IV

. L

atti

ce x

(z)

= C

1q-Z

+ C

3 --

U

AZ2

+B

z+

C

C +

Dq

-Z +

Eq

-2z

!

U +

T\7X

l A

z2 +

Dz+

E

C +

Bq

-Z +

Aq-

2Z

iT A

C;2

(X -

C3?

+ tC

;I(B

+ D

)(x

-C

3) +

HC

+ E

) C

+ tC

l1(B

+ D

)(x

-C

3) +

tCI2

(A +

E)(

x -

cd

Tm

C;1

[(2

Am

+ D

-B

)z +

Am

2 +

Dm

+ E

-C

] c 1

1 (1

-q)-lq~[(Aq-2m

_ E

)q-Z

+ (B

q-m

-D

)]

>."

/lC

;2[B

-D

-A

(/I

-1)

] q

! (1

-q

)-2

CI

2 (1

-q

-")(

A -

Eq

"-l)

t1 ~ ... " 1:1 !;l I:Il

'<

'0 " ... 1 .... :3.

n ~

~ C".

o ~ ""' ....

CA

NO

NIC

AL

lYP

ES

OF

EQ

UA

TIO

N (

2.1)

.

+ C

2z +

CJ

:

II.

La

ttic

e x(z)=C~

+

C3

: IV

. L

att

ice

x(z

)=C

1Q-Z

+

C

3 :

w

w z !s:: :! r. e: ~ til p.: J


References

[AS 1] Atakishiyev, N.M. and Suslov, S.K. About one class of special functions, Revista Mexicana de Fisica, vol. 34, No.2, (1988), p. 152-167.

[AS2] Atakishiyev, N.M. and Suslov, S.K. Difference hypergeometric functions, and Construction of solutions of the hypergeometrictype difference equation on non-uniform lattices, Physics Institute preprints No. 319 and 323, Baku, Azerbaijan SSR, 1989 (in Russian).

[AS3] Atakishiyev, N.M. and Suslov, S.K. On the moments of classical and related polynomials, Revista Mexicana de Fisica, vol. 34, No. 2, (1988), p. 147-151.

[AWl] Askey, R. and Wilson, J .A., A set of orthogonal polynomials that generalize the Racah coefficients or 6j-symbols, SIAM J. Math. Anal., vol. 10, No. 5,(1979), p. 1008-1016.

[AW2] Askey, R. and Wilson, J .A., Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials, 1985, Memoirs Amer. Math. Soc., No. 319.

[Ba] Bailey, W., Generalized Hypergeometric Series, Cambridge: At the University Press, 1935.

[B1] Boole, G., A Treatise on the Calculus of Finite Differences, 2nd ed. London: Macmillan, 1872; New York: Dover, 1960, p. 236-263.

[B2] Boole, G., A Treatise on Differential Equations, 5th ed., New York: Chelsea, 1959.

[D] Drinfel'd, V.G., Quantum Groups, Proceedings of the International Congress of Mathematicians, Berkeley, CA, 1986: American Mathematical Society, 1987, p.798-820.

[EMOT] Erdelyi, A., Magnus, W., Oberhettinger, F., and Tricomi, F., Higher transcendental functions, McGraw-Hill, New York, Vol. I, 11,(1953), Vol. III, (1955).

[GR] Gasper, G. and Rahman, M., Basic hypergeometric series, Cambridge University Press, Cambridge, 1990.

[HI] Hahn, W., Beitrage zur Theorie der Heineschen Reihen. Die 24 Integrale der hypergeometrischen q-Differenzengleichung. Das qAnalogon der Laplace-Transformation, Math. Nachr. Vol. 2, No. 6,(1949), p. 340-379.


[H2] Hahn, W., Uber Orthogonalpolynome, die q-DitJerenzengleichungen genugen, Math. Nachr., Vol. 2, No. 1,(1949), p. 4-34.

[H3] Hahn, W., Uber Polynome, die gleichzeitig zwei verschiedenen Orthogonalsystemen angehoren, Math. Nachr., Vol. 2, No.5, (1949), p.263-278.

[He1] Heine, E., Handbuch der Kugelfunctionen, VoU, Berlin: Druck und Verlag von G. Reimer, (1878), p.97-125, 273-285.

[H 2] H . E Ub d' R °h 1 (.I'·-11(q'-I) e eme,., er Ie ea e + 1q-l)(f"Y-lf . :c + (qa+l_11(qa_l)(~+1_1)(q'_I) 2 .

(q~ l)(q-l)(q"Y+1 1)(q"Y 1) .:C +"', J. reme u. angew. Math, Vol. 32, No.3, (1846), p. 210-212.

[H 3] H · E U t h b d' R 'h 1 (1-,a1(1-q') e eme" n ersuc ungen 'II. er Ie el e + (l-q)(i-q"Yf . :c + (l-qa)fl-qa+l)(I-~')(I-~+l) 2 J' M th

{1-;(I-q')(I-q-1 (l-q-1 I) .:C +"', . reme u. angew. a ., Vol. 34, No.4, (1847), p. 285-328.

[ILVW] Ismail, M.E.H., Letessier, J., Valent, G., and Wimp, J., Two families of associated Wilson polynomials, Canad. J. Math., Vol. 42, (1990), p. 659-695.

[IR] Ismail, M.E.H., and Rahman, M., The associated Askey- Wilson polynomials, Trans. Amer. Math. Soc., (1991), to appear.

[KR] Kirillov, A.N., and Reshetikhin, N. Yu., Representations of the algebra Uq (s1(2», q-orthogonal polynomials and invariants of links, LOMI Preprint E-9-88, Leningrad, 1988.

[K] Koornwinder, T.H., Orthogonal polynomials in connection with quantum groups, Orthogonal Polynomials: Theory and Practice, ed. by P. Nevai, NATO ASI Series C, VoL294, Kluwer Academic Publishers, 1990, p. 257-292.

[NR] Nassrallah,B. and Rahman, M., Projection formulas, a reproducing kernel and a generating function for q- Wilson polynomials, SIAM J. Math. AnaL, Vol. 16, No.1, (1985), p. 186-197.

[NSU] Nikiforov, A.F., Suslov, S.K., and Uvarov, V.B., Classicalorthogonal polynomials of a discrete variable, Nauka, Moscow, 1985 (in Russian).

[NUl] Nikiforov, A.F. and Uvarov, V.B., Classical orthogonal polynomials of a dicrete variable on non-uniform lattices, Preprint No. 17, Keldysh Inst. Appl. Math., Moscow, 1983 (in Russian).

[NU2] Nikiforov, A.F. and Uvarov, V.B., Special Functions of Mathematical Physics, 2nd ed., Nauka, Moscow, 1984 (in Russian).


[R] Rahman, M., An integral representations of a 10<[>9 and continuous bi-orthogonal 10<[>9 rational functions, Can. J. Math., Vol. 38, No.3, (1986), p. 605-618.

[S] Suslov, S.K., The theory of difference analogues of special functions of hypergeometric type, Russian Math. Surveys, The London Mathematical Society, Vol. 44, No.2, (1989), p. 227-278, correction in ibid Vol. 45, No.3, (1990).

[T1] Thomae, J., Beitrage zur Theone der durch Heinesche Reihe: 1 + 1_q B • l_ qb . X + 1_qB • 1_qB;1 . l_qb . l_qb:~ . x 2 + ... darstellbaren 1-q l-q< 1-q 1-q 1-q< l-q<

Functionen, J. reine u. angew. Math., Vol. 70, No.3, (1869), p. 258-281.

[T2] Thomae, J., Integmtion der Differenzengleichung (n+ re+1)(n + .A + 1)62cp(n) + (a + bn)6cp(n) + ccp(n) = 0, Zeitschrift f. Mathematik u. Physik, 1971, Vol. 16, No.2, p. 146-158; No.5, p. 428-439.

[VK] Vaksman, L.L. and Korogodsky, L.I., Algebra of bounded functions on the quantum group of plane motions and q-analogues of Bessel functions, Dokl. Akad. Nauk SSSR, Vol. 304, (1989), p. 1036-1040 (in Russian), English translation in Soviet MathematICS.

[VS] Vaksman, L.L. and Soibel'man, Algebm of functions on the quantum group SU(2), Functional Anal. Appl., Vol. 22, (1988), p. 170-181.

N.M. Atakishiyev Physics Institute N arimanov pr. 33 Baku 370143 AZERBAIJAN

S.K. Suslov Kurchatov Institute of Atomic Energy Moscow 123182 RUSSIA

Pade Approximants for Some q-Hypergeometric Functions

M.E.H. Ismail* R. Perline J. Wimpt

ABSTRACT We show that a large number of explicit formulas for Pade approximants for the ratios of basic hypergeometric functions result from an explicit expression given by Ismail and Rahman for the associated AskeyWilson polynomials. By specializing this result and using a new transformation for basic hypergeometric series, we are able to recover a result due to Andrews, Goulden and Jackson. We also show how Pade approximants off the main diagonal can be constructed in this latter case.

1 Pade Approximants and q-Functions

Let 00

f(z) := L lizi , (1.1) ;=0

be a formal series, where z is an indeterminate and the /j are complex numbers.

The set {plL/Ml(f, z)j qlL/Ml(f, z)}, where L, M are integers;::: 0 and

L M

p[L/Ml(f, z) = LP)L/Ml(f)zi, q[L/M](f, z) = L qJL/Ml(f) zi , j=O j=O

(1.2)

is called an [L/ M] Pade approximant to f if

p[L/M](f, z) - f(z)q[L/Ml(f, z) = O(zL+M+1) .

Often the depeI!dence on L, M and f in p and q will be suppressed if their meaning is clear from context.

"This author's work was partially supported by the National Science Founda-tion under grants DMS 8814026 and DMS 8912423. .

tThis author's work was partially supported by the National Science Foundation under grant DMS 8802381.

PROGRESS IN APPROXIMATION THEORY (A.A. Gonchar and E.B. Saff, eds.), ©Springer-Verlag (1992) 37-50. 37

38 M.E.H. Ismail, R. Perline, J. Wimp

Using the traditional algebra of formal power series it is easily found that the qj must satisfy

M

Eflih+1+i-j = 0, i = 0,1, ... ,M -1, j=O

and the Pj must satisfy

min(i,M)

Pi = E fli/i-;' i = 0,1,2, ... , L , j=O

(1.3)

(1.4)

empty sums being interpreted as zero and h = ° for j < 0. Some authors take Po = 1, but we shall not necessarily do that. We shall be primarily interested in the polynomial q(z), which we designated a [LIM} Pade denominator lor /, since once q is known P may be found from (1.4). The reference [5] provides a good background for the theory of Pade approximants.

Now let q be a fixed real number, ° < q < 1, a a complex number, and define

j-l

(a)j = II (1- aqi) ,j = 0,1,2, ... ,00, i=O

the q-Pochhammer symbol, or the q shifted factorial, where empty products are interpreted as unity. We shall also use the notation from [8]

A:

(al,a2, ... ,aA:)n = II(aj)n, n= 0,1,2, ... ,00. j=l

We will consider the case where I is a basic hypergeometric series of the kind

I( ) - "'" (a 1 , a2, a3, ... , ar +l ., ) ._ ~ (al, a2, ... , ar +l)j (' v z - r+l'1'r ,AZ .- L..J ( ) AZ,.

b b b b ·-0 q,bbb2 , ••• br j 1, 2, 3,···, r 1-

For the theory of such series, see [8] or [4]. Throughout this work we will invoke many of the properties of the q-Pochhammer symbols, as given on page 6 of reference [8]. 'Note that series of the kind mtPn for any m, n are easily obtained from the above series by putting numerator or denominator parameters equal to zero.

It turns out that for such a series there is a very simple relationship between the Pade approximants [L'IM] and [LIM] when L', L ~ M - 1 (the sub-diagonal, diagonal, and above the diagonal elements of the Pade table.)

Pade Approximants for Some q-Hypergeometric Functions 39

Theorem. Let L',L ~ M -1 and ar+l = q. Then

ai ---+ aiqL'-L

Note: Since Pade coefficients are not unique, the above should be interpreted to read: if the quantities on the left are [LIM] Pade denominator coefficients for f, then the quantities on the right with the indicated substitutions are [L' I M] Pade denominator coefficients for f.

Proof. It suffices to consider the case where /j = (a)j, ,\ = 1. Formula (1.3) gives

M

E qj(a)L+1+i-j = 0, L + 1 - M ~ 0, j=O

and using the fact that

shows that M

Eqj(aqL+1)i_i = O. i=O

Making the substitution a ---+ aqL'-L yields the result. • We can obtain the Pade approximants for the important case of the

q-analog of Guass's continued fraction by using the q-analog of the PfaffSaalschiitz formula (see [4], p. 68 (1) or [8]). The function we treat is

( q,a ) 00 (a). .

f(z) = 2¢Jl ; qz :=?= (b)~ (qz).1 . b J=O J

(1.5)

Let * _ (q-M)M_j(bqM-l)M_i f. _ (a)i..i

qi - () ( ) , J - (b) 'r . a M-i q M-j j

First we consider [M - 11M] Pade approximants. The sum (1.3) becomes

M M () M ( -M) (b M-l) ( i) '"' */ .. _ '"' */ .. _ ~ '"' q i q i aq i Hi L..Jqi L+1+I-J - L..Jqi M+I-J - (b). L..J (a).( ).(b i). q . i=O i=O 1 i=O J q J q J


This sum is proportional to the balanced (Saalschutzian)

which is proportional, by the Bailey result, to (q-i)M, which is zero for o ~ i < M. Thus the quantities qj are coefficients for a [M - 1/ M] Pade denominator approximant for f. (This is essentially the result of Andrews, Goulden and Jackson, [2]). Our Theorem then shows that coefficients for a [L/ M] denominator for the function fin (1.5) are

(q-M)M_j(bqL)M_j qj = (aqL+I-M)M_;(q)M_; ,

i.e.,

( q-M bqL )

q(z) = zM 2tPl 'j l/z aqL+1-M

(1.6)

The corresponding numerator coefficients can be constructed from (1.4). An interesting special case of (1.1)-(1.6) occurs with the choices

z ~ -z/qa,a ~ 00, b = O.

We get 00

fez) = Eqi(j-1)/2zj , (1.7) j=O

and for L 2': M - 1a [L/ M] denominator approximant to this function is

M (q-M) _q-;(L-M)-;(j-l)/2 q(z)=zML: J _

;=0 (q); zJ (1.8)

The function (7) is called a partial theta function. Lubinsky and Saff have discussed this function in [14].

It is interesting that the [M -1/ M] Pade approximant for (1.1) occurs as a special case of a master formula recently obtained by Ismail and Rahman [12] for the associated Askey-Wilson polynomials, although in a heavily disguised form. In fact, to reconcile the two requires a identity for 3tP2

function which seems to be new. In Section 2 we shall have to consider Pad6 approximants about 00,

but only for the case [M - 1/M]. These are defined as follows. The form {p(z)jq(z)} is a [M - l/M] Pad6 approximant about 00 for the formal Laurent series

00

"I- -;-1 L...J JZ ,

j=O


if ZM-1p(Z-1) = rJM- 1/Ml(f, z)j zM q(z-1) = q[M-1/Ml(f, z) ,

all the quantities on the right above being as in (1.1), (1.2). Pade approximants about 00 arise most naturally from the theory of

orthogonal polynomials. Our approach is that of the Bateman manuscript volumes, [7]. Let Q(t) be a distribution function ( a monotone increasing function with an infinite number of points of increase and all of whose moments exist) on the real line. We define the moment em, m = 0,1,2, ... , by

Cm = fa tmdQ(t) .

A distribution function Q generates a sequence of orthogonal polynomials Pn(z) which satisfy "a three-term recurrence relation,

(Actually, the polynomials may be generated by certain positive linear funetionals acting on the space of real polynomials. This is the approach, for instance, taken in Brezinski [6].)

We now generate two linearly independent solutions of (1.9), rn, 8 n ,

corresponding to the initial values

ro = 0, 80 = 1, r1 = 1, 81 = aoz + boo

Note rn is a polynomial of degree n -1, 8 n a polynomial of degree n. It is easy to show that the ratio rM IBM has a power series expansion about 00 whose first 2M coefficients are cjlaoco. Thus the ratio is the [M -11M] Pade approximant to the formal series,

1 [ dQ(t) 1 ~ Cj

aoco iRF. z-t ~ aoco ~ z;+1· J=O

The point is, when we encounter a system of orthogonal polynomials whose members Pn can be expressed in closed form, we can identify the denominator of the [M -11M] Pade,approximant with PM. ( By a closed form we generally mean a simple single sum, although what this means is open to interpretation. In fact the polynomials Pn may always be expressed as Gram determinants of the moments, but seldom can such an expression be reduced to anything simple.) To date two of the most general systems of orthogonal polynomials which can be written in closed form are the two families of associated Askey-Wilson polynomials, [12].


2 The Associated Askey-Wilson Polynomials

We consider the recurrence relation

where

P!: - p!:(zja,b,e,d/q), P~l = 0, p~ = 1,

a-l (1 - abqn+Q)(1 - aeqn+Q)(1 - adqn+Q)(1 - abedqn+Q-l) (1- abedq2n+2Q-l)(I_ abedq2n+2Q)

a(l- beqn+Q-l)(I_ bdqn+Q-l)(I_ edqn+Q-l)(I_ qn+Q)

(1 - abedq2n+2Q-l )(1 - abedq2n+2Q-2)

The polynomial p~ are called the associated Askey-Wilson polynomials and were very recently studied by Ismail and Rahman, [12]. A second family of associated Askey-Wilson polynomials {q!: (z)} was also studied in [12]. They satisfy the same recurrence as the p~'s but with different initial conditions,

2z - a - a-I - Ao qg = 1, qf = ----:------'Ao

When a = 0 and abed i= q,q2, both families reduce to the Askey-Wilson polynomials t

Pn(zja,b,e,dlq) = 41/J3 jq , (q_n,abedqn-l,az,a/z )

ab,ae, ad

where, as throughout, we let

Z = (Z + Z-l )/2 .

The Askey-Wilson polynomials were discussed in [3].

tThe referee has pointed out tha.t for Askey-Wilson polynomials a.nd Wilson polynomials the cases abed = q, q2, a.nd a + b + e + d = 1,2 a.re exceptional cases. Ao a.nd Co (for a = 0) a.re then indetermina.te. These exceptional cases are discussed in detail in [9] a.nd [15]. To a.void theindetermina.cy, the restrictions on abed here a.nd on 8 la.ter on are necessary. The a.uthors thank the referee for this observa.tion.


When one makes substitution

a -+ qG, b -+ q'" c -+ qC, d -+ qd, Z -+ (q" + q-")/2, (1.10)

and takes the limit q -+ 1, one gets the two families of associated Wilson polynomials discussed in [10]. The passage to the limit q -+ 1 is trivial in the case of explicit representations or recurrence relations. On the other hand it not easy to perform the limit q -+ 1 in the integral defining the orthogonality relation.

Other limiting cases are also of interest. Making the substitutions in p~

(1.11)

and taking the limit as N -+ 00 produces two associated versions of the little q-Jacobi polynomials. The traditional version of these polynomials, studied in the references [1],[13], corresponds to putting a = o. They are defined by

and satisfy the recurrence relation

= A nYn+l + BnYn + CnYn-l, n = 0,1,2, ... ,

_qn(1 _ aqn+1 )(1 _ a{3qn+l)

(1 - a{3q2n+l )(1 - a{3q2n+2) ,

_aqn(l_ {3qn)(I- qn)

(1- a{3q2n+l)(I- a{3q2n) ,

taking the limit q -+ 1 in the associated little q-J acobi polynomials yields the associated Jacobi polynomials studied in [16]. A second version ofthese polynomials was discussed by Ismail and Masson in [11]. The Pade approximants formed with these polynomi.als via the process described in Section 1 converge linearly to a ratio of contiguous Gaussian hyper geometric functions. The Pade approximants were essentially the truncates of Gauss' continued fraction. This result generalized a famous result originally given by Laguerre, who gave closed-form expressions ratio for Pade approximants to a ratio of hypergeometric functions when one of the parameters of the numerator hypergeometric fun~tions equals 1. The result in [16] allows the parameters of the numerator hypergeometric function to be arbitrary.

The Pade approximants to be given here are those whose numerators and denominators are associated Askey-Wilson polynomials and their limiting cases. First we discuss the most general case.


We follow [12] and adopt Bailey's W-notation for the very well-poised series,

r+l Wr(aj at, ... , ar-2j y) = r+l<Pr j Y . (a, qa1/2, _qa1/2, at, ... , ar-2, )

1/2 1/2 / / a , -a , qa aI, ... , qa ar-2

It is a remarkable fact that the two families of associated Askey-Wilson polynomials can be given in closed form. The first expression is

En (q-:-n, abcdq2a+n-l, abcdq2a-l, aZ, a/ Z),. 11: ~= q

11:=0 (q,abqa,acqa,adqa,abcdqa-l)1I:

x 10W9(abcdq2a+1I:-2 j qa, bcqa-l, bdqa-l, cdqa-l, q1r:+1, abcdq2a+n+1r:-l, q1l:-n j a2)

and the second is

En (q-n,abcdq2a+n-l,abcdq2a-l,aZ,a/Z)1I: 11: q:: = q

1r:=0 (q, abqa, acqa, adqa, abcdqa-l),.

x 10W9(abcdq2a+1I:-2 j qa, bcqa-l, bdqa-l, cdqa-l, q1l:+1, abcdq2a+n+1I:-l, q1l:-n j a2)

x W. (abcdq2a+1I:. qa+l bcqa bdqa cdqa q1l:+1 10 9 """

abcdq2a+n+1r: , q1r:-n+l j a2).

Notice that because of the presence of the factors q1r:-n, q1l:-n+l, the above 10 W9'S terminate, so the above expressions are indeed finite, although they are written in terms of Z, not z. Nevertheless, it is clear that they are polynomials of degree n in z, since

II~~~(I- aZqi)(1 - aZ-1qi)

II~~~(1- a(Z + Z-I)qi + a2q2i)

II1r: 1( '2 2') = i~O 1 - azql + a q I •


CASE I: The associated Askey-Wilson polynomials. The set

{ a+l a} PM-l,PM ,

is the [M - 1/ M] Pade approximant about 00 to the function

Z(1 - bcdZq2a-l )(1 - bcdZq2a)(1 - abqa)(1 - acqa)

w(z) = a(l- bzqa)(I- czqa)(I- dZqa)(I- bcdZqa-l)

(1 - adqa)(I- abcdqa-l) x ~~~~~~~--~~~ (1 - abcdq2a-l )(1 - abcdq2a)

X sW7(bcdZq2a; bcqa, bdqa, cdqa, qa+1, Zq/a; aZ)

s W7(bcdZq2a-2; bcqa-l, bdqa-l, cdqa-l, qa, Zq/a; aZ)

(1.12)

This is very easy to show. We simply note that P'f/!l and PM satisfy the same recurrence formula, and satisfy the appropriate initial conditions for the numerator and denominator polynomials respectively of the [M -1/ M] Pade approximant. The function w(z) is given in [12]. Note ao = 2/ Ao. The s W7 may be expressed, via the formula [4], p. 69 (3) as a sum of functions of the form 4<P3.

A second class of Pade approximants may be obtained from the second system of associated Askey-Wilson polynomials studied in [12]. It turns out that the system

{ a+l a } qM-l' qM ,

is the [M - 1/ M] Pade approximant about 00 to an identifiable function. That function is a little complicated, but it can be worked out by using the formulas given in ([12], (5.9)-(5.12)).

CASE II: q -+ 1, a =P O. We make the substitution (1.10) and let q -+ 1. The effect on p~ is to

yield an explicit formula for the associated Wilson polynomial, which the authors of [10] failed to find. Note the coefficient 2z - a - a- 1 goes into qZ + q-Z _ qa _ q-a. When the recurrence is divided by (1- q)2 and q -+ 1 all the coefficients are defined and the coefficient of the polynomial on the left becomes z2 - a 2 . We will let

z = v(+a2

and our Pade approximants will be about ( = 00. The polynomial has the explicit representation

ra «() = t(-n,s+2a+n-),s+2a-l,a+z,a-z)k n k=O (a+b+a,a+c+a,a+d+a,s+a-l}A:k!


x 9FS . (k - n,s+2a+ k- 2,a+ ~,a,b+ c+a -1,

where

-n,a -1+ ~,s+a+k -I,s+2a+n-I,

b+d+a -I,c+d+ a -I,k + I,s+2a +n+k -1 ) '1

s+ 2a -2,a+b+a+ k,a+c+a+k,a + d+a+k '

A:-l s= a+b+c+d, (e)A: = II(e+i).

j=O

The latter quantity is the ordinary Pochhammer symbol, or shifted factorial, and whenever we are discussing the q -+ 1 case it is understood that (')A: indicates this rather than the q-Pochhammer symbol.

The set {r~l.\«(), rM«(H,

is the [M - 1/ M] Pade approximant about ( = 00 to the function

u«() = 2(b+c+d+z+2a-I)(b+c+d+z+2a)(a+b+a) (b+ z +a)(c+ z+ a)(d+ z + a)(b+ c+ d+z+ a-I)

(a+d+a)(s+a-I) F (s+2a-I)(s+2a) G'

x

where

( b + c + d + z + 2a, a + 1 + (b + c + d + z)/2, b + c + a,

F = 7FS

s+ a,a+ (b+ c+d+ z)/2,b+ z+a + I,c+ z + a + 1,

b+ d+a,c+d+a,a + I,z+ 1- a, ) ; 1 ,

d+z+ a+ I,b+ c+d+z+ a,

( b + c + d + z + 2a - 2, a + (b + c + d + z)/2, b + c + a-I,

G = 7FS

s+ a-1,a - 1 + (b+ c+d+ z)/2,b+ z+a,c+ z + a,

b+d+a -I,c+d+a-I,a,z + I-a, ) ; 1 ,

d+ z + a,b+ c+d+ z +a -1,


and z= ",(+a2 •

CASE III: q -1, Q = O. Here we must assume 8 11,2 (see the previous footnote.) This case is

easily deduced from Case II. The set

{rAt_l' rM},

is the [M - 1/ M] Pade approximants about 00 to the function

2(8-1) (Z+I-a,Z+l-b,C+d,l) vee) = 4F3 ; 1 , (a+b-1)(c+z)(d+z) c+z+1,d+z+1,2-a-b

z = ",(+a2 •

NOTE: the polynomials r~ = rn are called Wilson polynomials. They satisfy the recurrence

(8 + n - 1)(a + b + n)(a + c + n)(a + d + n) (8+2n-1)(8+2n)

n(b+c+ n -l)(b+ d+ n -1)(c+ d+n -1) (8 + 2n - 2)(8 + 2n - 1)

CASE IV: Confluent cases, Q I O. Here we make the substitutions (1.11) in the formula for p~ and let

N - 00. We take for Z the principal value ( positive for z large positive) of

Z = z +.Jz2='1, so Z ,... 2zq-N,

and use ([4], p. 69 (2». The resulting polynomial are the associated little q-J acobi polynomials,

(qa+1 abu2q2a-l)n _ n (abu2q2a+n-l q-n)k 8a =' q na '"' ' (2zq/u)k

n (q,abu2qa-l)n t:o (qa+l,abqa)k


The gW7 in the numerator on the right in (12) becomes in the limit

and we find the set { a+l a} 8M _I> 8M ,


A major problem is identifying this result with the approximation (1.5), (1.6) when a = O. This requires what seems to be new transformation of basic hypergeometric series. We have the

Theorem.

where K is given by

(2bzu2q2a+l,I/2bzu2q2a,2zq/b,b/2z,2zq/u,u/2z)oo

Proof. This is done by taking a limit of a result in Bailey ([4], p. 69 (23)). We make the identifications

A = bcdZq2a, D = qa+1, E = Zq/a, F = bcqa, G = bdqa,H = cdqa,

(capital letters denoted Bailey's quantities), and then make the substitution (1.11). To effect the limit, use the fact that

(aq-:)oo ~ (~)N (q/a, a)oo . ([3q-)oo [3 (q/ [3, (3)00

The first 4tPa on the right of Bailey's formula approaches the 2tPl above while the second approaches

ItPo(2zq/Uj u/2zq) = 0 .

(Actually, this argument requires lu /2zql < 1, but the result then holds by analytic continuation.) •


Using the above and the formula ([4], p. 68 (2» we may now rewrite the previous result. We find the set


Note that

and an obvious identification of parameters gives the [M - 1/ M] Pade approximant discussed in Section 1.

References [1] Andrews, G. E., and Askey, R. A., Classical orthogonal polynomials,

in" Polynomes Orthogonaux et Applications", Lecture Notes in Mathematics Vol. 1171, (Eds. C. Brezinski et. al.), Springer-Verlag, Berlin, pp. 36-62 (1985).

[2] Andrews, G. E., Goulden, I.P., Jackson, n.M., Shank's convergence acceleration transform, Pade approximants and partitions, J. Combin. Theory Ser. A 43(1986), pp. 70-84.

[3] Askey, R. A. and Wilson, J .A., Some basic hypergeometric orthogonal polynomials that generalize the Jacobi polynomials, No. 319, Memoirs Amer. Math. Soc., Providence (1985).

[4] Bailey, W. N., Generalized Hypergeometric Series, Cambridge University Press, Cambridge (1935).

[5] Baker, G. A.,Jr. and Graves-Morris, P., Pad!. Approximants. Part I: Basic Theory, v. 13, Encyclopedia of mathematics and its applications, Addison-Wesley, Reading, Mass. (1981).

[6] Brezinski, C., Pade Type Approximation and General Orthogonal Polynomials, Birkhauser, Boston (1980).


[7] Erdelyi, A.,et aI, Higher Transcendental Functions, Volumes 1, 2, 3, McGraw-Hill, New York (1953).

[8] Gasper, G. and Rahman, M., Basic Hypergeometric Series, Cambridge University Press, Cambridge (1990).

[9] Gupta, D.P. and Masson, D.R., Exceptional Askey- Wilson polynomials, Proc. Amer. Math. Soc., to appear.

[10] Ismail, M.E.H., Letessier, J., Valent, G., and Wimp, J., Two families of associated Wilson polynomials, Canad. J. Math. 42(1990), pp. 659-695.

[11] Ismail, M.E.H. and Masson, D. R., Two families of orthogonal polynomials related to Jacobi polynomials, Rocky Mountain J. Math. 21(1991), pp. 359-375.

[12] Ismail, M.E.H and Rahman, M., The associated Askey- Wilson polynomials, to appear, Trans. Amer. Math. Soc. (1991).

[13] Ismail, M.E.H and Wilson, J., Asymptotic and generating relations for the q-Jacobi and 4<P3 polynomials, J. Approx. Theory 36(1982), pp. 43-54.

[14] Lubinsky, D.S. and Saff, E.B., Convergence of Pade approximants of partial theta functions and the Rogers-Szego polynomials, Constr. Approx. 3(1987), pp. 331-361.

[15] Masson, D.R. Wilson polynomials and some continued fractions of Ramanujan, Rocky Mountain J. Math. 21(1991), pp. 489-499.

[16] Wimp, J., Explicit formulas for associated Jacobi polynomials and some applications, Can. J. Math. 39(1987), pp. 983-1000.

Mourad E.H. Ismail Department of Mathematics University of South Florida Tampa, FL 33620

Ron Perline and Jet Wimp Department of Mathematics and Computer Science Drexel University Philadelphia, PA 19104-2875

Summation Theorems for Basic Hypergeometric Series of Schur Function Argument

s.c. Milne*

ABSTRACT In this paper we prove a Ramanujan 1 tPl summation theorem for a Laurent series extension of I.G. Macdonald's (Schur function) multiple basic hypergeometric series of matrix argument. This result contains as special, limiting cases our Schur function extension of the q-binomial theorem and the Jacobi triple product identity. Just as in the classical case, we write our new q-binomial theorem and 1 tPl summation as elegant special cases of K. Kadell's and R. Askey's q-analogs of Selberg's multiple beta-integral. We also apply our q-binomial theorem and K. Kadell's Schur function q-analog of Selberg's beta-integral to derive a Heine transformation and q-Gauss summation theorem for Schur functions.

1 Introduction

In this paper we prove a Ramanujan 1""1 summation theorem for a Laurent series extension of Macdonald's [47] (Schur function) multiple basic hyper geometric series of matrix argument. This result contains as special, limiting cases our Schur function extension [51] of the q-binomial theorem and the Jacobi triple product identity. Just as in the classical case [2,10,13,14,21,35,61,62], we write our new q-binomial theorem and 1 tPl

summation as elegant special cases of Kadell's [40,41] and Askey's [15] q-analogs of Selberg'S [2,15,57] multiple beta-integral.

We also apply our q-binomial theorem and Kadell's [40,41] Schur function q-analog of Selberg's beta-integral to derive a Heine transformation and qGauss summation theorem for Schur functions. Our summation theorems and transformations in this paper continue the study of special functions. of matrix argument in [12,27,28,33,39,44,46,47,58]. In addition, our results should have applications to orthogonal polynomials in several variables, and their q-analogs, corresponding to some of the classical work in [2,5-11,13,14,16-22,25,26,29,30,35,52,61,62].

As motivation we first recall Ramanujan's 1""1 summation in

·Partially supported by joint NSF /NSA grant DMS-8904455.

PROGRESS IN APPROXIMATION THEORY (A.A. Gonchar and E.B. Saff, eds.), @Springer-Verlag (1992) 51-77. 51

52 S.C. Milne

Theorem 1.1. (Ramanujan). Let

° < Iql < 1 (1.2a)

and Ib/al < Ixl < 1. (1.2b)

Then,

~ (a)n. xn _ {(ax)oo(q/ax)oo} . {(b/a)oo(q)oo} (1.3) ~ (b)n - (x)oo(b/ax)oo (q/a)oo(b)oo' n __ oo

where (A)n and (A)oo are defined by

00

(A)oo = II (1 - Aqr), (1.4a) r=O

and (lAb)

for all real numbers n. In particular, note that

(A)n = (1- A)(l- Aq) ... (1- Aqn-l), (1.5a)

(A)o = 1, (1.5b)

and

(1.6a)

Observing that I, (qn+l)oo

(q)n = (q)oo ' (1.7)

vanishes when n = -1, -2, ... , it is immediate that the b = q case of Theorem 1.1 becomes the classical q-binomial theorem [1,24,26,59] in

Theorem 1.8. If Iql < 1, Ixl < 1, then

1 + f: (a)" . x" _ (ax)oo ,,=1 (q)" - (x)oo .

(1.9)

Furthermore, it is not difficult to see that replacing a, q, b, and x by -l/c,q,O, and qzc, respectively, in Theorem 1.1, and letting c -+ ° yields the fundamental Jacobi triple product identity [1,24,26,37,59] given by

Summation Theorems for Basic Hypergeometric Series 53

Theorem 1.10. (Jacobi) If z =I 0, Iql < I, then

+00 00 L: znq(n~+n)/2 = II (1 + zqn+1)(1 + z-l qn)(l_ qn+1) (LIla) n=-oo n=O

(1.11b)

In [3,4,8,14,20,25,26,30,36] clever rearrangements of series or q-difference equations are utilized to prove Theorem 1.1. The most elegant proof appears in [34] where (1.3) is obtained directly from the q-binomial theorem and the analyticity of (1.3) in 6 for 161 sufficiently small. This approach was extended in [48,49,50] to yield a U(n) generalization of Theorem 1.1. directly from a U(n) multiple series refinement of the q-binomial theorem. The multilateral 1,p1 summation contains a new generalization of the Macdonald identities for A~l), with extra free parameters. See [49,50] for a survey of this work as well as related results.

The q-binomial theorem in (1.9) is closely related to beta functions and q-analogs of classical orthogonal polynomials. Thomae [61,62]' and later Jackson [35] first observed that (1.9) is equivalent to

[1 11-1 (X1q)00 d _ rq(a)rq(6) Jo Xl ( X1qb)00 qXl - rq(a + 6) , (1.12)

where (5.6) and (5.8) hold. That is, (1.9) is an extension of the beta function as an integral on [0,1]. Applications of Theorem 1.8 to orthogonal polynomials defined by basic hypergeometric series are discussed in [2,5,7,9,13,16,19,21,26,30,52]. The connection coefficient problem for little q-J acobi polynomials is solved in [5], and Theorem 1.8 leads to a proof of their orthogonality in [7].

The Ramanujan l,pl summation in (1.3) has numerous applications to beta function integrals and q-orthogonal polynomials in [2,6,8-11,13,14,17-22,25,26,30,52]. Askey observed in [13,14] that (1.3) is equivalent to

100 11-1 (-CXlqll+b)oo d _ rq(a)rq(b) (-cqll)00(-C- l q1-1I)00 Xl ( ) qXl - . (1 ) , o -CXloo rq(a+b) (-c)oo-c-qoo

(1.13) where (4.20) and (5.6) hold. It is not hard to see that (1.13) is an extension ofthe beta function as an integral on [0,00). Theorem 1.1 is needed to pass from (1.12) to (1.13), since we can't change variables in a discrete sum determined by (4.20) or (5.8). The orthogonality of big q-J acobi polynomials follows from (1.3) in [6]. This analysis for both the big and little q-Jacobi polynomials is simplified in [10]. Finally, (1.3) leads in [17,18,21,22,26] to the evaluation of a number of extensions and analogs of beta function integrals.

54 S.C. Milne

Let 8.\(Z1, ... , zn) be a Schur function of the variables {Zl, ... , zn} in (2.4) corresponding to the partition A = (A1, ... ,An), with A1 ~ A2 ~ ... ~ An ~ O.

The main purpose of this paper is to prove the Schur function extension of Ramanujan's 1V;J summation provided by

Theorem 1.14. Let o ~ Iql < 1 (1.15a)

and (1.15b)

with (1.15c)

Take 8p (ZlJ ... , zn) to be a Schur function of Zt, ... , Zn. We then have

(1.16a)

= E {(fI(aq1- i )'\i) (-I)(;)q-(;) Al~A2~···~A.. 1=1 -oo<'\i<oo

(1.16b)

(1.16c)

where Pi = Ai + n - i, for 1 ~ i ~ n. (1.17)

Our proof of Theorem 1.14 in Section 4 is based upon a q-analog of much of the work in [44,58]. That is, we first put together a determinental formula in Section 3 for the multiple basic bilateral hypergeometric series r'l".+1 [(a)jb,(b)jzjl] of Definition 2.15, and then apply Theorem 1.1 and an elegant transformation of a Vandermonde determinant to obtain Theorem 1.14. . By Remark 2.17 and Lemma 2.20 it follows that the b = q case of (1.16) i.s the q-binomial theorem for Schur functions in

Theorem 1.18. If Iql < 1 and IZil < 1, then

lIn (azi)oo '" (a).\. qn(.\) (z.) = L..J H () 8.$Zt, ... , Zn),

i=l 1 00 A .\ q (1.19)

1(.$;Sn

where (a).\, n(A), 8.\ (z), H.\(q), and leA) are defined in Section e.

SlUIlmation Th~orems for Basic Hypergeometric Series 55

The n = 1 case of (1.19) is the classical q-binomial theorem in (1.9). Theorem 1.18 is a q-analog of the well-known identity in the theory of symmetric functions appearing in [42; section 7.2] and [45; Ex. 1, p. 36]. We first proved Theorem 1.18 in [51] by utilizing the Cauchy formula for HallLittlewood symmetric functions in equation (4.7) of [45; p. 117]. Macdonald has independently discovered the new q-binomial theorem in (1.19).

Just as the Jacobi triple product identity in (1.11) is a special, limiting case of (1.3) we observe in Section 4 that Theorem 1.14 yields the triple product identity for Schur functions given by

Theorem 1.20. Let ZlZ2··· Zn "1O, Iql < I, and SI'(Zl, ... , zn) be a Schur function of Zl, ... , Zn. We then have

(1.21a)

I: {( II (1 - q>.r->..+6-r») q[C+2"1 )+ .. +(1+2"")]} "1<!:"2<!:···<!:".. l~r<.~n -00<>';<00

(1.21b) . (Zl··· Zn)>' .. . S(Al->. ....... >."_l->. ... O)(Zl. ... , Zn) (1.21c)

Our first proof of Theorem 1.20 in [51] is similar to Cauchy's classical derivation of (1.11) from (1.9).

Theorems 1.14, 1.18, and 1.20 are quite natural. We illustrate this in Section 4 by writing (1.16) and (1.19) as elegant special cases of Kadell's [40,41] and Askey's [15] q-analogs of Selberg'S [2,15,57] multiple beta-integral. These integrals extend (1.12) and (1.13).

It turns out that iterating a second determinental formula of Biedenharn and Louck [44] immediately yields the q = 1 Schur function case of Macdonald's new general q-analog of the Gauss summation theorem. This is much easier than first proving Kadell's [40,41] Schur function extension of Selberg'S multiple beta-integral and then deducing this result in the standard way. Having the q = 1 case, it is not difficult to guess Macdonald's q-Gauss summation theorem for Schur functions.

In Section 5 we derive a Heine transformation and consequently a qGauss summation theorem for Schur functions from Theorem 1.18 and Kadell's [40,41] Schur function q-analog of Selberg'S beta-integral. We then state conjectured q-analogs of Biedenharn and Louck's second determinental formula, and the Pfaff-Saalschultz summation theorem for Schur functions. Either of these conjectures implies the new q-Gauss summation theorem in (5.4).

In Section 2 we survey several results involving Schur functions and then recall the definition of the new multiple (Schur function) basic hypergeo-

56 S.C. Milne

metric series of Macdonald, as well as our bilateral extension. This material is needed in· the rest of the paper.

Many of our results in this paper should extend to Macdonald's general multiple basic hypergeometric series involving his new symmetric functions P>.(Zj q, t) from [46].

2 Background Information

We start by reviewing several basic facts about the symmetric functions known as Schur functions.

Let A = (A1' A2, ... , Ar, ... ) be a partition, i.e., a (finite or infinite) sequence of nonnegative integers in decreasing order, A1 2: A2 2: ... 2: Ar . ", such that only finitely many of the Ai are nonzero. The number of nonzero Ai, denoted by I(A), is called the length of A. If E Ai = n, then A is called a partition of weight n, denoted IAI = n, and we write A I- n. The conjugate partition to A is denoted by A', where

I (' I \') ( ) A = A1, A2' ... , "(>'d ' 2.1

and A~ is the number of parts Aj in A that are 2: i. For example, (5,2,1) is the conjugate partition of (3,2,1,1,1).

Two useful statistics associated with partitions are

n(A) = I: (i - 1)~ (2.2) i~l

and

n(A') = I: e;i) (2.3) i~l

Given a partition A = (A1' ... , An) of length ~ n, the Schur functions SA

are defined by d ( >.;+n-j)

( ) _ et zi 1$i,j$n S>. Zb ••• , Zn - .

det(z?-J h$i,j$n (2.4)

The determinant in the numerator of (2.4) is divisible in Z[Zl' ... , Zn] by each of the differences (Zi - Zj), 1 ~ i < j ~ n, and hence by their product, which is the Vandermonde determinant

II (Zi - Zj) = det(z?-jh$i,j$n == an(Zb ... , zn) = an(z). (2.5) 1$i<j$n

Thus, the quotient in (2.4) is a symmetric polynomial in Zl, ... , Zn with coefficients in z. For example, S(n) = hn and S(l") = en, are, respectively, the nth homogeneous and elementary symmetric functions of Zl, ... , Zn.


Schur functions (also denoted S-functions) were first considered by J acobi [38] just as in (2.4). Their relevance to the representation theory of the symmetric groups and the general linear groups was discovered much later by Schur [55]. The name S-function (or Schur function) is due to Littlewood and Richardson [43]. A modern treatment of S-functions, including the combinatorial definition, can be found in [31,43,45,54,56,60].

We will need the explicit formulas for the specialized Schur functions

8>.(I,q, ... ,qn-1) and 8>.(I,q,q2, ... ), (2.6)

which appears in [42,45,60]. Consider the Ferrers diagram of A in which the rows and columns are arranged as in a matrix, with the ith row consisting of Ai cells. For a given cell z = (i,j) E A we define the hook length h(z) and content c(z) as follows.

h(z) = h(i,j) = (Ai - i) + (A; - j) + 1, (2.7)

where A' is the partition conjugate to A, and

c(z) = j - i. (2.8)

Note that h(z) is the number of cells to the right, on, or below the cell in the (i,j) position, and c(z) measures how far the cell (i,j) is from the main diagonal.

The hook polynomial H>.(q) is given by

H>.(q) = II (1 - qh(a:» = (ft (q)>'i+n-i) a:E>' i=1

(2.9)

. ( II (1 _ q>'i->'i-i+j ») -1

1~i<j~n

Given (2.2) and (2.7)-(2.9) vie have"

n-1 n(>') II (1 - qn+e(a:» 8>.(1, q, ... , q ) = q . (1 _ h(a:» ,

a:E>' q (2.10)

and (1 2 ) - n(>') H ( )-1 8>. , q, q , ... - q . >. q . (2.11)

The analysis in Sections 3 through 5 requires the Schur function case of Macdonald's multiple basic hypergeometric series [47] given by

Definition 2.12. Let Iql < 1 and {Zll ... , zn} be indeterminants. We then let r4>. [(a); (6); Z; 1] denote the multiple basic hypergeometric series

(2.13a)

58 S.C. Milne

=

where 1(>.), n(>.), s>.(x), H>.(q) are as above and

n

(a)>. = II(aq1-i)>'i

i=l

(2.13b)

(2.14a)

(2.14b)

It is clear that classical basic hypergeometric series are the n = 1 case of Definition 2.12.

Our new 1""1 summation theorem relies upon

Definition 2.15. Let 0 < Iql < 1 and {Xl, ... ,xn } be indeterminants. We then let rq;~+l [(a)j b, (b)j Xj 1] denote the multiple basic bilateral hypergeometric series

where

(2.16a)

(2.16b) (2.16c)

Pi = >'i + n - i, for 1 $ i $ n. (2.16d)

Remark 2.17. Factoring x; .. out ofthe ith row in the numerator determinant for s>.(, Xi,) we have

(2.18a)

- (Xl' .. Xn)>. .. S(>.l->. ....... >. .. _l->. ... O)(,Xi,). (2.18b)

Equation (2.18) is a convenient way to view S>.{X1, .. " xn) in (2.16c) where >. is any decreasing n-tuple >'1 ~ ... ~ >'n of integers. This makes the multiple Laurent series in (2.16) seem quite natural.

It is not hard to see that the b = q case of (2.16) equals (2.13). That is, we have

rW~+l [(a)jq,(b);x;l] = rc)~ [(a); (b); X; 1]. (2.19)

Just note that (q);:: = 0 if >'n < 0, and observe that equations (2.2), (2.5), and (2.9) immediately imply


Lemma 2.20.

where Pi = Ai + n - i, for 1 ~ i ~ n. (2.22)

This situation in (2.19) is analogous to the classical case.

3 Determinental Formulas

In this section we establish determinental formulas for r W ~+1 [(a); b, (b); z; 1] and r<J? [(a); (b); z; 1]. However, we first need

Lemma 3.1

where (A;r)m = (1- A)(I- Ar)· .. (1- Arm - l ). (3.3)

Proof. Consider the determinant

n-k zi

(3.4)

where i = row and k = column. The a = q-(n-l-l) and z = ziq case of Theorem 1.8 enables us to expand the (1+ l)th column of D~+l(z) by means of

(3.5a)

(3.5b)

For 0 ~ j ~ n -1- 2, the determinant in (3.5b) has two equal columns, and is thus zero. Only the j = n -1- 1 term survives to yield the recursion

(3.6)

Noting that

(3.7)

60 S.C. Milne

equation (3.2) follows by recursion from (3.6), starting with I = n - 1. Q.E.D.

We are now ready to prove the determinental formula in

Theorem 3.8 Let n ~ 1, 0 < Iql < 1, (z) denote {Zl. ... , zn}, ~n(z) be the Vandermonde determinant in (2.5), and ,.'IJ.+1 [(a)jb,(b)jzj1] be determined by Definition 2.15. We then have

(3.9a)

d t( n-I: .1. (1-1: 1-1: b b 1-1: b 1-1: » = e Zi ·,.Y'.+1 a1q , ... ,a,.q j, 1q , ... ,.q iZi 1~i.l:~n

where

Proof. Let the Schur function 8>.(Z) be written as

where

() ~n(PjZ) 8>. Z = ~(z) ,

n

(3.9b)

(3.11)

~n(Pi Z) = Iztlo+n-kl = E £(p) II Zfp(i), (3.12) pES.. i=1

with Pi = ~i + n - i. It is then not hard to see by Definition 2.15, and the relation

(A 1-n) (A 1-i) (A 1-n) q n-i q >'i = q >'i+n-i, (3.13)

that

(bj )(;)~(z)II a1q n-i ... a,.q n-i { n (1-n) (1-n)}

q (b1q1-n) .... (b q1-n) . i=1 n-., n-. (3.14a)

(3.14b)

= E {(-I)(;)q-(;)~n(blr-1, ... ,bqP .. -1)~n(PjZ)} (3.15a) .1>.2>"'>." -oo<Pi<oo

(3.15b)


Since (3.15b) is symmetric in {Pl, "',Pn} and both Lln(bqPl-l, ... , bqP .. -l) and Lln(pj z) are skew-symmetric, the summation appearing in (3.15a) can be replaced by the summation

1 1 00 00

,2:=, 2: ... 2: . n. n. (P) Pl=-OO p .. =-oo

(3.16)

Note that the ith factor in (3.15b) can be taken into the ith column of the determinant Lln(pj z).

It is now not hard to see by the Zi = bqPi- l case of Lemma 3.1 that we can write (3.14)-(3.15) in the form

= 2: l(bqPi-l j q-l)n_kl·l/p,,(Zi)l, (P)

where Ik(O) is the function

Expanding the two determinants transforms (3.18) into

n

(3.17a)

(3.17b)

(3.18)

(3.19)

2: 2: 2: c(u)c(p) II(bqPi-ljq-l)n_u(i)' IPi(zp(i), (3.20) (P) uES .. pES.. i=l

where u acts on columns and p acts on rows. But, (3.20) also equals

n

2: 2: c(up) II gn-u(i) (Zp(i)' (3.21) uES .. pES.. i=1

where 00

gn-k(O) = 2: (bq'-ljq-l)n_k ·1,(0). (3.22) '=-00

Now, fix u and look at the inner sum in (3.21). Replacing i by u-1(i) in the product gives

n

L: f(Up) IT gn-i(Zpu-1(i)' (3.23) pES.. i=1

62 S.C. Milne

Using ptT instead of p in (3.23) gives

n

E f(p) IIUn-i(Zp(i» = IUn-I:(Zi)h~i,l:~n, (3.24) pES. i=1

since f(tTptT) = f(p). Here, i = row and k = column. Since (3.24) is independent of tT, we have that (3.21) equals

n1Ign-l:(zi)hSi,I:Sn. (3.25)

All that remains is to consider gn-I:(Zi). We clearly have

00 (J' (a1q1-n), ... (arq1-n), gn_I:«(J) = ~ (b)'_n+l: . (b1q1-n), ... (b.q1-n), ' (3.26)

'_-00

(3.27a)

(3.27b)

= (In-I:. [(a1q1-n)n_I: ... (arq1- n)n_l:] (b1q1-n)n_I: ... (b,ql-n)n_1:

(3.28a)

. rt/J'+1 (a1q1-1:, ... , arq1-1:; b, b1q1-1:, ... , b,q1-1:; (J). (3.28b)

Substituting (3.26)-(3.28) into (3.25), and factoring the products in (3.28a) out of the kth column of the resulting determinant gives

(3.29a)

d t( n-I: .1. (1-1: 1-1: b b 1-1: b 1-1: » (3 29b) . e zi . rY'.+1 a1q , ... ,arq ;, lq , ... "q ;Zi· .

Equating (3.17) with (3.29), and then dividing both sides by (3.29a) completes the proof of Theorem 3.8. Q.E.D.

Keeping in mind (2.19) and Definitions 2.12 and 2.15 it is not hard to see that the b = q case of Theorem 3.8 gives the second determinental formula in

Theorem 3.30. Let n ~ 1, Iql. < 1, (z) denote {zt, ... , zn}, an(z) be the Vandennonde detenninant in (e.5). and re), [(a); (b); Z; 1] be detennined

Summation Th~rems for Basic Hypergeometric Series 63

by Definition 2.12. We then have

.<:In(z) r<l13 [at, ... ,ar;bl, ... ,b3 ;z; 1] (3.31a)

= d t( n-I: A. (1-1: 1-1: b 1-1: b 1-1: » e Zi . r'f'6 a1q , ... , arq ; 1q , ... , 6q ; Zi 19.1:~n

(3.31b)

where

(3.32)

Theorem 3.30 is a q-analog of the analogous formulas in [44,58].

4 A 1 W 1 Summation Theorem for Schur Functions

This section is devoted to proving Theorem 1.14, obtaining Theorems 1.18 and 1.20 as special, limiting cases, and then establishing certain elegant special cases of Kadell's and Askey'S q-analogs [15,40,41] of Selberg's [2,15,57] multiple beta-integral.

We first find that Theorem 1.14 is a direct consequence of Theorem 3.8, Theorem 1.1, and

Lenuna 4.1. Let a1 be arbitrary. Then

Lln(z) == Iz,-I:I = Iz,-I:(a1q1-l:zi)I:_1h9.I:~n

Proof. Let Q~)(z) be the determinant

Q(l)( )_ n Z1, "',Zn -

n-I: zi

(4.2)

(4.3)

for 1 ~ I ~ n + 1. By the a = q-(1-1) and Z = a1 Zi case of Theorem 1.8 we expand the Ith column of Q~) (z) to obtain

Q~)(z) = ~ { (q),-1 (_1)i a{q(1-1)i . qW} i=O (q)i(q),-1-j

(4.4a)

. (4.4b)

For 1 ~ j ~ 1- 1, the determinant in (4.4b) is O. Thus, the j = 0 term leads to the recursion

(4.5)

64 S.C. Milne

Equation (4.2) is immediate from

(4.6)

and the proof of Lemma 4.1 is complete. Q.E.D. Lemma 4.1 is a special case of Lemma 5 of [40]. Kadell utilized slightly

different special cases of his lemma. We prove Theorem 1.14 in the more compact form

IWI [ajb,-jxj1]

where (4.7a) is determined by Definition 2.15. Observe from Theorem 3.8 that

(4.7a)

(4.7b)

(4.8a)

(4.8b)

l.From the classical l.,pl summation in (1.3), elementary row and column operations, and equation (1.4a) it is not hard to see that (4.8b) becomes

(4.9a)

(4.9b)

Noting that

(4.10)

we have (b/aXi)1c_1 = (b/q)1c-1 (aqb-Iql-1cxi)1c_I. (4.11) (q/axi)1c-1 (axiql-1c)1c_l

It now follows from Lemma 4.1 and elementary column operations that ( 4.9b) equals

(b/q)(;)~n(x),

and the proof of (4.7) is complete. Q.E.D. Setting b = q in (4.7) yields

[ ] rrn (axi)oo I clo aj -j Xj 1 = (.)'

i=l X, 00

(4.12)

(4.13)


which is the q-binomial theorem for Schur functions in Theorem 1.18. Equa,tion (4.13) follows similarly from Theorem 3.30 and the classical q-binomial theorem.

Next, keeping in mind Definition 2.15, Remark 2.17, and the relations n n

II(-qlcqi)~i = c-I>'l q-n(>')q[(?)+oo+e2")] . II(-cqi->'i)>.p (4.14) i=l i=l

and

~n(, qPi,) = (_l)(;)qn(~)q(:) . ( II (1 - q>.r->..+._r»), (4.15) 1Sr<'Sn

it is not difficult to see that replacing a,q,b and Zi by -l/c,q,O, and qczi, respectively, in (4.7), and letting c -+ 0 yields the triple product identity for Schur functions in Theorem 1.20. This situation is the same as the classical case.

We now discuss q-beta integrals arising from Theorems 1.14 and 1.18. First, an elementary calculation shows that the

(4.16)

case of the Schur function q-binomial theorem in (4.13) is equivalent to the ~ = 0 case of the multiple q-beta integral in Theorem 5.5. To see this, utilize (1.4a), (2.2), (2.5), (2.9), (2.10), (2.13), (2.14), (2.21), (2.22), (5.6), (5.8), (5.15), and the change ofsummation argument in (3.16). The classical (n = 1) case was first observed by Thomae [61,62], later by Jackson [35], and recently studied further in [2,5,7,9,13,16,19,21,26,30,52].

Next, it is not difficult to show that the

Zi = qo+n-i, for 1 ::; i ::; n,

a = ~cqn-1

b = _ cqo+6+2n-2

case of (4.7) is equivalent to

(4. 17a)

(4.17b)

(4.17c)

Theorem 4.18. Let ~(z) be the Vandermonde determinant in (2.5) and fq(z) the q-gamma function in (5.6), with Iql < 1. We then have

100 100 ( n (_CZoq0+6+2n-2») ... II zl'-1 • 0 00 ~!(Z)dqZ1· .. dqzn (4.19a) o 0 i=1 (-cz.)oo

66 S.C. Milne

. (ftfq(a+n-i)fq(b+n-i»)} i=l f q(a+b+2n-i-l)

where the multiple q-integral in (4.19a) is defined by the sum

100 ••• 100

!(X1, ... , Xn)dqX1 ... dqxn

-OO<Pi<OO 1~i~n

(4.19b)

(4.19c)

To obtain (4.19) from the (4.17) case of (4.7), utilize (1.4a), (2.2), (2.5), (2.9), (2.10), (2.14), (2.16), (2.18), (4.15), (4.20), (5.6), (5.15), and the change of summation argument in (3.16).

Motivated by the fact that (4.19c) equals

n ( -1 2+i-2n-a) rr -c q a+n-1 ( n-i) ,

i=1 -cq a+n-1 (4.21)

it follows from (5.6) and the q-binomial theorem that the limiting case q -+ 1 of (4.19) can be written as the definite integral evaluation in

(4.22a)

= c-n(a+n-1) (ft(i)!) (rrn f(a + n - i)f(b ~ n - i») . . . f(a + b + 2n - a-I) 1=1 1=1

(4.22b)

Note that (4.22) also follows from the .\ = 0 and q = 1 case of (5.7) by the change of variables

Xi = cy;/(l + CYi), for 1 ~ i ~ n. (4.23)

The classical (n = 1) case of (4.17)-(4.23) was observed by Askey in section 5 of [13], and further discussed in [2,6,8,10,13,14,19,21,26,52]. See the Ie = 1 case of Conjecture ·3 of [15] for a nonsymmetrical version of Theorem 4.18. It may be possible to deduce a.\ f. 0 generalization of (4.7) from a suitable q-binomial theorem arising from Theorem 5.5 and then derive a corresponding generalization of Theorem 4.18 in which .\ is also nontrivial.

Summa.tion Theorems for Basic Hypergeometric Series 67

5 A q-Gauss Summation Theorem for Schur Functions

One of the fundamental results of classical basic hypergeometric series is the q-Gauss summation theorem in

Theorem 5.1. Let Iql < 1 and Icl < labl. Then

Here, we consider the extension of Theorem 5.1 provided by

Theorem 5.3. Let n ~ 1, Iql < 1, and Ic/abl < Iqln-l. We then have

(5.4)

The q = 1 case of Theorem 5.3 can be found in [47]. We put together a proof of (5.4) based upon Kadell's [40] q-analog of Selberg's integral given by

Theorem 5.5. (Kadell). Let ~ be a partition with l(~) :5 n, 8>.(Z) the Schur function in (e.4), and An(z) the Vandennonde detenninant in (e.5). Also take r.,(z) to be the q-gamma function

r (z) = (1- q)I-~ (q)oo ., (q~)oo'

(5.6)

with Iql < 1. We then have

n!q[n(>.)+a(;)+2(;)1 ( II (1- q>..->';+i-i)/(I_ q») (5.7b) l;Si<i;Sn

. IT r.,(a + n - i + ~i)r.,(b+ n - i) i=1 r.,(a+b+2n-i-l+~i) ,

where the multiple q-integral in (5.7a) is defined by the 8um

11 ... 11 f(zl, ... , Zn)d.,ZI ... d.,zn

(5.7c)

68 S.C. Milne

OSJ'j<OO

l~i~n

Remark 5.9 Theorem 5.5 is the k = 1 case of equation (7.1) of [40]. Here, Kadell defines 'lt1~k (i) by means of

(5.10)

which he discusses in equations (4.1)-(4.18) of his paper. The fact that (5.10) is a symmetric function of {tl, ... , tn} is important to Kadell's analysis in [40] as well as to our proof of Theorem 5.19 below. Our specialization of (5.10) with k = 1 and ti = Xi is what leads to the factor t1~(x) ofthe integrand in (5.7a). Consequently, as in (7.1) of [40] our integrand in (5.7a) is a symmetric function.

We first utilize Theorem 5.5 to derive an integral formula for the series in (2.13). To this end we define a related function of two sets of variables (x) and (y) by

Definition 5.11. Let Iql < 1 and {Xl, ... , xn} and {Yl, ... , Yn} be indeterminants. We then let r(). [(a);(b);x,y;l] denote the multiple basic hypergeometric series

(5.12a)

= ~ {(ad~··.(arh}. qn(~) . s~(Y1>""Yn)S~(.Xl, ... ,Xn). (5.12b) ~ (bt)~ ... (b.h H~(q) s~(, qn-.,)

l(~)~n

It is now not difficult to show that we have

Theorem5.l3. Let r+l().+l [(a); (b); x; 1] and r(). [(a);(b);x,y;l] be given by Definition 2.12 and 5.11, respectively. Take the same assumptions as in Theorem 5.5. We then have

;w;. [qa qQl qQ • . qfJ qb1 qb • . Y Y . 1] r+ 1 ':1.'. +1 , , ... , , , , ... , , 1,,,,, n, (5.14a)

(n r'l(,8+1-i) )

}] r'l(O: + 1- i)rq(,8 - 0: + 1- i) (5.14b)

. {( P (1 _ qi-i)/(l _ q)) -1 . ~! . q_(a_n+1)(;)_2(;)}

l~'<J~n

(5.14c)


. (g zf-n (Ziq~~~~:+l )00) . A~(z)dqZl ... dqzn } • (5.14d)

Proof. Interchange the order of integration and summation in (5.14d) and then apply the a = a - n + 1 and b = {3 - a - n + 1 case of Theorem 5.5. Utilizing equations (2.9), (2.10), (2.14), and the relations

IIn (q)>'i+n-i - rrn () . - II (1- ..i-i) (5.15) (qn+l-i) . - q n-I - 'i-

i=1 >'. i=1 1$i<i$n

now transforms (5.14b-d) into (5.14a). Q.E.D. Our proof of Theorem 5.3 requires the special case of Theorem 5.13 in

Corollary 5.16. Take the same assumptions as in Theorem 5.5. We then have

~ [qll< qfJ. q'Y. zq1-i . 1] 21"" " (5.17a)

(IT r q('Y+ 1 - i») (5.17b) i=1 rq({3 + 1- i)rq('Y - {3 + 1- i)

. {( II (1 _ qi-i)/(l _ q») -1 . ~! . q-(fJ-n+1)(;)-2(;) } l$i<i$n

(5.17c)

.11 ... 11 (ft zf-nt~q)~~)n) . A~(z)dqZ1···dqZn.(5.17d)

o 0 i=l z.zq Il<

Proof. Make the substitutions a -+ {3, a1 -+ a, {3 -+ 1', and Yi = zq1-i in (5.14). Observing that

S>.(, zq1-i,)S>.(,Zi,) s>. (, qn-i , )

( 1-n ) = S>., Zizq "

the a = qll< and Zi -+ zizq1-n case of (1.19) leads to (5.17). Q.E.D.

(5.18a)

(5.18b)

The classical q-Gauss summation theorem in (5.2) is an immediate consequence of Heine's transformation [1,26] for 2<P1 basic hypergeometric series. In the same way, Theorem 5.3 follows directly from

Theorem 5.19. (Heine Transformation for Schur Functions).

2~1 [a, b; c; , zq1-i, ; 1] (5.20a)

70 S.C. Milne

Proof. We show that (5.17) is equivalent to (5.20). Since the integrand in (5.17 d) is a symmetric function and An (X) = 0 if

any of the Xi are equal, it is clear from (5.8) that (5.17d) can be written as the multiple sum

( R n+1) (2) { (n ( 1+A.+n-i) ) r ,,- (1 - )n '"' II q '"f-p-n n.q q L...J . (zqA.+1-i)a

A .=1 I(A):5n

. q(p-n+1)IAI . A~(, qA.+n-i,) }. (5.21)

From equations (1.4), (2.9), (2.10), (2.14), (5.6), (5.15), and the relation

A~(,l.+n-i,) = q2n(A)+2(a) . ( II (1- qA._Aj+i-i») 2, (5.22)

1:5i<i:5n

it follows that (5.17a-c) and (5.21) is the a = qa, b = qP, c = q'"f case of (5.20). Q.E.D.

The left-hand side of (5.20) is symmetric in a and b while the right-hand side appears not to be. Consequently by alternating the application of this symmetry property with the transformation (5.20) we obtain

Theorem 5.23. (Euler Transformation for Schur Functions).

2cJ11 [a, b; c; , zq1-i, ; 1] (5.24a)

_ ijq 00 zq 00 a z. . c 1-i . { n (c 1-i) (b 1-i) } b - g (zq1-i)00(cq1-i)00 2cJ11 [b,~, bz,,;;q ,,1], (5.24b)

and

(5.25a)

(5.25b)

The proof of Theorem 5.3 is completed by setting z = c/ab in (5.24). Theorem 5.3 also follows directly from (5.20) and (1.19). In a similar

way, (5.4) can be derived from (5.17) and the A = 0 case of (5.7). We chose the above proof because we also wanted Theorem 5.19.

Note that equation (5.25) is an extension of the classical q-analog of Euler's transformation of a 2Ft hyper geometric series.


Exactly as in the classical case [2], equations (5.24) and (5.25) imply that

(5.26a)

= (ft (-'ell q1-I)OO(l'lIq1- I)oo) 2~1 [pe- iII , lie-iii j I'"j , -'eill q1-1, j 1] 1=1

(5.26b)

is symmetric in -',1', and II, and also symmetric in (J and -(J. Following Rogers, one should look for an expansion of wn(-', 1', II, q, (J) in terms of cos (J that is symmetric in -',1', II. To do this we need contiguous relations for 2~1 [(a)j (b)j Zj 1].

In the rest of this section we motivate Theorem 5.3 by showing that it is a natural consequence of each of two elegant conjectures.

The first is an alternate determinental formula for 2~1 [(a)j (b)j Zj 1] different than the one in Theorem 3.30. That is,

Conjecture 5.27. Take the same assumptions as in Theorem 9.90. We then have

~n(Z) 2~1 [a,bjcjzj1] (5.28a)

The n = 2 case of (5.28) is a consequence of Theorem 3.30 and the q-contiguous relation

[1- ab z] 2¢h (a, bj cq-1 j z) = 2tPl (aq-1, bq-1 j cq-1 j z) C

[b(1- aqjc) + a(1- bjq )] .J.. ( b . . ) + q(1- cq-1) z· 2Y'1 a, , c, z , (5.29)

which we use to expand the second column of the determinant in (5.28b). Equation (5.29) can be quickly verified by computing the coefficient of zm on each side. See [23,26,29,32,53,63] for a more systematic treatment of q-contiguous relations such as (5.29). However, (5.29) can't be iterated to lead to a proof of (5.28) in general. Thus, (5.28) remains a conjecture for n~3.

This situation is intriguing since the q = 1 case of (5.29) can be iterated to yield a proof of the q = 1 case of Conjecture 5.27. This is exactly what Biedenharn and Louck did in [44] as part of their proof of a general Euler transformation for their 2:F1 hypergeometric series.

72 S.C. Milne

Now, Theorem 5.3 follows by induction from Conjecture 5.27 and Theorem 5.l.

Setting Xl = clab in (5.28), it is not hard to see that

(5.30a)

~ {(g (:6 - .,) ) -, ~:'('2' ... "'). (:6)"-' . ,~,(a,6;c; :b)} (5.30b)

I n-k (ab ) (-k+1 I . Xi -Xi . 2tP1 a, bj cq j Xi) 2<i k<n C -' -k-1

(5.30c)

. Ix?-l-(k-l)(q a: Xi)(k_l)_l . 2tPl (a, bj cq-lq-(k-1)+l j xi)12~i.k~n }

(5.31)

= 2tPl (a, bj Cj :b) . 2<)1 [a, bj cq-lj X2, ••• , Xnj 1]. (5.32)

If we take Xi = (clab)q1-i, for 1 ~ i ~ n, in (5.30)-(5.32) then (5.4) follows by induction from (5.2):

The q = 1 case of this induction also works and establishes the corresponding summation theorem from [47].

The second conjecture is an extension of the classical q-analog of the Pfaff-Saalschutz summation theorem. We have

Conjecture 5.33. Let N be a nonnegative integer. Then

....... [a b q-N. C d· q1+n-i . 1] 3'¥2 " "" "

(5.34a)

(5.34b)

where cd = abqn-N. (5.35)

If we write everything out explicitly as in Definition 2.12, and utilize (2.10) and the homogeneity of 8A(X) we find that (5.34) becomes

" {(ah(b)A(q-Nh}. qIAI+2n(A) . II(l- qn+c(y») ~ (c)A(d)A HA(q)HA(q) yEA

I(~)~ .. Al~N

(5.36a)


= (c/ a )(N)" (c/h )(N)" (c)(N),,(c/ah)(N)" '

(5.36b)

where (5.35) holds and (N)n denotes the partition with n parts equal to N.

Observing that

(q-N)>. _ ( C ) 1>'1 n { (qN . qi->'i)>'i }

«ab/c)qn-N)>. - abqn . n «c/ab)qN. qi-n->'i)>'i ' (5.37)

it is clear that letting N -+ 00 in (5.34) yields (5.4). The q = 1 case of (5.34) is in [47]. The analysis in this section should be pursued further.

References [1]

[2]

[3]

[4]

[5]

[6]

[7]

[8]

[9]

G.E. Andrews, "The Theory of Partitions", Vol. 2, "Encyclopedia of Mathematics and Its Applications", (G.-C. Rota, Ed.), AddisonWesley, Reading, Mass., 1976.

G.E. Andrews, "q-Series: Their Development and Application in Analysis, Number Theory, Combinatorics, Physics, and Computer Algebra", CBMS Regional Conference Lecture Series 66 (1986), Amer. Math. Soc., Providence, R.1.

G.E. Andrews, On Ramanujan's summation of ItPl(a, h, z), Proc. Amer. Math. Soc. 22 (1969), 552-553.

G.E. Andrews, On a transformation of bilateral series with applications, Proc. Amer. Math. Soc., 25 (1970), 554-558.

G.E. Andrews, Connection coefficient problems and partitions, AMS Proc. Sympos. Pure Math. 34 (1979), 1-24.

G.E. Andrews and R. Askey, "The Classical and Discrete Orthogonal Polynomials and Their q-Analogues", in preparation.

G.E. Andrews and R. Askey, Enumeration of partitions: the role of Eulerian series and q-orthogonal polynomials, "Higher Combinatories" (M. Aigner, ed.) Reidel, Boston, (1977), p. 3-26.

G.E. Andrews and R. Askey, A simple proof of Ramanujan's summation of the ItPl, Aequationes Math. 18 (1978), 333-337.

G.E. Andrews and R. Askey, Another q-eztension of the beta function, Proc. Amer. Math. Soc. 81 (1981), 97-100.

74 S.C. Milne

[10] G.E. Andrews and R. Askey, Classical orthogonal polynomials, "Polynomes Orthogonaux et Applications", Lecture Notes in Math. 1171 (1985), Springer, Berlin and New York, p. 36-62.

[11] G.E. Andrews, R. Askey, B.C. Berndt, K.G. Ramanathan, and R.A. Rankin, eds., "Ramanujan Revisited", Academic Press, New York (1988).

[12] K. Aomoto, Jacobi polynomials associated with Selberg'S integral, SIAM J. Math. Anal. 18 (1987), 545-549.

[13] R. Askey, The q-gamma and q-beta functions, Appl. Anal. 8 (1978), 125-141.

[14] R. Askey, Ramanujan's extensions of the gamma and beta functions, Amer. Math. Monthly 87 (1980), 346-359.

[15] R. Askey, Some basic hyper geometric extensions of integrals of Selberg and Andrews, SIAM J. Math. Anal. 11 (1980), 938-951.

[16] R. Askey, Two integrals of Ramanujan, Proc. Amer. Math. Soc. 85 (1982), 192-194.

[17] R. Askey, A q-beta integral associated with BCl , SIAM J. Math. Anal. 13 (1982), 1008-1010.

[18] R. Askey, An elementary evaluation of a beta type integral, Indian J. Pure Appl. Math. 14 (1983), 892-895.

[19] R. Askey, Orthogonal polynomials old and new, and some combinatorial connections, "Enumeration and Design" (D.M. Jackson and S.A. Vanstone, eds.,), Academic Press, New York (1984), p. 67-84.

[20] R. Askey, Ramanujan's l1/Jl and formal Laurent series, Indian J. Math. 29 (1987), 101-105.

[21] R. Askey, Beta integrals in Ramanujan's papers, his unpublished work and further examples, "Ramanujan Revisited" (G.E. Andrews et al., eds.), Academic Press, New York (1988), p. 561-590.

[22] R. Askey and R. Roy, More q-beta integrals, Rocky Mountain J. Math. 16 (1986), 365-372.


[23] R. Askey and J. Wilson, A set of orthogonal polynomials that generalize the Racah coefficients or 6 - j symbols, SIAM J. Math. Anal. 10 1979), 1008-1016.

[24] W.N. Bailey, "Generalized Hypergeometric Series", Cambridge Mathematical Tract No. 32, Cambridge University Press, Cambridge, 1935.

[25] N.J. Fine, "Basic Hypergeometric Series and Applications", Mathematical Surveys and Monographs, Vol. 27 (1988), Amer. Math. Soc., Providence, R.I.

[26] G. Gasper and M. Rahman, "Basic Hypergeometric Series", Vol. 35, "Encyclopedia of Mathematics and Its Applications", (G.-C. Rota, Ed.), Cambridge University Press, Cambridge, 1990.

[27] S.G. Gindikin, Analysis on homogeneous spaces, Russian Math. Surveys 19 (1964), 1-90.

[28] K.1. Gross and D. St. P. Richards, Special functions of matrix argument. I: Alebraic induction, zonal polynomials, and hypergeometric /unctions, Trans. Amer. Math. Soc. 301 (1987), 781-8U.

[29] W. Hahn, tiber orthogonal polynome, die Differenzengleichungen geniigen, Math. Nachr 2 (1949),4-34.

[30] W. Hahn, Beitrage zur Theorie der Heineschen Reihen, Math. Nachr. 2 (1949), 340-379.

[31] P. Hall, The algebra of partitions, in Proceedings, 4th Canadian Math. Congress, Banff (1959), p. 147-159.

[32] E. Heine, Untersuchungen iiber die Reihe ... , J. Reine Angew. Math. 34 (1847), 285-328.

[33] C.S. Herz, Bessel functions of matrix argument, Ann. of Math. (2) 61 (1955), 474-523.

[34] M.E.H. Ismail, A simple proof of Ramanujan's ltPl sum, Proc. Amer. Math. Soc. 63 (1977), 185-186.

[35] F.H. Jackson, Transformations of q-series, Messenger of Math. 39 (1910), 145-153.

[36] M. Jackson, On Lerch's transcendent and the basic bilateral hypergeometric series 2tP2, J. London Math. Soc. 25 (1950), 189-196.

76 S.C. Milne

[37] C.G.J. Jacobi, "Fundamenta nova theoriae /unctionum ellipticarum", (1829), Rigiomnoti, fratrum Borntrager (reprinted in "Gesammelte Werke", Vol. 1, pp. 49-239, Reimer, Berlin, 1881).

[38] C.G. Jacobi, De /unctionibus alternantibus ... , Crelle's Journal 22 (1841), 360-371 [Werke 3, 439-452].

[39] A.T. James, Distribution of matrix variates and latent roots derived from normal samples, Ann. Math. Statist. 35 (1964), 475-501.

[40] K. Kadell, A proof of some q-analogs of Selberg's integral for k = 1, SIAM J. Math. Analysis 19 (1988), pp. 944-968.

[41] K. Kadell, The Selberg-Jack polynomials, to appear.

[42] D.E. Littlewood, "The Theory of Group Characters", 2nd ed. Oxford at the Clarendon Press, 1940.

[43] D.E. Littlewood and A.R. Richardson, Group characters and algebra, Philos. Trans. Roy. Soc. London Ser. A 233 (1934), 99-141.

[44] J.D. Louck and L.C. Biedenharn, A generalization of the Gauss hypergeometric /unction, J. Math. Anal. Appl. 59 (1977), 423-431.

[45] I.G. Macdonald, "Symmetric Functions and Hall Polynomials", Oxford Univ. Press, London/New York, 1979.

[46] I.G. Macdonald, A new class of symmetric /unctions, Publ. I.R.M.A. Strasbourg, 1988, 372/s-20, Actes 20e Seminaire Lotharingien, p. 131-171.

[47] I.G. Macdonald, Lecture notes from his talk, Univer. of Michigan, June 1989.

[48] S.C. Milne, A U(n) generalization of Ramanujan's ItPl summation, J. Math. Anal. Appl. 118 (1986), 263-277.

[49] S.C. Milne, Multiple q-series and U(n) generalizations of Ramanujan's I WI sum, "Ramanujan Revisited", (G.E. Andrews et al. eds.), Academic Press, New York (1988), p. 473-524.

[50] S.C. Milne, The multidimensional I WI sum and Macdonald identities for AP>, Proc. Sympos. Pure Math. 49 (part 2)(1989), 323-359.


[51] S.C. Milne, A triple product identity for Schur junctions, J. Math. Anal. Appl., in press.

[52] M. Rahman, Some extensions of the beta integral and the hypergeometric function, to appear.

[53] E.D. Rainville, "Special Functions", Macmillan Co., New York, 1960.

[54] J. Remmel and R. Whitney, Multiplying Schur junctions, J. Algorithms 5 (1984),471-487.

[55] 1. Schur, "Uber ein Klasse von Matrizen die sich einer gegebenen Matrix zuordnen lassen", Dissertation, Berlin, 1901. [Ges Abhandlungen I, 1-72].

[56] M.-P. Schiitzenberger, La correspondance de Robinson, in "Combinatoire et representation du groupe symetrique", Strasbourg, 1976. Lecture Notes in Mathematics No. 579, Springer-Verlag, New York/Berlin, 1977.

[57] A. Selberg, Bemerkninger om et multpelt integral, Norske Mat. Tidsskr. 26 (1944), 71-78.

[58] D.P. Shukla, Certain transformations of generalized hypergeometric series, Indian J. Pure Appl. Math. 12 (8) (1981), 994-1000.

[59] L.J. Slater, "Generalized Hypergeometric Functions", Cambridge University Press, London and New York, 1966.

[60] R.P. Stanley, Theory and applications of plane partitions, Studies in Applied Mathematics 50 (1971), 167-188, 259-279.

[61] J. Thomae, Beitriige zur Theorie der durch die Heinesche Reihe ... , J. Reine Angew. Math. 70 (1869), 258-281.

[62] J. Thomae, Les series Heineennes superieures, 0'11. les series de la forme ... , Annali di Matematica Pura de Applicata 4 (1870), 105-138.

[63] J .A. Wilson, "Hypergeometric Series, Recurrence Relations and Some New Orthogonal Polynomials", Thesis (1978), Univ. of Wisconsin, Madison.

S.C. Milne, Department of Mathematics, Ohio State University, 231 West 18th Ave., Columbus, OH 43210

Orthogonal Polynomials, Recurrences, Jacobi Matrices, and Measures

P. Nevai

ABSTRACT This is a compact bare bone survey of "orne aspects 01 orthogonal polynomials addressed primarily to nonspecialists. Special attention is paid to characterization theorems and to spectral properties of Jacobi matrices.

1. INTRODUCTION

There are a number of excellent recent surveys on this subject (cf. References), in particular, the proceedings of the first NATO Advanced Study Institute on "Orthogonal Polynomials and Their Applications" [45]. What differentiates this short survey from the above ones is that this is a compact bare bone survey of some aspects %rthogonal polynomials addressed primarily to nonspecialists. I wrote it in the hope that it is going to be a sufficiently light meal easily digestible by those analysts who have neither time nor patience to read surveys of more substantial character. Naturally, I would like to see the readers of this paper to be sufficiently attracted to orthogonal polynomials so eventually they will take the time and effort for learning more about this beautiful part of mathematics. In the references section I list a short collection of books and papers which will enable the reader to continue this excursion.

2. NOTATION ON THE REAL LINE THE RECURRENCE:

ZPn(Z) = an+1Pn+1(Z) + bnPn(z) + a"p"_I(Z)

n = 0,1, ... , P-l = 0, and Po = const > O. Frequently, though not always, Po = 1 which amounts to considering probability measures.

THE MEASURE: a is a positive Borel measure on the real line R with finite moments and infinite support.

THE JACOBI MATRIX:

J= (~ where aj > 0 and bj E R.

41 0 0 0 61 42 0 0 42 62 430 o 436a44

... ... )

This material is based upon reseal'Ch supported by the National Science Foundation under Grant No. DMS-8814488 and by NATO under Grant No. CRG.870806.

PROGRESS IN APPROXIMATION THEORY (A.A. Gonchar and E.B. Saff, eds.), ~Springer-Verlag (1992) 79-104. 79

80 P. Nevai

THE ORTHOGONAL POLYNOMIALS:

Pn(X) = Pn(a, X) = Pn(da, X) = "Ynxn + lower degree terms

and

3. NOTATION ON THE UNIT CIRCLE

THE RECURRENCE:

n = 1,2, ... , where C)O = 1. Here • is the "reverse" operation defined by

(azn + bzn- 1 + ... + c)* = a + bz + ... + czn

and c)n = K.n¢n are the monic orthogonal polynomials.

THE MEASURE: I' is a positive Borel measure on [0,211") with infinite support.

THE ORTHOGONAL POLYNOMIALS:

and

~ f27r ¢n(eit)¢m(eit)dl'(t) = onm. 211" Jo

THE SZEGO FUNCTION:

( 1 127r eit + z ) D(I',z) = exp -4 -'-t -logl"(t)dt , 11" 0 e' - z

In particular, D2(1', 0) is the geometric mean of 1".

4. THE BEGINNING OF RECENT TIMES

Izl < 1.

In addition to the tremendous amount of work on continued fractions and the moment problem, the "Blumenthal" and, most significantly, the "Szego" theorem are the underlying results of most research in the general theory of orthogonal polynomials in the past quarter of a century.

Orthogonal Polynomials, Recurrences, Jacobi Matrices and Measures 81

THEOREM. (Blumenthal-Weyl) lfthe diagonal and subdiagonal of the Jacobi matrix converge to finite limits, say,

lim an = a > 0 & lim bn = b E R n~oo - n-+oo

then the derived set of the support of the corresponding spectral measure is [b-2a,b+2a].

This can be proved either by using Poincare's theorem on ratio asymptoties for solutions of linear homogeneous difference equations with convergent recursion coefficients, or by noticing that the conditions of the theorem are precisely those guaranteeing that the corresponding Jacobi matrix is a compact perturbation of the constant Jacobi matrix, and thus, by Weyl's theorem, they have the same continuous spectrum (cf. [6,38,39]).

EXAMPLE: Chebyshev polynomials (of the second kind): an = ~ & bn = 0, Jacobi polynomials: an = ~+O(~~) & bn = O(~2)' Pollaczek polynomials: an = ~ + O(~) & bn = O(~), and da(x) = Ixlfdx or something similar: an = ~ + const (-~)" + O(~) & bn = const (-~)" + O(~) (for the last example see [39]).

THEOREM. (Szego [57], Kolmogorov, Smirnov, Ahiezer, M. G. Krein, Shohat, Geronimus et al.)

00

logJJ'EL1 ¢:::::> "'lc)~(JJ,O)I<oo ¢:::::> lim #Cn(JJ) = D-1(JJ,0) < 00. L...J n-+oo n=O

I believe that the following four components played a crucial role in the seventies in molding an exceptionally creative atmosphere for the development of the general theory of orthogonal polynomials.

1. RECURRENCES AND ORTHOGONAL POLYNOMIALS ON THE UNIT CIRCLE: Here the main problem lies in finding properties of measures associated with orthogonal polynomials in terms of the coefficients of the underlying recursion formula. The main tools are those of classical analysis. For instance,

THEOREM. (Geronimus [24])

00

~ Ic)n(JJ, 0)1 < 00 ==> JJ is absolutely continuous. n=O

OUTLINE OF THE PROOF: By the recurrence formula, the convergence of the series L:=o Ic)n(JJ, 0)1 implies

'82 P. Nevai

uniformly on the unit circle. Hence, get) = limn_oo 1</1: (Il , eit)I-2 is a nonvanishing continuous function. On the other hand, for every measure Il and for every continuous function f, we have . 12r f(t)dt 12r

lim 1</1* ( it)12 = f(t)dll(t). n_oo 0 n Il,e 0

Therefore, Il must be absolutely continuous. As a matter of fact, Il' = g .•

In addition, there was Baxter's characterization of E::o l~n(ll, 0)1 < 00

in terms of the Fourier coefficients of log Il' (cf. [4)). A particularly striking case of it is the following

THEOREM. (Baxter) Let Il be absolutely continuous, and let Il' be positive. Then

n=O

2. DISCRETE SCATTERING THEORY: (Initiated by Case & Kac) Here methods of continuous scattering theory were adapted to investigate the relationship between the real and complex recursions and the corresponding orthogonal polynomials. For instance,

THEOREM. (Geronimo-Case [20)) If

~ n (lan(a) - ~I + Ibn(a)l) < 00 (4.1)

then a has at most a finite number of mass points, and a is absolutely continuous at ±1.

PROOF: (by Chihara & Nevai [7)) First one proves that if (4.1) holds then there is A and B such that AB 1: 0 and limn_oo r;£~'1) = 1. This is done by a careful asymptotic analysis of the recurrence equation. Then a zero counting argument shows that each orthogonal polynomial has a uniformly bounded number of zeros to the right of 1, and, therefore, the support of corresponding measure cannot contain infinitely many points to the right of 1 (since each point in the support attracts zeros of all Pn'S). In addition, the above limit relation guarantees the divergence of E:=op~(a, 1), and, hence, by a routine argument frequently used in the theory of moments, a must be continuous at 1. Another proof can be achieved by showing that there is a "tail" of the corresponding Jacobi matrix which has no eigenvalues at all. This was the original proof by Geronimo and Case .•

It is interesting to note that Gelfand and Levitan [18] investigated continuous scattering by adapting methods from orthogonal polynomials. "The idea of this work is simple. Similar to the way in which polynomials, orthogonal with respect to a given weight function, are constructed by orthogonalizing the powers of %, we construct from the spectral function p( %) the eigenfunctions </1(%,.\) by orthogonalizing the functions cos(v'Xt) with respect to p(.\ l."


3. ABSOLUTE CONTINUITY OF PHASE OPERATORS: Here perturbation methods and operator theory are used to determine certain properties of spectral measures of Jacobi matrices. For instance,

THEOREM. (Dombrowski [8])

an(a) '\. I, bn(a) == 0 &, supp(a) = [-2,2] :::::} a e AC.

4. QUALITATIVE BEHAVIOR OF RECURSION COEFFICIENTS: After Szego's theory of orthogonal polynomials this was a major step towards studying orthogonal polynomials outside the Szego class. For instance,

FUNDAMENTAL THEOREM. (Rahmanov)

Ji > 0 a.e. :::::} lim 4>n(P,O) = O. n_oo

At the present time there are four valid proofs of this theorem [51, 33, 52,46], the most recent and perhaps the simplest one is given in [46].

5. CHARACTERIZATION THEOREMS

It is natural to ask if the condition p' > 0 almost everywhere is equivalent to limn_oo 4>n(P, 0) = 0, but, alas, the answer to this question is that there are plenty of purely singular measures for which Iiffin-+oo 4>n(P, 0) = 0 holds (cf. works of Von Neumann, Simon et al., Lubinsky, Magnus &, Van Assche and Totik).

OPEN PROBLEM: "How many" singular measures are there for which the condition Iimn_ oo 4>n(P, 0) = 0 holds?

A very promising approach to this problem is given by

THEOREM. (Totik [58]) Let dm(t) = 2~dt, and let (T < -1 be a given number. Suppose for every j = 1,2, ... we have a sequence of Borel measures {Pj,t }r=l on [0,211') such that limt_oo Pj,t = r m in the weak'" -topology. Then there is a subsequence {kj} of natural numbers such that for the measure P = Ei=l Pj,t; we have Iiffin-+oo 4>n(P, 0) = O.

THEOREM. (Mate &, Nevai &, Totik [33], Nevai [43], Li &, Saft' [27])

. 1211' II4>n(p, ei ')12 1 p' > 0 a.e. <=> hm sup 14> ( ")12 - 1 dt = O.

n-oo l~l 0 n+l p, e l

THEOREM. (Nevai [46])

lim 4>n(P,O) = 0 n_oo Ii . f Y'n p, e _ 1 dt - 0 1

211' IIA. (it)12 1 n~ ~l 0 l4>n+'(P, eit)12 - .

OPEN PROBLEM: Are there any natural ''intermediate" classes of orthogonal polynomials?

EXAMPLE: If 4>n(P, 0) = e, where lei < 1 then supp(p) is a proper subinterval of[O, 211')( cr. [24]).

Another type of characterization theorems is the following which can be proved by using the methods developed in [35, 36, 37, 39].

84 P. Nevai

THEOREM. (Ma.te & Nevai & Totik)

Ii > 0 a.e. <==> liminf [2ft I/(t)ll~n(l', eif)I" 1"(t)cIt > 0 n-oo 10

Vp> 0 & VI s.t. 12ft I/(t)ldt > 0, (5.1)

and an analogous result holds for orthogonal polynomials in [-1,1].

DEFINITION. The positive Borel measure measure I' defined on (0,211") is called regular if lim sUPn_oo II:n (I') t $ 1. Similarly, the measure cr on [-1,1] is called regular if limsuPn_oo 'Yn(cr)t $ 2.

Regular measures can be characterized in terms of nth root asymptotics as follows.

THEOREM. (ErdOs & Turan [16], Li & Saff & Sha [28])

cr is regular li I ( )It _ Iz + Vz2=11 n_~ Pn cr,z - 2

locally uniformly outside the unit interval, and

I' is regular

locally uniformly outside the unit disk. In addition,

lim sup II:n(l')t $ 1 n_oo

1 n

=> lim - 2: ()k(l', 0) = O. n-oo n k=O

A particularly simple characterization of regular measures is given by the following

THEOREM. (Pan & Saff [50])

I' is regular 12ft [ ~~(I',eit) ];;;r lim sup ./.. (if) - 1 cit = O.

n-oo 1~1 0 Y'n+l 1', e

Here the branch of [ ~!:I ];;;r is chosen to be positive at O.

PROOF: (by Nevai) Let I ~ 1. Since [~!:/'~~o ] ~ = [,,::~{;)];;;r, we can use Cauchy's formula to obtain

_ [ 1I:n(1') ];;;r _ -.!..12ft [ _ [ ~~(I',ei') ] ~l 1 ( ) - 2 1~. (.,) dt.

II:n+l I' 11" 0 n+l 1', e'


On the other hand, first by Schwarz's and then by Holder's inequality,

for every n, I = 1,2,... . Without loss of generality we can assume that 11':0(1') ~ 1, since this amounts to a renormalization of the measure. But then we can use the inequality 1 :::; II':n(I')~ for n, 1= 1,2, ... , in the numerator on the right-hand side. In the middle we can take the supremum over all I ~ 1, and then we have to do the same in the denominator on the right-hand side. After taking these supremums, we can choose an increas-

ing subsequence Ij such that limj_oo II':n+l;(I');;:f:r; = limsuPn_oo II':n(l')t.

Then limj _ 00 II':n (I' );;:f:r; = 1. Hence, if 11':0 (I') ~ 1, then

{ . ~}-l 1 1211" [ ;~(I',eit) ].;r 1- hmsuplI':n(I')" :::;suP -2 ;* (it) -1 dt

n-oo '~l 1r 0 n+' 1', e

86 P. Nevai

for every n = 1,2, .... Now the theorem follows directly from these inequalities .•

In [50], Pan and Saff give an analogous characterization of the Szegc5 class as well.

For further information on regular measures I recommend Stahl and Totik's book [55] on "General Orthogonal Polynomials."

OPEN PROBLEM: Is it possible to characterize regular measures in the spirit of (5.1)?

The above results suggest that while regular measures can be characterized in terms of nth root asymptotics, ratio asymptotics are intimately connected with the recursion coefficients. This is indeed the case as shown by

THEOREM. (Nevai [39], Mate & Nevai & Totik [31])

lim an(a) = 1/2 & lim bn(a) = 0 <==> lim Pn+l(a, z) = z+v:;a=t n ..... oo n ..... oo n ..... oo Pn(a, z)

locally uniformly for z ;. supp( a) and

locally uniformly for Izl ~ 1.

6. DERIVATIVES OF ORTHOGONAL POLYNOMIALS

The (unit interval version of the) characterization theorem (5.1) is an immediate consequence of the following result.

THEOREM. (Mate & Nevai & Totik [35]) Let 0 < p:$; 00. Then there is a constant C with the property that for every measure a supported in [-1,1] such that a' > 0 almost everywhere there, the inequality

{11 I(t) P dt}t :5c~~{11 I/(t)pn(a,t)IPdt}; -1 J al(t).;r-:::t:i -1

holds for every measurable function I in [-1,1]. For instance, the constant C = .Ji2maxH-i.o} can be used in the above inequality.

OPEN PROBLEM: What is the optimal value of the constant C in the above theorem?

This theorem has a number of applications in problems related to Fourier series in orthogonal polynomials and in various interpolation and quadrature processes. Recently, it turned out that similar inequalities involving derivatives of orthogonal polynomials are equally useful. In conjunction with this, I propose the following


CONJECTURE. Let 0 0 almost everywhere there, the inequality

{jl I /(t) ll" dt}~ $Climinf.!.{jll/(t)p~(a,t)l"dt}* -1 Va'(t) (1 - t2) n-oo n -1.

holds for every measurable function / in [-1,1].

For (sufficiently smooth) generalized Jacobi weight functions it is possible to prove the above conjecture.

THEOREM. Let m

U(z) = II Iz - VI:IAIJ (6.1) 1:=1

and m

W(z) = g(z) II Iz - VI:IBIJ (6.2) 1:=1

where VI: E [-1,1], AI: > -1, BI: > -1, and 9 is a nonnegative function. If r ~ 0, p ~ 1 and 0 < Cl $ g(z) $ C2 < 00 for z E [-1,1], then there is a positive constant C such that

11 U(z)dz < -1 W(z)~(I- z2) r"t3 , -

C lim inf ..!..jl IPn(W,z)np~(W, z)I"U(z)dz. (6.3) n-oo nP -1

Here Pn (W) denotes the orthonormal polynomials associated with the weight function W. In view of this theorem, it is natural to expect that not only the above Conjecture but also an appropriate extension involving an arbitrary number of derivatives may hold as well.

The rest of this section deals with the proof of the above theorem. Since the details are somewhat technical, the reader may safely skip to the next section. The following lemma generalizing the first theorem in this section is needed in the proof.

LEMMA. Let supp(a) = [-1,1]' a'(z) > 0 almost everywhere in [-1,1], and let 0 < P $ 00. Then for every sequence {In} of Lebesgue-measurable functions in [-1,1]

88 P. Nevai

In particular,

liminf In = ° a.e. n-oo

PROOF OF THE THEOREM: In what follows, for the sake of simplicity in the notations, we will write Pn for Pn(W). Let {Zjn}j=l denote the zeros of Pn in decreasing order, and let Zo = 1 and Zn+l = -1. Then

(6.4)

for k = 0,1, ... , n (cf. [39, Theorem 9.22, p. 166]). Let Un be defined by

Given a > 0, define the set En(a) by

En(a) =

[-1, 1] \ { [U [Yk - ;, Yk + ~] 1 U [-1, -1 + :2] U [1, 1 - :2] } . IYkl<l

Then by (6.4), for every sufficiently small 0 < a < ao = ao(U),

sup __ Z_ < 00 [ U( )] ±l

,;EE .. (a) Un(z) &

In what follows, we fix ° < a < 1 so that (6.5) holds. Even though it is natural to assume that Ak > -1 for k = 1,2, ... , m, in (6.1), the proof of (6.3) does not require this condition. So assume that r ~ 0, P ~ 1 and 0< Cl :$ g(z) :$ C2 < 00 for Z E [-1,1]. Then, since Pn vanishes at Zkn, ~h~ -

for Z E [Zk+l,n, Zk-l,n], k = 1,2, ... ,n. Thus, by Holder's inequality,


where ~ + ~ = 1, that is,

for all Xk+1,n $ x $ Xk-1,n. Therefore, multiplying both sides by Un and using the second inequality in (6.5), we obtain

IPn(x)lr+PUn(X)[Xk_1,n - Xk+1,nP-P $ C11"'k-l·"IPn(tWlp~(t)IPUn(t)dt :C.+1,.

for all Xk+1,n $ X $ Xk-1,n where C1 = C1(r,p,U). Now by (6.4),

for all Xk+1,n $ X $ Xk-1,n where C2 = C2(r,p, U, Wl. Integrating the latter over [Xk-1,n, Xk+1,n] we obtain

for k = 1,2, ... ,n. Let 1D denote the characteristic function of the set D, and let Dn (a) be the set defined by

Dn(a) = U ([Xk-1,n, Xk+1, .. ] : [Xk-1,n, Xk+1,n] S; En(a)}.

Adding together inequalities (6.6) for all k such that [Xk-1,n, Xk+1,n] C

Dn(a), we obtain

11 IPn(xW+P1D"(<I)(X)~n(x)dx $ C2 [ IPn(tWlp~(t)IPUn(t)dt. -1 [y\;",~ + ;~] JD,.(a)

(6.7) Since Dn(a) S; En(a), we can use the first relation in (6.5) to replace Un by U in the right-hand side of (6.7). Therefore,

where Ca = Ca(r,p, U, W). Note that

for a.e. x E [-1,1].

90 P. Nevai

Hence, by (6.8) and by the Lemma, inequality (6.3) follows when r ~ 0, P ~ 1 and 0 < Cl :5 g(z) :5 C2 < 00 for z E [-1,1]' •

In ajoint project with Yuan Xu [47], we hope to find some generalizations and applications of (6.3). Nevertheless, I do not know how to prove similar results for general weights or measures.

7. JACOBI MATRICES AND SELF-ADJOINT OPERATORS

Let H be infinite-dimensional and separable Hilbert space, J be a bounded linear self-adjoint operator on H. Assume there is a cyclic vector 'r/J for J, that is, Span({Jn'r/J}~=1) = H, and let {'r/Jn}~=o be the Gram-Schmidt orthogonalization of {In+1 'r/J } ~=o. Then J has a tridiagonal matrix representation

(~ a1 0 0 0

""") a1 61 a2 0 0 ...

J= ~ a2 62 a3 0 0 a3 63 a4

such that 'r/Jn = Pn (J)'r/J where

ZPn(Z) = an+1Pn+1(z) + 6nPn(z) + anPn_1(Z)

n = 0,1, ... , P-1 = 0, and Po = 1. If J = J >"dE)., is the spectral resolution of J and Q is defined by Q(B) = IIE(B)'r/J1I2 for every Borel set B then J is unitarily equivalent to a multiplication operator on L2( Q, R) and

1. PnPmdQ = 6nm ·

PROBLEM: Find (analytic and geometric) properties of the spectral measure Q in terms of J, that is, in terms of the recursion coefficients {an} and {6n}, and vice versa.

Discrete Spectrum of the Jacobi Matrix.

EXAMPLE: If an = 1 + ~ & 6n = 0 then

supp(Q) = [-2,2] U {oo # of mass-points}.

If an = 1 + ~ & 6n = 0 then

supp(Q) = [-2,2] U {depending on A there is a finite or infinite # of mass-points}.

If an = 1 + ~ & 6n = 0 then

supp(Q) = [-2,2] U {finite # of mass-points}.

In the following two propositions J(Che6yshev) means the Jacobi matrix with 0 on the main diagonal and with ~ on the subdiagonal and the superdiagonal.


THEOREM. (Mate & Nevai [32], Bargmann, Geronimo & Case, Nikishin et al.) Let k ~ 1, and let Tk denote the Chebyshev polynomial of degree k. If

00 l+k k L . L 1[[Tj:(J(a» - Tj:(J(Chebyshev»]]jd < 36(n + 2k) l=nJ=I-k

for every n ~ no, then

supp(a) = [-1,1] U {finite # of mass-points}.

This theorem is a generalization of the Geronimo-Case Theorem in Section 4. The above inequality describes a quantitative perturbation which is stronger than the trace class!

EXAMPLE: Even though the Jacobi matrix associated with the Jacobi polynomials does not necessarily satisfy the above condition, anything "just" a little bit better than that already will.

It should be possible to generalize these results to asymptotically periodic Jacobi matrices which are intimately related to orthogonal polynomials on several intervals. The role of Chebyshev polynomials is taken over by the orthogonal polynomials of the corresponding equilibrium measures (cf. works of Geronimo & Van Assche, Mate & Nevai, Saff & Totik, Stahl & Totik, and Gonchar & Rahmanov).

EXAMPLE: a2n = 1880 + ~,a2n+1 = 1895 - ,t., and bn = O. In view of the fast rate of convergence of the recursion coefficients, there must be a finite number of mass-points only! Prove it!

All this is intimately related to continued fractions and chain sequences!

Spectral Density of the Jacobi Matrix.

PROBLEM: When is a' E Szego, that is, log a' (cos 0) ELI?

THEOREM. (Szego & Shohat [54]) Assume supp(a) = [-1,1]. Then

log a' (cos t) E Ll ([0,11"]) ¢:::::}

00 00 00 00

~)2an - 1] < 00 & ~)2an - 1]2 < 00 & L bn < 00 & L b~ < 00.

n=l n=l n=O n=O

This can be proved by transplanting it to the unit circle, and then by using Szego's theorem. There are hardly any exciting results when supp(a) = [-1,1] is replaced by an = 1/2 + 0(1) & bn = 0(1), that is, by a compact perturbation of J(Chebyshev).

92 P. Nevai

CONJECTURE. For every Jacobi matrix J

00

L [l2an - 11 + Ibn I] < 00 ==> log a'(cos 0) E L1· n=O

THEOREM. (Nevai [41], Geronimo & Van Assche [21]) For every Jacobi matrix J

00

L log n [l2an - 11 + Ibn 11 < 00 ==> log a' (cos 0) ELI. n=1

THEOREM. (Dombrowski & Nevai [10]) For every Jacobi matrix J

2an '\. 1 & bn = 0 ==> a'(z);::: const ~ (-1 ~ z ~ 1).

The paper [10] contains a number of related results on the connection between Jacobi matrices and their spectral measures. Among them is one of my favorite results which is the following 7race Fonnula.

THEOREM. (Dombrowski & Mate & Nevai [10]) If the recursion coefficients {an(a)} and {bn(a)} satisfy

lim an(a) = -21 & lim bn(a) = 0 n ...... oo n-+oo

and 00

L lan+l(a) - an(a)1 + Ibn+l(a) - bn(a)1 < 00,

n=O then

00

L {[a!+I(a) - a!(a)] p!(a,z) n=O

holds uniformly on all compact sets in (-1,1). In addition, the measure a is absolutely continuous in the open interval (-1,1), a'(z) > 0 for all z E (-1,1), and a' is continuous in (-1,1).

OUTLINE OF THE PROOF: Step 1. One starts with the Dombrowski formula [9] (cf. [48] as well):

n

L {[a~+I(a) - a~(a)] pHa, z) 1:=0

+ a1:(a) [b1:(a) - b1:-1(a)]p1:-1(a, z)p1:(a,z)}

= a!+l(a) [p!(a, z) - z - b'«) Pn(a, Z)Pn+l(a, z) + p!+I(a, z)] . an +1 a


This is proved by induction. Recently, B. Osilenker [49] found a trilinear extension whose implications are still unclear. I have great expectations from Osilenker's formula. Step 2. One proves that the orthogonal polynomials are uniformly bounded on all compact sets in (-1, 1). This is shown by using a discrete Gronwall-type inequality in conjunction with Dombrowski's formula. Therefore,

00

E {[a~+l (0:) - a~(o:)] p~(o:, x) n=O

where c) is a continuous function in (-1, 1). Step 3. One proves that c)( x) > 0 for x E (-1, 1). This is shown by another discrete Gronwall-type estimate. Step 4. Computation of c). This is done by proving that the right hand side in Dombrowski's formula converges weakly whenever limn_oo an(o:) = ~ and limn _ oo bn(o:) = O. The latter follows from that fact that if J is a compact perturbation of the constant Jacobi matrix J(Chebyshev), then for every fixed polynomial P, the matrix P(J) is banded, and it is a compact perturbation of P(J(Chebyshev)). Step 5. One proves absolute continuity of 0: in (-1,1). This follows from the uniform convergence of the 1race Formula to a positive function.

Gronwall-type inequalities are a very useful and convenient tool when using methods related to successive iterations. For instance, a Gronwalltype inequality is the following

PROPOSITION.

I/(x)l ~ Ig(x)1 + l X I/(t)h(t)ldt ("Ix ~ 0) ~ I/(x)1 ~ max Ig(s)le!o" Ih(t)ldt ("Ix ~ 0).

0:53:5X

A good source for it's difference analogues is Atkinson's book [3] on "Discrete and Continuous Boundary Problems."

OPEN PROBLEM: Extend the 1race Formula to orthogonal polynomials on several intervals and to asymptotically periodic Jacobi matrices which are their closely related counterparts.

EXAMPLE: In what follows is the graphl of the approximation of a sieved 4-para.meter Pollaczek weight by the 104 th partial sum of the 1race Formula. Given k E N+ and a, b, c, A E R, we define the k-sieved 4-parameter

lThis graph was produced by filtering and smoothing the output of the original Light.peetffM Pa.cal version of my The Orthogonal Polynomial Machine™ through P]CTEX.

94 P. Nevai

Pollaczek polynomials as the characteristic polynomials associated with the Jacobi matrix J = J(II:, a, 6, c,~) where

n= 1,2, ....

Here n

An = k+c+2~ 2n

& Bn = T + 2a + 2c + 2~

& Cn = -26

for n = 0 (mod 11:) and

n & Dn = k + C + 2~ - 1

An = 1 & Bn = 2 & Cn = 0 & Dn = 1

otherwise. Naturally, the Pollaczek polynomials are orthogonal with respect to a positive measure if and only if all parameters are chosen in such a way that all sub diagonals in J are positive. For the 4-parameter Pollaczek polynomials the reader is referred to Chihara's book [6, p. 185], whereas for the sieving process itself to the works of AI-Salam, Askey, Ismail and their coauthors such as [1, 5]. Here we consider the 3-sieved 4-parameter Pollaczek weight with a = 0, 6 = -1, C = 0, and ~ = t.

0.20

0.15

0.10

0.05

0.00 +-L.---+-O::::;"'--r------4---====:t -1.0 -0.5 0.0 0.5 1.0

Since the recurrence coefficients associated with the above orthogonal polynomials are not of bounded variation, the Trace Formula for this case has not been proved yet. Therefore this example should be viewed as explorational mathematics. It is not very difficult to prove though that the discrete spectrum of the corresponding Jacobi matrix is an infinite set.2

8. GERONIMO & VAN ASSCHE VS. TURAN

ASYMPTOTICALLY PERIODIC JACOBI MATRICES: Given two periodic sequences {a~O)}~=o and {6~O)}~=o with period N ~ 1, the Jacobi matrix J

2 A tivial but useful observation is as follows: if the integral of the density of a probability measure is less than 1 then there must be a singular component.


is called asymptotically N-periodic (J EANP) if

lim [Ian - a~O) I + Ibn - b~O) I] = o. n_oo

Periodic and asymptotically periodic Jacobi matrices have been investigated by Aptekarev, Geronimus, Grosjean, and by Geronimo and Van Assche. The spectrum of such Jacobi matrices can be determined by perturbation methods.

THEOREM. (Geronimo & Van Assche [21, 23]) Let J be an asymptotically N -periodic Jacobi matrix. If

00

L: [Ian - a~O)1 + Ibn - b~O)I] < 00

n=O

then the spectral measure a is absolutely continuous on the essential spectrum of J. In addition, a' is also continuous and positive inside the essential spectrum of J.

OPEN PROBLEM: It should be possible to replace convergence of above series by

00

L: InQ(J(a)) - Q(J({a~O),b~O)}))]]ikl < 00

i,k=O

for an arbitrary polynomial Q (though one mayor may not be able to determine what happens at a finite number of points depending on Q).

TURAN DETERMINANTS: They are defined by Dn = p~ - Pn+1Pn-1, and they are a useful tool for recovering measures which (essentially) live on one interval.

THEOREM. (Mate-Nevai-Totik [37]) If supp(a) = [-1,1] and a' > 0 almost everywhere in (-1,1), then

lim 11 IDn(a, x)a'(x) - ~~I dx = O. n-oo -1 ~

THEOREM. (Mate-Nevai-Totik [34]) If limn_ oo an = !, limn_ oo bn = 0 and

00

L: lan+1 - ani + Ibn+ 1 - bnl < 00,

n=O

then

uniformly on all compact sets in (-1,1).

These results were used by Askey, Ismail and their coauthors for finding weight functions of Pollaczek-type and related polynomials.

96 P. Nevai

In what follows is the graph of an approximation of the "truncated" or ''finite'' Hermite weight function where a is given by da(z) = J~l e-t2 dt on the interval [-1,1], by Thran Determinants.3

1.2

0.2

-1. -0.5 0.5 1.

Here the solid black line is the weight function a' whereas the gray line represents the approximation by

Note how perfect the fit is on closed subintervals of (-1,1) after only 6 iterations of the recurrence formula. Since the Jacobi matrix of the "truncated" Hermite weight function cannot be evaluated explicitly, one needs to use methods developed in [39] to obtain a recurrence formula for the recursion coefficients. But that is another story ...

Unfortunately, Thran Determinants are of no help for orthogonal polynomials on several intervals! Enter Geronimo £3 Van Assche!

SHIFTED TURAN, THAT IS, GERONIMO & VAN ASSCHE DETERMINANTS4

[23]: ( ) an+l

Dn N = PnPn-N+l - Pn+lPn-N· an-N+l

As shown by the following theorem, these are great for finding spectral measures of asymptotically periodic Jacobi matrices.

3This PostScript™ graph was prepared on the Mathematica™ version of my The Orthogonal Polynomial Machine™. 4For asymptotically N-periodic Jacobi matrices the factor an+t!an-N+l tends to 1 as n -+ 00, and, therefore, it is not 80 important. As Walter Van A88che says: "It just made (our) proofs easier, that's all."


THEOREM. (Geronimo & Van Assche [23]) If J is asymptotically N-periodic and of bounded N -variation, that is,

00

l: [Ian - an+NI + Ibn - bn+NI] < 00

n=O

then for every j = 0,1,2, ... ,N - 1,

lim DnN+j(N,z) = Cj ~«z» n-oo CI:' Z

uniformly on all compact sets inside the essential spectrum of J\ {some

exceptional points}. Here Cj = l/a}~l' j = 0,1,2, ... ,N -1, and V is a function which depends on the spectral measure of the periodic Jacobi matrix Jo = J( {a~O), b~O)}), and it can be computed explicitly.

EXAMPLE:5 First we fix T(z) = CT4(Z) where T4(Z) is the 4th degree Chebyshev polynomial ofthe first kind and c ~ 1. Then, taking the four intervals E = T-l([-l, 1]), we consider the orthogonal polynomials Pn(w,z) on E associated with the weight function W = w(c) defined by

c w(z) = 2U3(z) ' z E E,

which are obtained from the Legendre polynomials by a polynomial transformation [22, Sect. VI]. Here Us denotes the second kind Chebysev polynomial of degree 3. These polynomials satisfy the recurrence formula

zPn(w, z) = an+1Pn+l(W,Z) + anPn-l(W,Z)

with Po = 1, P-l = 0 and

1 1 2 a~O) qn-l(C) 2 1 2 a4n+2 = 2 & a4n+S = 2 & a4n = 2c qn(c) & a4n+1 = 2 - a4n,

where qn(z) are the orthonormal Legendre polynomials with weight function wo(z) = 1/2 on [-1,1]. The following is a graph of the approximation to the weight w(t) using shifted Thran, that is, Geronimo & Van Assche Determinants.

2.5

2.0

1.5

1.0

0.5

8

0.0 -f-,....;..-r-+-r-r-.,....,..L.,--r-I~ ...... ,...,.-'r--..-i-...,... -1.0 -0.5 0.0 0.5 1.0

51 am grateful to Walter Van Assche for working out this example for me at a very short notice, and for introducing me to the wonderful world of P}CIEX.

98 P. Nevai

Here the original weight w = w(~) is indicated by "0" whereas the solid line shows the Geronimo & Van Assche approximation to it by

which uses the orthogonal polynomials Pn up to degree 10 at most. The graph of the approximated weight function in the limiting case c = 1 is given in [23].

REMARK. In general, orthogonal polynomials on several intervals are not equivalent to asymptotically periodic Jacobi matrices, independently of the smoothness properties of the corresponding measure. It is possible to have orthogonal polynomials on several intervals for which no periodicity can be found in the recurrence coefficients. A necessary and sufficient condition for the intervals to lead to asymptotically periodic recurrence coefficients is that the equilibrium measure has equal mass on each of the intervals (cf. Aptekarev [2]).

It is quite reasonable to expect that the following extension of Rahmanov's and Aptekarev's theorems holds.

CONJECTURE. (Van Assche) If the support ofthe measure is a finite union of intervals which are obtained as the inverses of an interval through a polynomial mapping, and if the derivative of the measure is positive almost everywhere on the intervals then the corresponding Jacobi matrix is asymptotically periodic.

Somewhat unexpectedly, if J is allowed to have complex entries then the situation becomes much more complicated. Nevertheless, I believe that the following conjecture is true.

CONJECTURE. H J is complex valued and it is asymptotically N-periodic and of bounded N -variation then limn_co DnN + j (N, z) exists uniformly on all compact sets inside the essential spectrum of J\{some exceptional points}.

Unfortunately, all attempts to deal with higher order equations have so far been unsuccessful.6 Therefore, I propose the following

OPEN PROBLEM: Investigate asymptotic behavior of solutions of higher order linear and possibly nonhomogeneous difference equations with asymptotically N-periodic coefficients of bounded N-variation.

6During my visit to lovely WITS University in the Summer of 1990, Doron Lubinsky and I embarked on a project examining solutions of fairly general operator equations. We hope to complete our initial work "Su},exponential growth of solutions of difference equations" soon. (This note was added at proof reading on September 12, 1990.)


9. GENERALIZED POLYNOMIALS

A Nikol'skil-type inequality is an inequality between two (equivalent) "norms" for polynomials of a fixed degree. For instance, given an integer m> 0, real numbers r i > -1, i = 1,2, ... ,m, and -1:5 t1 :5 t 2 :5 ... :5 tm :5 1 we can define a "generalized Jacobi weight function" w associated with these parameters by

m

w(z) = II Iz - tl:lrj. (9.1) 1:=1

Then it is reasonably well known that given 0 < q :5 p :5 00, there is a positive constant C(w,p,q) such that the Nikol'skil-type inequality

{ 1 }1/P { 1 }1/9

11IRn(t)IPw(t)dt :5 C(w,p,q)n~-* 11IRn(t)19w(t)dt

(9.2) holds for all polynomials Rn of degree at most n (cf. [39]). Applying this inequality with p = 00 and q = 2, and using the identity

[tp~(a, X)]-1 = min IRn~ )1 2 J1 IRn(tWda(t), 1:=0 R .. EP.. x -1

z E C, (9.3)

one immediately obtains (C, 1) bounds for the corresponding orthogonal polynomials.

Both the Cotes numbers from Gaussian quadratures and the Christoffel functions which are their continuous extensions are closely related to the optimal constants in Nikol'skil-type inequalities. As a matter of fact, the former can be thought of as "Nikol'skil functions" in Loo to L2 type Nikol'skil inequalities. Thus the relationship between orthogonal polynomials and Nikol'skil constants becomes imminent. The question is how Nikol'skil-type inequalities can be used to prove new results on orthogonal polynomials. In particular, one would like to be able to investigate orthogonal polynomials associated with variable weight functions.

If one could find the precise relationship between the constant C( w, p, q) in (9.2), the parameters p, q and the weight w, then such inequalities could be extended to the case when neither m nor tj are fixed in (9.1). Variable weight functions and measures are becoming a useful tool in solving numerous problems not only in orthogonal polynomials but also in numerical analysis and related areas.

This was the underlying motive for introducing

GENERALIZED POLYNOMIALS: If

m

I(x) = II Iz - tklrk , 1:=1

100 P. Nevai

where tic E C and ric ~ 0, then f is called (the absolute value of) a generalized complex algebraic polynomial and r is its generalized degree.

Generalized polynomials (both algebraic and trigonometric) were investigated by T. Erdelyi and his collaborators. One result in this direction is the following generalization of (9.2).

THEOREM. (Erdelyi & Mate & Nevai [14]) Let X be a nonnegative, nondecreasing function defined in [0,00) such that ~ is nonincreasing in [0,00). Then there is an absolute constant C such that for all 0 < q ::5 p ::5 00

{ 1 }1/P { 1 }1/1J 11 x(J(t»)Pdt ::5 [C(1 + qr)]t-~ 11 x(J(t))9dt (9.4)

holds for all generalized complex algebraic polynomials f of degree at most r. IfX(x) = x then ;: is a suitable choice for C.

PROBLEM: What is the optimal value of the constant C above?

An analogous inequality holds for generalized trigonometric polynomials, except that [C(1 + qr)]t-~ is replaced by [C(1 + qr)]t-.;. In the trigonometric case the constant C can be chosen 4~ when X(x) = x. This trigonometric analogue and (9.3) yield the following

COROLLARY. Let w be a weight function in [-1,1] given by (9.1) where tic E C and ric ~ O. Let r = E~=1 rl:' Then

n

~ w(x) I: Iplc(W, xW ::5 4e1["(3 + 4n + 2r) (9.5) 1c=0

uniformly for n = 1,2, ... , and x E [-1,1].

This estimate is sharp. As a matter of fact, the following theorem holds too, though its proof is way more involved than that of (9.5).

THEOREM. (Erdelyi & Nevai [15]) Let W be a weight function in [-1,1] given by (9.1) where tic E C and ric ~ O. Let r = E~=1 ric and M = 1 + 1~r' and let WM be defined by

-1::5 x ::5 1.

Then there exist two positive absolute constants C1 and C2 such that

n

CHr < W (x) "'p2(w x) < CHr 1 - M L...J Ic , - 2 1:=0

uniformly for n = 1,2, ... and x E [-1,1].


OUTLINE OF THE PROOF OF (9.4): First (9.4) is proved for p = 00 by applying T. Erdelyi's Remez-type inequality

m( {z : z E [-1,1] and xf(f(z» ~ exp (-qfyS) IIx(f)II~}) ~ C3S

where 0 < s < 2 (cr. [13]) with s = (1 + qf)-2. Here m denotes the Lebesgue measure, C3 is an absolute constant and I is an arbitrary generalized complex algebraic polynomial of (generalized) degree at most f. By this inequality and by integrating the characteristic function of the subset I ~ [-1,1] where e xf(f(z» ~ IIx(f)II~ one obtains

IIx(f)II~ $ ~3 (1 + qf)21 XIJ(f(t»dt $ ~3 (1 + qf)2I1x(f)II;

which proves (9.4) for p = 00. For p < 00 inequality (9.4) is proved by writing IIx(f)II~ = IIxP (f)lh = IIxP- f+f (f)lh $ IIx(f)II~-fllx(f)II: and then by using the previously proved case of p = 00 to estimate IIx(f)II~f .•

Similarly to this theorem, a number of other classical polynomial inequalities have been transplanted by T. Erdelyi and his collaborators to generalized polynomials. In particular, the inequalities of Bernstein, Chebyshev, Markov, Nikol'skil, Remez, and Schur in weighted Lp and Orlicz spaces remain valid for generalized polynomials for every p E (0,00]. Unfortunately, the best constants are still unknown. In addition, one can obtain estimates of zeros, Cotes numbers and related quantities for generalized Jacobi polynomials and for other widespread classes of orthogonal polynomials. The latter have applications to numerical integration and numerical solution of integral equations. For instance, we have the following

THEOREM. (Erdelyi [12]) Let 0 < l < 1 and let w be a weight function in [-1, 1]. Assume w satisfies

m( {t : t E [0,11"] and I log- w(cost)1 ~ ..\}) $ K..\1-t, ..\ >0,

with a suitable constant K where m denotes the Lebesgue measure. Let {cos tkn}~=1 denote the zeros of the corresponding nth degree orthogonal polynomial in decreasing order. In addition, let tOn = 0 and tn+t,n = 11". Then

(n=1,2, ... , and k=1,2, ... ,n)

where C(l,K) is a constant depending on l and K only.

Iflog w(cost) E L1([O,1I"]), that is, if w satisfies the Szego condition, then this theorem can be applied with l = ! (ef. [39]).

Finally, it may be a shocking news to many people working in approximation theory that the following is still an

102 P. Nevai

OPEN PROBLEM: Given p E (0,00), find the optimal constant C(n,p) in the Markov inequality

for all algebraic polynomials Qn of degree at most n.

The existence of such a constant C( n, p) with sUPn> 1 C( n, p) = C(p) < 00 was proved in [26] for 1 $ p $ 00 and in [40] for 0 <:: p < 1. I believe that the optimal constant is only known for very few values of p such as p = 2 (cf. [11]) and p = 00. As a matter of fact, for p = 2, we have

(f5 1 C(I,2) = va> C(2,2) = V"4 > C(3,2) > ... > C( n, 2) ! ;:

for n - 00 (cf. [25]).

10. EPILOGUE

I stop here, and the readers are asked to continue either by studying some of the works listed in the references or by contacting me so I could give (possibly very opinionated) advice as to what else is happening on this side of the general theory orthogonal polynomials.

It is my pleasure to thank Tom Erdelyi, Ed Saff, Walter Van Assche, and the referee for reading the manuscript and making numerous suggestions regarding the organization and presentation of the material covered in this survey.

REFERENCES

1. W. A. AI-Salam, W. R. Allaway and R. Askey, Sieved u/tralh,perical orthogonal polynomial., Trans. Amer. Math. Soc. 284 (1984), 39-55.

2. A. I. Aptelat.rev, A.ymptotic propertie. of polynomial. orthogonal on a .y.tem of contoure., and periodic motion. of Toda lattice., Math. USSR-Sb. 53 (1986), 233-260.

3. F. V. Atkinson, "Discrete and Continuous Boundary Problems," Academic Press, Inc., Boston, 1964.

4. G. Baxter, A convergence equivalence related to orthogonal polynomial. on the unit circle, Trans. Amer. Math. Soc. 99 (1961),471-487.

5. J. A. Charris and M. E. H. Ismail, On .ieved orthogonal polynomial., V, Sieved PollaczeA: polynomial., SIAM J. Math. ADal. 18 (1987), 1177-1218.

6. T. S. Chihara, "An Introduction to Orthogonal Polynomials," Gordon and Breach, New York-London-Paris, 1978.

7. T. S. Chihara and P. Nevai, Orthogonal polynomial. and mea,ure. with finitely many point mallie" J. Approx. Theory 35 (1982), 370-380.

8. J. Dombrowski, Spectral propertie, of pha,e operator" J. Math. Phys. 15 (1974), 576-577.

9. J. Dombrowski, Tridiagonal matri:!: repre,entation, of cyclic ,elf-adjoint operator" II, Pacific J. Math 120 (1985),47-53.

Orthogonal Polynomials, Recurrences, Ja.c::ohi Matrices and Measures 103

10. J. Dombrowski and P. Nevai, Ol1ho,o,,1II pol,,,omial., mea.1Ire. a"d recurre"ce relatio"., SIAM J. Math. Anal. 11 (1986), 752-759.

11. P. DOrfler, New i"elJualitie. oJ Markov t,pe, SIAM J. Math. Anal. 18 (1981), 490-494.

12. T. Erdelyi, Niko! dia-t,pe i"eIJUIIlitie. Jor ,e"eralized pol,,,omial. a"d zero. oJ ol1ho,o,,1II pol'"omial., preprint, 1989.

13. T. Erdelyi, Remez-t,pe i"eIJ1Ialitie. 0" the .ize oJ ,e"eralized pol,,,omilll., preprint, 1989.

14. T. ErdeIyi, A. Mate and P. Nevai, I"egualitie. Jor ,e"eralized "o"-,,e,ative pol,"omial., preprint, 1990.

15. T. Erdelyi and P. Nevai, Ge"eralized Jacobi wei,ht., Chri.toDel J .. ctio". a"d zero. oJ ol1ho,o"al pol,,,omial., in preparation, 1990.

16. P. Erd& and P. Torh, 0" i"terpolatio", III, Annals of Math. 41 (1940), 510-555. 17. G. Freud, ''Orthogonal Polynomials," Pergamon PleBS, Oxford, 1971. 19. I. M. Gelfand and B. M. Levitan, 0" the determi"atio" oJ a diDere"tial elJUlltio"

/rom it • • pectral Ju"ctio", in Russian, kv. Alcad. Nauk. SSSR, Ser. Mat. 15 (1951), 309-360.

19. J. S. Geronimo, 0" the .pectra oJ i"fi"ite-dime".io"al Jacobi matrice., J. Approx. Theory 53 (1988), 251-265.

20. J. S. Geronimo and K. N. Case, Scatteri", theoTJ a"d pol,,,omial. ol1ho,o"al 0" the relll li"e, Tr&ll8. Amer. Math. Soc. 258 (1980), 467-494.

21. J. S. Geronimo and W. Van AlIIICbe, OrtAo,o"al pol,,,omial. with a.,mpttiticall, periodic recurre"ce coet1icie"t., J. Approx. Theory 46 (1986), 251-283.

22. J. S. Geronimo and W. Van A88Che, Ol1ho,o"al pol,,,omial. 0" ,everal i"tertlal, via a pol,,,omial mappi"" Tr&ll8. Amer. Math. Soc. 308 (1988), 559-581.

23. J. S. Geronimo and W. Van AlIIICbe, Approzimati", the wei,ht Ju"ctio" Jor ol1ho,o"al pol,.lomial. 0" ,everal i"tertlal., J. Approx. Theory 65 (1991),341-371.

24. L. Ya. Geronimus, "Orthogonal Polynomials," Consultants Bureau, New York, 1971.

25. P. Goetgheluck, 0" the Markov i"eIJUIIlit, i" V-.pace., J. Approx. Theory 62 (1990), 197-205.

26. E. Hille, G. Szegl) and J. D. TlIIIl8I"kin, 0" .ome ,e"eralizatio" oJ a theorem oJ A. MarkoD, Duke Math. J. 3 (1937), 729-739.

27. X. Li and E. B. Saff, 0" Nevai', characterizatio" oJ mea.ure. with pOlitive derivative, J. Approx. Theory 63 (1990),191-197.

28. X. Li, E. B. Saff and Z. Sha, Behavior oJ be,t Lp pol,,,omialapprozima,,t. 0" the u"it i"tertlala"d 0" the .. it circle, J. Approx. Theory 63 (1990),170-190.

29. D. S. Lubinsky, A '1Irtle, oJ ,e"eral ol1ho,o"al pol,,,omial, Jor wei,ht, 0" fi"ite a"d i"fi"ite i"tertlal" Acta Appl. Math. 10 (1987); 237-296.

30. D. S. Lubinsky, "Strong Asymptotics for Extremal ErrOD and Polynomials AIIBOciated with Erd&-type Weights," Pitman Research Notes in Mathematics Series, Longman Scientific & Technical, Vol. 202, Harlow, United Kingdom, 1988.

31. AI. Magnus, Toeplitz matriz tech"ilJue. a"d co"verge"ce oJ complez wei,ht PaU approzima"t., J. Comput. Appl. Math. 19 (1987), 23-38.

32. A. Mate and P. Nevai, Ei,e"vaI1le. oJ fi"ite band-width Hilbert .pace operator. and their applicatio", to ol1ho,onal pol,nomial., Can. J. Math. 41 (1989), 106-122.

33. A. Mate, P. Nevai, and V. Totik, A"mptotic. Jor the ratio oJ leading coet1icie"t. oJ ol1ho"ormal pol,,,omial, 0" the .. it circle, Constr. Approx. 1 (1985), 63-69.

34. A. Mate, P. Nevai, and V. Totik, A"mptotic. Jor ol1hogonal pol,nomial, defined b, a recurrence relatio", Constr. Approx. 1 (1985), 231-248,

35. A. Mate, P. Nevai and V. Totik, NecellaTJ condition, Jor mean convergence oJ F01lrier .erie, i" ol1hogo"al pol,nomial" J. Approx. Theory 46 (1986), 314-322.

36. A. Mate, P. Nevai, and V. Totik, Strong and wed convergence oJ ol1hogonal pol,"omial., Amer. J. Math. 109 (1987), 239-281.

104 P. Nevai

37. A. Mate, P. Nevai and V. Totik, Ezten.ion. of Szego'. theo,", of orthogonal polynomial., II & III, Constr. Approx. 3 (1987),51-72 & 73-96.

38. A. Mate, P. Nevai and W. Van Assche, The .upport of meaaure. auociated with orthogonal polynomial. and the .pectra of the related .elf-adjoint operator., Rocky Mountain J. Math. 21 (1991 501-527).

39. P. Nevai, "Orthogonal Polynomials," Mem. Amer. Math. Soc., Providence, Rhode Island, 1979.

40. P. Nevai, Bern.tein'. InelJuality in LP for 0 < p < 1, J. Approx. Theory 27 (1979), 239-243.

41. P. Nevai, Orthogonal polynomial. defined 6y a recurrence relation, Trans. Amer. Math. Soc. 250 (1979),369-384.

42. P. Nevai, Giza Freud, orthogonal polynomial. and Chri.toDel function •• A ca.e .tudy, J. Approx. Theory 48 (1986),3-167.

43. P. Nevai, Characterization of mea.ure. a •• ociated with orthogonal polynomial. on the unit circle, Rocky Mountain J. Math. 19 (1989), 293-302.

44. P. Nevai, Re.earch problem. in orthogonal polynomial., in "Approximation Theory VI," Vol. II, C.K. Chui, L. L. Schumaker and J. D. Wlll'd, eds., Academic Press, Inc., Boston, 1989, pp. 449-489.

45. P. Nevai, "Orthogonal Polynomials: Theory and Practice," NATO AS! Series C: Mathematical and Physical Sciences, Vol. 294, Kluwer Academic Publishers, Dordrecht-Boston-London, 1990.

46. P. Nevai, Wedly convergent .elJuence. of function. and orthogonal polynomial., J. Approx. Theory 65 (1991), 322-340.

47. P. Nevai and Y. Xu, Mean convergence of Hermite interpolation, in preparation, 1990.

48. B. P. Osilenker, On .ummation of polynomial Fourier .erie. e:l:pan.ion. of function. oj the cla .. e. L!:<.>:) (p ~ 1), Soviet Math. Dokl. 13 (1972), 140-142.

49. B. P. Osilenker, The repre.entation oj the trilinear kernel in general orthogonal polynomial. and .ome application., J. Approx. Theory 67 (1991), 93-114.

50. K. Pan and E. B. Saff, Some characterization theorem. Jor mea.ure. a .. ociated with orthogonal polynomial. on the unit circle, in "Approximation Theory and Functional Analysis," C. K. Chui, ed., Academic Press, Boston, 1991.

51. E. A. Rahmanov, On the a.ymptotic. of the ratio oj orthogonal polynomial., II, Math. USSR-Sb. 46 (1983), 105-117.

52. E. A. Rahmanov, On the a.ymptotic. of polynomial. orthogonal on the unit circle with weight. not lI41illfying Szeg;;'11 condition, Math. USSR-Sb. 58 (1987), 149-167.

53. E. B. Saff and V. Totik, "Logarithmic Potentials with External Fields," in preparation.

54. Shohat, "Theorie Generale des Polinomes Orthogonaux de Tchebichef," Memorial des Sciences Mathematiques, Vol. 66, Paris, 1934, pp. 1-69.

55. H. Stahl and V. Totik, "General Orthogonal Polynomials," manuscript, 1989. 56. M. H. Stone, "Linear Transformations in Hilbert Space and Their Applications to

Analysis," Colloquium Publications, Vol. 15, Amer. Math. Soc., Providence, Rhode Island, 1932.

57. G. Szego, "Orthogonal Polynomials," Colloquium Publications, Vol. 23, fourth edition, Amer. Math. Soc., Providence, Rhode Island, 1975.

58. V. Totik, Orthogonal polynomial. with ratio a.ymptoticlI, Proc. Amer. Math. Soc. (to appear).

59. W. Van Assche, "Asymptotics for Orthogonal Polynomials," Lecture Notes in Mathematics, Vol. 1265, Springer-Verlag, Berlin-New York-London, 1987.

P. O. Box 3341, Columbus, Ohio 43210-0341, U.S.A. E-mail: nevaiOmp •. ohio-lltate.edu and nevaiOohlltpy.6itnetj Fax: 1-614-459-5615

Szego Type Asymptotics for Minimal Blaschke Products

A.L. Levin* E.B. SafF

ABSTRACT Let p. be a positive, finite Borel measure on [0,2l1'). For o < r < 1, 0 < p < 00, let

where the infimum is taken over all Blaschke products of order n having zeros in Izl < 1. Let B: denote a minimal Blaschke product and let G(p.') denote the geometric mean of the derivative of the absolutely continuous part of p.. In the first part of the paper we present a self-contained proof of a result due to Parfenov; namely En,p '" r nG(p.')l/P as n -+ 00. In the second part we describe the extension of the classical SzeglS function D(z) and prove that B:(z) '" z" {G(p.')1/PID(z)2/p} as n -+ 00, uniformly on compact subsets of the annulus r < Izl < I/r. Some generalizations and applications are also discussed.

1 Introduction

Let Bn denote a monic Blaschke product of order n with zeros in Izi < 1 :

n

II z - nk Bn(z) = 1 '

k=1 - nk Z Inkl < 1, k = 1, ... ,n.

Let J.t be an arbitrary positive, finite Borel measure on [0,211') whose support contains infinitely many points. For 0 < r < 1, 0 < p < 00, define

(1.1)

A standard argument shows that the infimum in (1.1) is attained for some

-Research was conducted while visiting the Institute for Constructive Mathematics, Department of Mathematics, University of South Florida.

tResearch supported in part by the National Science Foundation under grant DMS-881-4026.


106 A.L. Levin, E.B. Saff

B~, but the question of uniqueness of this minimal Blaschke product remains open. In the sequel B~ will denote any minimal Blaschke product of order n, that is

(1.2)

Our aim is to describe the asymptotic behavior (as n - 00) of En,p and B~(z). Since l(z-ak)/(I-akz)1 represents the hyperbolic distance between z and ak, the results to be presented may be viewed as the extension to the non-Euclidean setting of the classical strong Szego theory. So let us first recall some basic facts of this theory.

Let

p> 0, (1.3)

where the infimum is taken over all monic algebraic polynomials Pn(z) = zn + ... of degree n. This infimum is attained for the unique monic polynomial which we denote by <f'n,p(z).

Given any f E LdO,211"], f ~ ° a.e., define its geometric mean GU) by

GU) := exp {2~ 127r log f( O)dO } . (1.4)

The case] log f = -00 is not excluded - we then set GU) = 0. Returning

to the measure Il, let Il = Ila + Il. be its canonical decomposition into the absolutely continuous and the singular parts (with respect to the Lebesgue measure dO). We denote by 1l'(O) the Radon-Nikodym derivative dlla/dO of Ila with respect to dO. Since, by the definition, Il' E LdO, 211"], we may consider G(Il'). If logll' E LdO; 211"] (or, equivalently, if G(Il') > 0) we say that Il satisfies the Szego condition. We then define the SzeglJ function of Il by

{ I 1211" eiB + z } D(dllj z) := exp --4 10gll'(O)-·-B -dO , 11" 0 e' - z

Izl > 1. (1.5)

This function has the following properties (cf. [Sz, p. 276], but notice that D(d"j z) in (1.5) and D(z) defined in [Sz, p. 277] are related by D(dllj z) = D(I/z)) :

(i) D(dJ.tj z) is analytic and non-vanishing in Izl > Ij (ii) D(dllj 00) = G(Il')1/2 j (iii) limp ..... l+ D(dlljpe i9 ) =: D(dJ.tjei9 ) exists for a.e. 0 in [0,211"] and

ID(dlljeiB )12 = 1l'(O) a.e. on [0,211"].

The following results (due to Szego, Kolmogorov and Krein) describe the behavior of fn,p and <f'n,2(Z).

Szeg<S Type Asymptotics for Minimal Blaschke Products 107

Theorelll 1.1 For every 0 1, (1.7) "-+00 )

with the limit being uniform for Izl 2: R > 1.

The proofs of these results can be found in [GS, Ch. III]. The main ingredient in the proof of Theorem 1.1 is the relation

inf {G(j)-1-21 12~ f(O)dl'(O)} = G(I"), (1.8) IEC[0,2~1, 1>0 7r 0

to which (1.6) is easily reduced. (This reduction is carried out for p = 2, but a similar argument applies to any p > 0.) The method of proof of Theorem 1.2 is a purely L2 argument. Yet, it can be modified to deal with any p > 1. This was done by Geronimus and more recently by X. Li and K. Pan:

Theorelll 1.3 ([G], [LP, Thm 2.2]) If I' satisfies the Szeglf condition, then forp> 1

with the limit being uniform for Izl 2: R > 1.

We return now to our Blaschke product setting. The following result (essentially due to o. Parfenov) is analogous to Theorem 1.1.

Theorelll 1.4 ([Pal, Thm 2]) For every 0 < p < 00, 0 < r < 1,

(1.10)

and, moreover,

(1.11)

In his proof, Parfenov also utilizes the relation (1.8). Below we give a direct proof of Theorem 1.4.

In applications the following version of Theorem 1.4 is sometimes more convenient. This version will enable us to consider the weighted Loo-norm.

Theorelll 1.5 Let w(O) E Lp[0,27r]. Assume w 2: 0 a.e. and for p = 00

assume additionally that w(O) is upper semi-continuous. For 0 < p ~ 00,

o < r < 1, define

(1.12)


(with obvious modification for p = 00). Then

lim r-nEn,p(w;r) = G(w) n-+oo

(1.13)

and, moreover,

r-nEn,p(w;r)~G(w), n=0,1,2,.... (1.14)

Remark 1. The infimum in (1.12) (like that in (1.1» is attained for some B:. The uniqueness of such a minimal monic Blaschke product is known only for p = oo,w == 1, in which case D.J. Newman showed that B:(z) = zn.

Remark 2. The existence of the limits in (1.10) and (1.13) is trivial; for if B: is minimal, then on choosing Bn+1(z) = zB:(z) we obviously obtain En+1,p :5 rEn,p. Hence the sequence r-n En,p is monotonically decreasing and thus converges.

We turn now to the description ofthe behavior of B:(z). It is easy to see (cf. [FM, Cor. 14] and the method of proof of Theorem 2 in [Sa]) that the zeros of any B: lie in the disk Izl < r (and, consequently, the poles of B: lie in Izl > l/r). Hence one may expect that the analogue of the limit relation (1.7) holds in the annulus r < Izl < l/r. But first we must answer the question: what is the "Szego function" for this setting? In a forthcoming paper we will describe this modification of the classical Szegc5 function as well as the asymptotics for B: (z) in full generality (log p' E Ll [0, 211"]). Here we confine ourselves to a simpler situation, namely logp' E C[0,211"].

Theorem. 1.6 Let f(6) be a positive, continuous 211"-periodic function. For o < r < 1 there exists a unique function D(J;r;z) =: D(z) (the Szeg6 function of f for the annulus r < Izi < l/r) that satisfies the following conditions: (i) D(z) is analytic and non-vanishing in r < Izl < l/r and satisfies there

D(z)D(l/z) = G(J).

In particular, ID(z)12 = G(J) for Izl = 1;

(ii) ID(z)1 is continuous in r :5 Izl :5 l/r and

ID(reill )12 = f(6), 0:5 6 :5 211";

(1.15)

(1.16)

(1.17)

(iii) log D(z) is single-valued in r < Izl < l/r and there is a branch of log D( z) that satisfies

21 f logD(z)ldzl is real. 11" J1z1=1

(1.18)

Szeg6 Type Asymptotics for Minimal Blaschke Products 109

Remark 3. Notice that (1.18) and (1.16) imply that the integral in (1.18) is equal to (1/2) log G(f). This corresponds to the normalization (ii) of the classical Szego function defined above.

Remark 4. An integral representation for D(fj rj z) (similar to (1.5» can also be given. We shall do this in a future paper.

Remark 5. Given ex> 0, we denote by D(z)o< the function exp(ex logD(z», where the branch oflogD(z) is chosen to satisfy (1.18).

Remark 6. For the general case, namely log f E Ll[0,211"], we define D(fj rj z) in the same fashion, except that property (ii) is replaced by

(ii) D(z) is an outer function in the Hardy space H2 of the annulus r < Izl < 1, and its limiting values on Izl = r satisfy (1.17).

We can now formulate our main result.

Theorem 1.7 (a) Let 1" be a positive, continuous 211"-periodic function and let D(dp.j rj z) denote the Szeg6 function of p.'«(J) for the annulus r < Izl < l/r. Given 0 < p < 00, let B~ denote a Blaschke product that realizes the infimum in (1.1). Then

lim z-n B~(z) = G(p.')l/P /{D(dp.jrjz)}2/P (1.19) n_oo

uniformly on compact subsets of the annulus r < Izl < l/r.

(b) Let w( (J) be a positive, continuous 211"-periodic function and let D(wj rj z) denote the Szeg6 function of w«(J) for the annulus r < Izl < l/r. Given 0 < p ::; 00, let B~ denote a Blaschke product that realizes the infimum in (1.12). Then

lim z-nB~(z) = G(w)/{D(wjrjz)}2 (1.20) n_oo

uniformly on compact subsets of the annulus r < Izl < l/r.

We remark that the method of the proof of Theorem 1.7 is a new one and it can be applied to the classical polynomial situation. This will enable us to extend Theorem 1.3 to any p > 0 :

Theorem 1.8 If I' satisfies the Szeg6 condition, then for p > 0

nl~ z-nIPn,p(z) = G(p.')l/p /{D(dp.jz)}2/P

locally uniformly for Izl > 1.

This paper is organized as follows. In Section 2 we prove some auxiliary results. In Section 3 we prove Theorems 1.4 and 1.5. The Szego function is discussed in Section 4. In Section 5, Theorems 1.7 and 1.8 are proven. Finally, in Section 6, we consider a more general situation and discuss the relation between En,p and the n-widths of certain classes of analytic functions.


2 Auxiliary Results

(a) The proof of the lower bounds (1.11) and (1.14) Let Bn(z) = I1~=1(z-ak)/(a-akz), lakl < 1, k = 1,2, ... ,n. Let

dJl = Jl'(O)dO+dJl3 be the canonical decomposition of 1'. Since I' is positive, so is 1'3. Hence

2~ fo21r IBn(rei/J)lPdJl ~ 2~ fo21r IBn(rei/J)lPJl'(O)dO. (2.1)

Assuming flog 1" > -00 (otherwise (1.11) is obvious) and using the Jensen inequality, we obtain

(2.2)

n 1 fo21r. . 1 L: 2" {log Ire,/J - ak I -log 11 - akre,/J l}dO + -log G(Jl'). k=1 ~ 0 P

Since lakl < 1, log 11- akzl is harmonic in Izl < 1, and so the mean value theorem yields

(2.3)

Furthermore,

> logr (2.4)

(see Lemma 14.4.1 in [HD. Inserting (2.3), (2.4) into (2.2) and using (2.1) we obtain the lower bound (1.11). Applying it with dJl = [w(O)]PdO, we also obtain the lower bound (1.14) for p < 00. The case p = 00 then follows by passing to the limit as p i 00.

(b) A nnihilating the singular part of a measure In [N, Lemma 4], P. Nevai introduced a simple but very effective device

to deal with a singular part of a measure. Following is a version of his result with one ingredient added.

Lemm.a 2.1 Let u be a positive, finite Borel measure on [0, 2~) that is singular with respect to dO. Then there is a sequence {hn } of continuous 2~-periodic functions such that

(2.5)

Szego Type Asymptotics for Minimal Blaschke Products

hr. (0) -+ 1 a. e. with respect to dO,

12,..

lim hn(O)du(O) = 0, n ..... oo 0

lim G(hn ) = l. n ..... oo

111

(2.6)

(2.7)

(2.8)

Proof. Let S C [0,211") be a Borel set such that Is dO = 0 and u(S) = u([O, 211"». Let {Sn} be a decreasing sequence of open subsets of [0,211") containing S such that

r dO <.!. and r du = 0, (2.9) 15,. n 1[0,2,..)\5"

00

and set S:= n Sn. For every n, let en be a compact set such that en C Sn n=l

and 1 1 du < -

[O,2,..)\C,. n (2.10)

(such en exists since a finite Borel measure is regular). Let hn be a continuous function from [0,211"] into [lin, 1] such that

1 hn(O) = - on en, hn(O) = 1 on [0,211"]\ Sn. (2.11)

n

Then hn -+ 1 on [0,211"] \ S. Hence (2.6) holds. Also, by (2.10) and (2.11),

° ~ 12,.. hn(O)du(O) ~ r du(O) +.!. r du(O)

1[o,2,..)\C,. n lc ..

Whence, (2.7) holds. Furthermore, by (2.9),

o ~ 12,.. log hn(O)dO = is .. log hn(O)dO ~ (-log n) is .. dO

> logn

n

Thus, lim r2,.. log hn(O)dO = 0 and (2.8) follows.

n ..... oo 10 If hn satisfies hn(O) = hn(211"), we are done. If not, redefine hn by setting

hn(O) = hn(O)tn(O), where

{ 1, 0 ~ 0 < 211" - lin

tn(O) := hn(O) + (211" - 0)[1 - hn(O)]n, 211" - ~ ~ 0 ~ 211".


By construction, h,,(211") = 1 and so h,,(O) = h,,(2·"-). Moreover, the h,,'s obviously satisfy (2.5) with lin replaced by 1/n2, as well as (2.6), (2.7) and~~. •

(c) A special class of weights In this paragraph we prove Theorem 1.5 for a special class of weights.

These will be used later to approximate arbitrary J.&'(O).

Example. Let (2.12)

where A > 0 and

lakl<r, k=l, ... ,M. (2.13)

For n ~ 1 define a Blaschke product B"M of order nM by

(2.14)

Since latl < r, we have

1 B M(Z) - __ Z(,,+l)M (1 + 0(1)) r ~ Izi ~ 1, (2.15)

" - BM(Z) ,

where 0(1) --+ 0 as n --+ 00, uniformly in r ~ Izi ~ 1. In particular, we obtain

M

IB"M(reill )I = Ar"M ;(0) (1 + 0(1)).

Since lak I < r, equations (2.3) and (2.4) imply that

1 1211" I reill - ak I -2 log 1 'II dO = logr 1r 0 - akre'

and therefore G(w) = ArM. Thus (2.16) may be rewritten as

(2.16)

(2.17)

uniformly for 0 E [O,21r]. In view of (1.14) and the existence of the limit (1.13), this implies Theorem 1.5 for w(O) of the form (2.12).

(d) Approximation of continuous weights The passage from w(O) of the form (2.12) to general continuous weights

is furnished by the following


Lemma 2.2 ([Pal, Lemma 1]) Let w(O) be a non-negative continuous 27r-periodic function. Then w(O) can be approximated on [0,27r] arbitrarily closely in the uniform norm, by functions of form (2.12), that is by ..\IBM(rei')I, where ..\ > 0 and BM has all its zeros in Izl < r.

Proof. Since w ~ 0 is continuous and 27r-periodic, it may be approximated uniformly on [0, 27r] by positive trigonometric polynomials. Such a polynomial may be written as Ig(ei9 )12 , where g(z) is a polynomial in z, whose zeros all lie in the unit disk Izl < 1. So it suffices to approximate lei9 - PI, IPI < 1, on [0,27r] by functions of the form (2.12) or, equiva.lently, to approximate Iz - ai, lal < r, on Izi = r by functions of the form ..\IBM(Z)I, where all zeros of BM lie in Izi < r.

For m = 1,2, ... let

(2.18)

Since lal < r, the product in (2.18) represents a Blaschke product with zeros in Izi < r. It is readily verified that for Izl = r we have

Since r < 1, ..\mIBm(z)l-+ Iz - al as m -+ 00, uniformly on Izl = r. This completes the proof. •

3 Proof of Theorems 1.4 and 1.5

We start with the proof of Theorem 1.4. Let dl' = 1"(O)dO + dl'., where 1" E L1 [0, 27r], and I'~ = 0 a.e.

STEP 1. We first show that it suffices to assume that for some a> 0,

1"(0) ~. a, o ~ 0 < 27r, (3.1)

(and consequently, flogl"(O)dO > -(0). Indeed, assume that Theorem 1.4 holds for such 1'. Given any 1', define

fn(O) := { 1"(0),

l/n,

if 1"(0) > l/n,

if 1"(0):5 l/n.

By the Monotone Convergence Theorem,

12r 12r lim 10gfn(0)dO = 10gl"(0)dO n-oo 0 0


(the latter integral may be equal to -00) and consequently

lim G(fn) = G(Jl). n ..... oo

(3.2)

Let dl'n := fn(O)dO + dl' •. Since 1"(0) ~ fn(O), and Theorem 1.4 holds (by assumption) for each dl'n, we have

Passing to the limit as n -+ 00 and applying (3.2) yields li.rnm ..... oo r-m Em,p(dl'; r) ~ G(I"). Since the reverse inequality has already been proved, Theorem 1.4 holds for 1'. Thus, from now on we assume that (3.1) holds.

STEP 2. Fix any f > O. Then there is a trigonometric polynomial Qe that satisfies

121< 11"(0) - Qe(O)ldO < f. (3.3)

In view of (3.1) we may also assume (see e.g. [Sz, Thm 1.5.3]) that

o ~ 0 ~ 271". (3.4)

(3.5)

(3.6)

STEP 3. For k = 1,2,3, ... , define

(3.7)

where the hie are constructed by Lemma 2.1 (for the measure (j = 1'.). The assertion (2.8) of that lemma then gives

(3.8)

STEP 4. Applying Lemma 2.2 to the continuous, 271"-periodic weight w(O) = [/Ic,f(O)]l/P we can find a sequence {Wl,le,e(O)}~l that satisfies

l lim Wl,le,e(O) = [fle,f(O)P/P, uniformly on [0,271"], (3.9) ..... 00

where (3.10)

Szego Type Asymptotics for Minimal Blaschke Products 115

and all the zeros of BMl lie in Izl < r. From (3.9) and the fact that lleAO) 2:: ka > 0 on [0,211"] we obtain

lim w: Ie f(O) = lie f(O), uniformly on [0,211"] (3.11) l--+oo 'J '

and, also, that (3.12)

STEP 5. Applying the result of Section 2(c) (see (2.17)) to the weight (3.10) we construct, for each n = 1,2,3, ... , a Blaschke product BnMl such that

r-nMlIBnMAreill)I'= G(wl,le,f)wi,L(O)(1 + 0(1)), (3.13)

where 0(1) -+ 0 as n -+ 00, uniformly on [0,211"]. Now set

L:= lim r-n En,p(dJ.'; r). n-oo

(3.14)

Applying (3.6) with <p = [left hand side of (3.13) ]P and letting n -+ 00,

we obviously obtain

Now, let l-+ 00. Then (3.11) and (3.12) imply that

[ 1 rr f 1 r2r h (0) ] L ~ G(fle,f) 211" Jo hie (O)dO + 211""1;':"00 + 211" Jo Q:(0)dJ.'6.

Since hie ~ 1, we obtain from (3.7) and (3.4) that

11/;':1100 ~ ~.

Hence, letting k -+ 00 and applying (2.6), (2.7) of Lemma 2.1 and (3.8), we get

L ~ G(Qf) [1 + 2:a] . Finally, let f -+ 0 and apply (3.5) to obtain L ~ G(J."). Since the reverse inequality has already been proved, the proof of Theorem 1.4 is complete .

• Proof of Theorem 1.5. Applying Theorem 1.4 with dJ.' = [w(O)]PdO, we obtain Theorem 1.5 for p < 00.


For p = 00 assume first that w( 0) is continuous and 211"-periodic. Then we repeat the proof of Theorem 1.4 omitting steps 2 and 3. We thus obtain a sequence {wt} of functions of the form (3.10) that satisfies

lim Wl(O) = w(O), uniformly on [0,211"]. l-+oo

From this we proceed to (3.13) (with Wl instead Wl,i:,€) and obviously obtain

lim r-n En oo(Wj r) $ G(Wl). n-+oo '

Letting l -+ 00 and recalling the assumption w(O) ~ a > 0 of Step 1 we obtain that liIDn-+oor-n En,oo(w, r) $ G(w). The reverse inequality was proved in Section 2(a).

If W is merely upper semi-continuous and 211"-periodic we find a decreasing sequence {fi:} of continuous 211"-periodic functions that converges to W

pointwise. The Monotone Convergence Theorem implies that lim GUi:) = i:-+oo

G( w). Since W $ fi:, we obtain

lim r-n En oo(Wj r) $ lim r-n En 00(!J:, r) = GUi:). n-+oo ' n ...... oo '

The result now follows by passing to the limit as k -+ 00.

Finally, if w(O) i= w(211") we consider instead W defined by W(O) = w(211") = max{w(O), w(211")} and w(O) = w(O) for 0 < 0 < 211". Then w is upper semi-continuous, 211"-periodic and satisfies En,oo(wjr) = En,oo(wjr). Thus, the previous case applies. •

4 The Szego Function for the Annulus

In this section we give the proof of Theorem 1.6.

Lemma 4.1 Let u(z) be harmonic in r < Izl $ 1 and continuous in r $Izl $ 1. If

1 11 uds=- uds 1031=1 r Izl=r

(4.1)

(ds denotes the element of arc length), then u(z) has a single-valued conjugate v(z) in r < Izl $ 1.

Proof. Let p := ../Z2 + y2. For f > 0 small enough, u(z) and log 1/ p are harmonic in r + f $ Izi $ 1. By Green's theorem we then have:

( f + f ) {u! log! _ (log!) ~u } ds = 0, (4.2) 11031=1 l lzl=r+f un p p un


where () I {)n denotes differentiation along the inward normal with respect to the annulus r + f < Izl < 1. Since

() 1 { -log- = {)n p

1 on Izl = 1

-(r + f)-I, on Izi = r + f,

we obtain from (4.2) that

f uds __ 1 f uds= (log_I) f {)u ds . JI~I=l r + f JI~I=r+f r + f JI~I=r+f ()n

(4.3)

Since u is harmonic, the integral in the right-hand side of (4.3) is independent of f. Letting f - 0 in (4.3) and using the continuity of u in r ~ Izl ~ 1 we obtain (see (4.1» that

1 {)u -() ds = 0, r < rl ~ 1.

1~I=rl n·

Since the last integral represents (up to the factor log l/rl) the period about Izl = r of a harmonic conjugate ofu(z) (cf. [F, pp. 79-80]), the result follows. •

Proof of Theorem 1.6. Let f«(J) be a positive, continuous 211'-periodic function. Then logf«(J) is continuous and 211'-periodic and G(f) > O. Let u(z) be the solution of the Dirichlet problem in r < Izl < 1, with boundary values

(4.4)

Since u = const. on Izi = 1, u( z) has a harmonic extension (by the reflection principle) to r < Izl < l/r. Next, (4.4) and the definition (1.4) ofG(J) yield

12'11: u(ei9)d(J = 12'11: u(rei9)dO.

Hence u(z) satisfies (4.1) of Lemma 4.1. Applying this lemma, pick any single-valued harmonic conjugate v(z). Since u is harmonic in r < Izl < l/r, so is v. Let

and define

"'( := -21 f v(z)ldzl 11' JI~I=l

(4.5)

D(z) := ei'Y eu(~)+itl(~), r < Izl < l/r. (4.6)

By its construction, D(z) obviously satisfies (i) and (ii) of Theorem 1.6 (the relation (1.15) follows by the reflection principle). Defining the (singlevalued) branch of log D( z) by

logD(z) := u(z) + i("'( + v(z», (4.7)

we obtain by (4.5), that (iii) of Theorem 1.6 is also satisfied.


The uniqueness of such D is easily established. For if Dl, D2 both satisfy the conditions (i), (ii) of Theorem 1.6, the Maximum Principle (for the harmonic function log IDt/ D21) yields that Dt/ D2 is a unimodular constant. Then (iii) of Theorem 1.6 yields that (for a suitable branch of log) the integral ~zl=llog(Dt/ D2 )ldzl is real. Hence Dl = D2 • This completes the proof of Theorem 1.6. •

Example. Let

(4.8)

where the zeros al, ... , aM of BM lie in Izl < r (recall the example of Section 2 (c». We claim that there is a single-valued branch oflog(BM(Z)/zM) in r < Izl < l/r that satisfies

flog BM~Z) Idzl = O. (4.9) llzl=l z

Indeed, consider the branches

and

and define

I Z-al:._ ~(al:)j 1 og--.- - L.J - -:-,

z j=l z J

M

Izl > r, (4.10)

BM(Z) '" { Z - al: } log ~ := L.J log -z- -log(1 - Ql:z) , r < Izl < l/r. (4.12) 1:=1

Notice that the Laurent expansion (in r < Izl < l/r) of log(BM(z)/zM) does not contain a constant term. Hence (4.9) follows.

We also know that, for the case considered, G(w) = rM. Thus (4.8) and (4.9) imply that the function

{rM BM(Z)/zM} 1/2 := rM/2 exp {! log (BM(Z)/zM)}

is the Szego function D(w;r;z) of w(O) for the annulus r < Izl < l/r. Recalling (2.15) of Section 2 we obtain that the (asymptotically) minimal Blaschke product of order nM satisfies

M

BnM(Z) = znM rM B~(z)/zM (1 + 0(1»

. nM G(w) ( » = z {D(w;r;z)p(1 +01 ,

SzeglS Type Asymptotics for Minimal Blaschke Products 119

where 0(1) -+ 0 as n -+ 00, uniformly on compact subsets of r < Izl < l/r. This illustrates Theorem 1.7 for p = 1.

5 Proof of Theorems 1.7 and 1.8

We start with the proof of part (a) of Theorem 1.7. Under our assumptions on 1"(6) one can define the Szego function D(I"j rj z) =: D(z). Set

._ B~(z) D2/P(z) <pn(z) .- -;n- . Gl/p(I") , (5.1)

and observe that <Pn is analytic in r < Izi < l/r and l<Pnl is continuous in the closed annulus. Since (by (1.16»

l<Pn(z)1 = 1 for Izl == 1, (5.2)

it suffices to prove that limn-+oo <Pn(z) = 1, uniformly on compact subsets of 0 := {z : r < Izl :$ I}.

From the proof of the lower bound (see Section 2(a», we know that

r-np 2~ fo21r IB~(rei')IP 1"(6)d6 ~ G(I").

This and (1.10) imply that

lim r-np ..!. f2X; IB~(rei')IP 1"(6)d6 = G(I"), n-+oo 211" Jo

or, equivalently (by (5.1) and (1.17»

(5.3)

We have already mentioned in the Introduction that the zeros of B~ lie in Izl < r. Hence there is a branch of log (B~(z)/zn) that is single-valued in o and satisfies

flog B~~z) Idzl = 0 J1Z1=1 z

(see the Example at the end of Section 4). Also, (see Remark 3 in Section 1), there is a branch of log D2/P(Z) that satisfies

..!. f logD2/P(z)ldzl = logG(P')l/P. 211" J1z1=1

Thus we can define the single-valued branch oflog<Pn(z) in 0, such that

f log <pn(z)ldzl = o. J1Z1=1

(5.4)


For any n = 1,2, ... , fix this branch and define

~(Z) := exp(PlogCPn(z», Z E O.

Given ( E 0, we apply the Cauchy formula for r + £ < Izi < 1 (for £ > 0 small enough) to ~(z) and deduce (see (5.2» that

I~«()I $ c{ 1 ~ 1(1 + 1(1_1r_ £ lZI=r+£ Icpn(Z)IPldZI}, (5.5)

where c is a constant independent of £ and n. Since ICPn I is continuous in r $ Izi $ 1 we may pass to the limit as £ -+ 0 and then use (5.3) to obtain that {~} (and, consequently, {CPn}) form a normal family in O. Choose any convergent subsequence {CPn }neA :

l~ CPn(z) =: cp(z) . (5.6) .. eA

It remains to show that cp(z) == 1 in O. For this purpose we introduce the function

Un(Z) := Icpn.(z)1" - plog ICPn(z)l- 1. (5.7)

Let us examine some properties of Un. Since ICPn I is continuous in ri and ICPnl > 0 in ri (recall that the zeros of B~ lie in Izl < r), we obtain that gn is continuous in 0 and

(5.8)

Next, since CPn is analytic and nonvanishing in 0, it follows that ICPn IP is subharmonic in 0 and log ICPn I is harmonic in O. Thus, Un is subharmonic inO.

By the logarithmic convexity of the integral means of subharmonic nmction (cf. [HK, Theorem 2.12]) we may write for a given r 0 small enough:

1 {2'1f logl/p 1 (2'1f i' 27r 10 Un (pei')dfJ $ log{I/(r + £)} 27r 10 Un (r + £)e ) dfJ

+ log{p/(r + £)} 1.. (2'1f Un (ei')dfJ (5.9) log{I/(r + £)} 27r 10

log l/p I r'lf i' = log{l/(r + £)} 27r 10 Un (r + £)e ) dO,

where, in the last step, we used the property (see (5.2» that

Un(e i ') = O. (5.10)

Szeg<> Type Asymptotics for Minimal Blaschke Products 121

Next, (5.4) and (5.3) yield

1 1211" . lim - gn (re,fJ )d(J = O. n_oo 211' 0

(5.11)

Passing in (5.9) to the limit first as f -+ 0+ and then as n -+ 00, n E A, we obtain (by (5.8), (5.11), (5.6) and (5.7)) that

r 0 for x > 0, x =P 1 (5.14)

implies that g(z) ~ 0 in n. Then (5.12) yields g(z) = 0, zEn. By (5.13), (5.14) we thus obtain:

1<p(z)1 = 1, zEn,

and therefore (5.15)

for some -11' ~ "'f < 11'. It remains to show that "'f = O. This follows directly from (5.6), (5.4).

Indeed, the branch oflog we fixed above for n = 1,2, ... can be written in the form

(5.16)

where Log denotes the principal branch, kn is an integer and a is a fixed point. As n -+ 00, n E A, the integral in (5.16) approaches 0 and Log <Pn(a) -+ i"'f (by (5.15)). Thus, (5.4) yields: kn = 0 for n ~ Nand "'f = O. The proof of part (a) of Theorem 1.7 is now complete.

Proof of part (b) of Theorem 1.7. For 0 < p < 00, apply Theorem 1.7(a) with d,,«(J):= wP«(J)d(J. For p= 00, set

( \ ._ B~(z) D2(z) <Pn z t .- ----;n- G( w) .

Then (5.2) holds. Also, (5.3) holds with p = 1 and with the equality sign replaced by ~ . Proceeding as in the proof of Theorem 1.7(a), we get the result. •

Remark 7. The same proof applies for the general case, namely log ,,' E L1 [O, 211']. In this case D(,,'; r; z) belongs to the Hardy space H2 in the annulus n (see Remark 6 in Section 1) and therefore ~ E H~(n). Hence,


in (5.5), we may pass to the limit, as E: -+ O. Also, D is an outer function and therefore

(5.17)

Thus, (5.8) is valid. The rest of the proof remains unchanged.

Proof of TheoreUl 1.8. We give the sketch of the proof and leave the details to the reader. Set

( ) .~ CPn,p(z) D(dl'i'Z)2/P CPn z.- zn _ G(I")1/p'

where CPn,p is the minimal polynomial and D(dl'i z) is defined by (1.5). Since CPn,p is monic and has its zeros in Izl < 1, we obtain by the properties of D(dl'iZ), that

CPn(OO) = 1 (5.18)

and that there is a branch of log CPn(z) in Izl > 1 that satisfies (5.4). Since CPn satisfies (5.3) with r = 1, the normality of {CPn} follows. As before, define Un and use the monotonicity (rather than logarithmic convexity) of its integral means, to obtain

127r Un (peifJ)dO ~ 127r Un (1 + E:)eifJ ) dO,

for p > 1 and E: > 0 small enough (cf. (HK, Theorem 2.12], for the case r1 = 0). Passing to the limit, first as £ -+ 0 and then as n -+ 00, n E A, and using (5.4) and (5.3), we deduce as before that cp(z) = ei'Y, Izl > 1. Since cp(oo) = 1, we get 'Y = 0 and the result follows. •

6 Generalizations. Application to n-widths.

Let us consider a more general case, when the circle Izl = r is replaced by a compact set K in the open unit disk 6. Given such a K and given a positive, finite Borel measure I' on K, we set for 0 < p < 00

(6.1)

where the infimum is taken over all Blaschke products of order n, with zeros in 6.

We shall need some basic notions from the potential theory (cf. [T, pp. 94-104]). Let

V := infJ' f log I z - S 1-1 duzdu{ =: inf l(u),

D iK 1-(z D


where the infimum is taken over all probability measures on K. Then o < V :5 00. Provided V < 00, there exists a unique probability measure II on K (the equilibrium distribution for K) such that V = 1(11). The equilibrium potential u for K is defined by

It satisfies

u(z):= flog' z - i ,_1 dll(. iK 1- (z

u(z) :5 V for z E b.

(6.2)

(6.3)

u(z) = V II a.e. for z in K. (6.4)

We set c(K;b.) := I/V, and call c(K;b.) the capacity of K relative to b. (cf. [W] and notice that the capacity in [T] is defined to be e-V ).

Proceeding as in Section 2(a) and using (6.2), (6.3), we obtain (cf. [FM2]) the lower bound

(6.5)

where

G(I") := exp L (log ~~) dll, (6.6)

and dl'/dll denotes the Radon-Nikodym derivative with respect to II of the part of I' that is absolutely continuous with respect to II. The crude upper bound, namely

En,p(dl';K):5 e-n/(1+£)e(K;.6.), n> n(€), € > 0,

can also be obtained. Together with (6.5) this gives (cf. [FM1,2])

lim E!t;(dl'; K) = e-1/e(K;.6.). n .... oo

(6.7)

Now, let us tum our attention to another quantity. For 1 :5 q :5 00, let Aq denote the restriction to K of the unit ball of the usual Hardy space Hq in b.:

(6.8)

For 1 :5 p < 00, the Kolmogorov n-width of Aq in the space Lp(dl'; K) is defined by

where Xn denotes an arbitrary n-dimensional subspace of Lp(dl'; K). Fisher and Micchelli have proved (see [FMl]) that

(6.10)


In view of (6.7) this gives the n-th root asymptotics for dn. A similar result holds for weighted Loo norms. These asymptotics were established by Widom [W] (a simpler proof is given in [FM1]), but many special cases were known earlier (see [LT] for the history of this problem).

The first result concerning the strong asymptotics of dn was established by Parlenov. He considered the case p = q = 2, K is a smooth closed curve, and dl' = wldzl, where w E C{K), w > 0, and Idzl is the arc length on K. Although stated in different terms, the result of Parfenov reads (cf. [paa] for w = 1 and [Pa2] for general w):

(6.11)

Since Aoo C A2, we have dn(Aoo, L2(dl'i K» $ dn (A2' L2(dl'i K» (see (6.8), (6.9». Therefore (6.5), (6.10) and (6.11) yield:

Theorem 6.1 Let K be a simple closed Jordan cUnJe of the class C1+ f , t: > O. Let dl' = wldzl, where wE C(K), w> 0, and Idzl denotes the arc length on K. Then

lim e-n/c(Kjt:.) En 2{dl'i K) = G(I")1/2. n-+oo '

(6.12)

To describe the behavior of minimal Blaschke products B: we have to first defin:e the appropriate Szegl) function. Let 0 denote the doubly connected domain bounded by a curve K and by its reflection about Izl = 1. The Szego function D( dl'i K i z) of 1" for the "annulus" 0 is defined as in Theorem 1.6 (with obvious alterations) and with (1.18) replaced by

L log D( z )dv is real.

Having defined D(dl'i Ki z), we observe that the relation (6.12) implies that the zeros of {B:}~l have no limit points outside K. Hence the method of the proof of Theorem 1.7 applies and we obtain

Theorem 6.2 Assume the conditions of Theorem 6.1. Let B~ denote a Blaschke product that realizes the infimum in (6.1), for p = 2. Let u be the equilibrium potential for K defined by (6.2) and let v be its conjugate. Then

lim e(u(z)+iv(z»n B* (z) = G{I") , n_oo n D{dl'i Ki z)

uniformly on compact subsets of 0.

The details of the proof as well as some generalizations will be given in a future paper.


References

[F] Fisher, S.D., Function Theory on Planar Domains, John Wiley & Sons, New York, 1983.

[FM1] Fisher, S.D., Micchelli, C.A., The n-widths of sets of analytic functions, Duke Math. J., 47(1980), 789-80l.

[FM2] Fisher, S.D., Micchelli, C.A., Optimal sampling of holomorphic functions, Amer. J. Math., 106(1984),593-609.

[G] Geronimus, J., On extremal problems in the space LCj», Math Sbornik, 31(1952), 3-26. (Russian)

[GS] Grenander, V., Szego, G., Toeplitz Forms and their Applications, Chelsea, New York, 1984.

[H] Hille, E., Analytic Function Theory, vol. 2, Ginn and Company, Boston, 1962.

[HK] Hayman, W.K., Kennedy, P.B., Subharmonic Functions, Academic Press, London, 1976.

[K] Koosis, Paul, Introduction to Hp Spaces, London Math Soc. Lecture Notes Series 40, Cambridge Vniv. Press, Cambridge, 1980.

[LT] Levin, A.L., Tikhomirov, V.M., On a theorem of Erokhin, Appendix to V.D. Erokhin, Best linear approximations of functions analytically continuable from a given continuum into a given region, Russ. Math. Surveys, 23(1968), 93-135.

[LP] Li, X., Pan, K., Asymptotics of Lp extremal polynomials on the unit circle, to appear in J. Approx. Theory.

[N] Nevai, P., Weakly convergent sequences of functions and orthogonal polynomials, J. Approx. Theory 65(1991), 322-340.

[Pal] Parfenov, O.G., Widths of a class of analytic functions, Math. VSSR Sbornik, 45(1983), 283-289.

[Pa2] Parfenov, O.G., The singular numbers of imbedding operators for certain classes of analytic and harmonic functions, J. Soviet Math., 35(1986), 2193-2200.

[Pa3] Parfenov, O.G., Asymptotics of the singular numbers of imbedding operators for certain classes of analytic functions, Math. VSSR Sbornik, 43(1982), 563-57l.

[P] Pinkus, Allan, n- Widths in Approximation Theory, SpringerVerlag, Heidelberg, 1985.

126 A.L. Levin, E.B. Sa.ff

[Sa] Saff, E.B., Orthogonal polynomials from a complex perspective, In: Orthogonal Polynomials: Theory and Practice (Paul Nevai, ed.), Kluwer Acad. Pub., Dordrecht (1990), 363-393.

[Sz] Szego, G., Orthogonal Polynomials, ColI. Pub I. , vol. 23, Amer. Math. Soc., Providence, R.I., 1975.

[T] Tsuji, M., Potential Theory in Modern Function Theory, Dover, New York, 1959.

[W] Widom, H. Rational approximation and n-dimensional diameter, J. Approx. Theory, 5(1972), 343-361.

A. L. Levin Department of Mathematics Open University Max Rowe Educational Center 16 Klausner Street P.O.B. 39328, Ramat. Aviv Tel-Aviv 61392 ISRAEL

E. B. Saff Institute for Constructive Math. Department of Mathematics University of South Florida Tampa, FL 33620 USA

Asymptotics of Hermite-Pade Polynomials

A.I. Aptekarev H. Stahl*

ABSTRACT We review results about the asymptotic behavior (in the strong and weak sense) of Hermite-Pade polynomials oftype II (also known as German polynomials). The polynomials appear as numerators and denominators of simultaneous rational approximants. The survey begins with general remarks on Hermite-Pade polynomials and a short summary of the state of the theory in this field.

1 Introduction

1.1 DEFINITIONS

Hermite-Pade polynomials are associated with a vector

1= (fo, Jr, ... , 1m) E A( {0})m+1 (1.1)

of m + 1 functions which are assumed to be analytic in a neighborhood of zero. These polynomials are generalizations of Pade polynomials (numerators and denominators of Pade approximants) in the following sense: Set m = I, 10 == -I, Jr = I E A( {OJ). Then for every pair (no, n1) E N 2 there exist polynomials Pno and Qn, (Qn, ~ 0) of degree at most no and n1, respectively, such that

Qn, (z)Jr(z) + Pno(z)/o(z) = Qn, (z)/(z) - Pno(z) (1.2)

Here 0 denotes Landau's symbol. The pair (Pno ' Qn,) is called a Pade form. The two components of the form are Pade polynomials, and the quotient Pno/Qn, is the (no, nd - Pade approximant to the function I. For m > I, this definition can be generalized in two different directions:

a) Hermite-Pade polynomials 01 type I (Latin polynomials in Mahler's ter-

"Research supported in part by the Deutsche Forschungsgemeinschaft (AZ: Sta 299 14-1).

AMS (MOS) subject classification: 33A65, 30E10, 41A28.


128 A.1. Aptekarev, H. Stahl

minology): For any multi-index n = (no, ... , nm ) E N m +1 and any vector function (1.1) there exists a vector of polynomials

(Po, ... ,Pm) E G30 ITn;-l) \ (0, ... ,0), (1.3)

such that m

Epi(z)fi(z) =: R,.(z) = O(zlnl-l) as z -- 0, (1.4) i=l

where Inl := E::l ni and ITk denotes the set of all polynomials of degree at most k. The vector (1.3) is called an Hermite-Pade form of type I and its m + 1 components Hermite-Pade polynomials of type 1.

b) Hermite-Pade polynomials of type II (German polynomials in Mahler's terminology): For any multi-index n = (no, ... , nm ) E N m +1 and any vector function (1.1) there exists a vector of polynomials

(1.5)

with (No, ... , Nm ) := (Inl- no,·· ., Inl- nm )

such that

for all i,j = 0, ... , m, i i= j. The vector (1.5) is called an Hermite-Pade form of type II and its m + 1 components Hermite-Pade polynomials of type II.

lt is easy to see that in case of m = 1 and fo == -1 (respectively fo == 1) condition (1.4) (respectively (1.6» transforms to the defining condition (1.2) of ordinary Pade polynomials.

System (1.6) contains (m + 1)m/2 relations of which, however, only m relations are linearly independent. An independent set of relations is, for instance, given by

qo(z)/j(z) - qj(z)fo(z) = O(zlnl+l), j = 1, ... ,m. (1.7)

The other relations then follow since

implies

Asymptotics of Hermite-Pade Polynomials 129

If /0(0) :F 0, then without loss of generality we can assume that /0 == 1. We only have to divide all components of (1.1) by /0. In this latter case it is natural to consider the vector of rational functions

( ql , ... , qm) qo qo

(1.8)

as an approximant to the vector (h, .. . /m) of functions. Its components are called simultaneous rational approximants (with common denominator) to the functions h, ... , /m. The relations (1.7) show that these are mdimensional generalizations of Pade approximants.

The possibility to construct approximants exists not only for HermitePade polynomials of type II; for type I polynomials there exist algebraic and integral approximants (see [B-GrM]). However, we will not discuss these latter approximants in the present paper. Our main interest is directed toward asymptotics of Hermite-Pade polynomials of type II. Only if type II and type I polynomials cannot be considered separately, will we extend our discussion to include asymptotics of type I polynomials.

1.2 ORTHOGONALITY

As in the case of Pade approximants, the zero of high order at z = 0 in (1.7) implies certain orthogonality relations for the inverses of HermitePade polynomials of type II, and analogously (1.4) implies an orthogonality relation for linear combinations of polynomials of type I with the functions /0, ... , /m as coefficients.

Denote the inverse polynomials of Pj and fJj by

Pj(z) := Zn j -lpj (l/z),

Qj(Z) := zNjfJj (l/z), j=O, ... ,m,

and assume that /0 == 1. Then

Ie (lc t Pj«()(no-n j /j(l/()d( = 0 for k = 0, ... , No - 2 C j=l

and

(1.9)

(1.10)

t (lcQo«()(no-n j /j (1/() d( = 0 for k = 0, ... , nj - 1, j = 1, ... , m.

(1.11) In both integrals, C is a closed integration path that encircles infinity and is close enough to infinity such that /j(l/z), j = 1, ... ,m, is analytic on and in the exterior to C.

There exist orthogonality relations for the polynomials Qj, j = 1, ... , m, analogous to (1.11), but it is necessary that /j == 1, which can be obtained by dividing all components of (1.1) by /j.

130 A.I. Aptekarev, H. Stahl

The relations (1.10) and (1.11) reduce to the classical concept of orthogonality if m = 1. Especially the relations (1.11) are basic for the investigation of the asymptotic behavior of polynomials of type II. The orthogonality property defined by (1.11) is called multiple orthogonality.

Hermite-Pade polynomials inherit many formal properties of orthogonal polynomials, but as a general rule the situation becomes more complicated. For example, Hermite-Pade polynomials can be expressed by determinantal formulas involving the power series coefficients of the components of (1.1). Simultaneous rational approximants (1.8) can be represented as convergents of m-dimensional vector continued c(m) (or p(m» - fractions. Hermite-Pade polynomials of type II satisfy m + 2 - term recurrence relations, which generalize the well-known 3-term recurrence relations of orthogonal polynomials, and so on.

1.3 DIFFERENT ASPECTS OF THE HERMITE-PADE

POLYNOMIAL THEORY

There exist two basic aspects or directions of research in Hermite-Pade polynomial theory. They correspond to similar directions in the theories of Pade approximants, continued fractions, or orthogonal polynomials, and they are

1) the formal or algebraic aspect and 2) the analytic or asymptotic aspect.

The algebraic aspect of the theory is concerned with the various relations between polynomials of type I and II (Mahler relations), formulas of the theory of C<m) (or p(m» - fractions and so on. To this aspect also belongs the question of uniqueness of the polynomials.

Multi-indices n E N m +1 for' which the systems (1.4) and (1.7) have a unique solution up to normalization are called normal indices. A system (1.1) of functions for which all indices are normal is called a perfect system.

The algebraic aspect of the theory and especially the existence of explicit formulas for Hermite-Pade polynomials has found wide applications. The most classical ones are concerned with applications in analytic number theory.

The second (analytic) aspect ofthe theory deals with the behavior ofthe Hermite-Pade polynomials when Inl -+ 00. This behavior implies convergence results for the different kinds of approximants based on Hermite-Pade polynomials of both types.

For the description of the asymptotic behavior of sequences of polynomials {QnldegQn = n}~=l it is useful to distinguish the following three types of asymptotic formulas:

v'IQn(z)I-I<lI(z)1 as n -+ 00 (1.12)


(1.13)

Q,,(z) 4>(z)" --+ F(z) as n -+ 00, (1.14)

with convergence in some domain, and an appropriately defined function 4>. These three types of asymptotic behavior form a hierarchy and are called n-th root (or weak), ratio, and power (or strong) asymptotics, respectively.

N-th root asymptotics represent the weakest form in the hierarchy. But for many applications these asymptotics are sufficient. They show the global geometric structure of the asymptotic behavior; they are sufficient, for instance, to determine the lines on which the zeros of the polynomials become dense, and to deduce the density of their distribution. With their help we can answer questions about the convergence of approximants at least in capacity.

Ratio asymptotics provide a more precise description. They are connected with the existence of limits for the coefficients in recurrence relations defining the sequence {Q .. }. This connection has been exploited in the theory of continued fractions and in the spectral theory of difference operators.

An even higher degree of accuracy is given by the strong asymptotics (1.14). These formulas possess a more delicate analytic character than the other two. For example, with their help it is possible to localize individual positions of the zeros.

Passing through the hierarchy from (1.12) through (1.14) we obtain increasingly more precise formulas. However, the class of polynomial sequences for which asymptotic formulas can be proved becomes more restrictive at the same time.

1.4 A BRIEF SURVEY OF THE HISTORY AND THE

PRESENT STATE OF OUR TOPIC

In the development of the theory of Hermite-Pade polynomials it is possible to distinguish two different lines of investigations. One direction is concerned with Hermite-Pade polynomials for special classes of functions like exponentials, binomial sequences, logarithms etc.; the other direction with polynomials for general classes of functions. In this respect the situation is the same as that of orthogonal polynomials, where we have on the one hand the polynomials orthogonal with respect to special weights, like Jacobi, Hermite and Laguere polynomials, and on the other hand polynomials orthogonal with respect to general weight measures.


The theory of Hermite-Pade polynomials for special systems (1.1) of functions has a long history that goes back more than 100 years. It starts with the famous results by Hermite about the transcendence of the number e ([Hel]'[He2]). Hermite constructed explicit expressions for simultaneous rational approximants for the system of functions (&Z)j=o. It is easy to verify that polynomials of type II to this system are given by

j=O, ... ,m

for any multi-index n E Nm+l, where the polynomials are normalized in such a way that qo(O) = 1. The remainder terms in (1.7) are then given by the explicit formulas

j=O, ... ,m

In the appendix of [KI] a very simple and direct deduction of the transcendency of e is given based on Hermite's results and a simplification by Hilbert [Hi].

After this beautiful success by Hermite, the algebraic aspect of the theory for special classes of functions has been developed by Pade ([Pal], [Pa2], [Pa3]), J. Mall [Mal] (a student of O. Perron), and especially by K. Mahler and his successors ([Mah1], [Mah2], [Mah3], [Ja], ... ). The main interest of Mahler was directed towards number theory. The research of the algebraic aspect has continued up to the present day ([Ba], [deBrl], [deBr2], [deBr3], [Co], [Ni4], ... ).

As far as analytic aspects of the theory of Hermite-Pade polynomials are concerned, investigations have begun, even for special classes of functions other than the exponential function, only in recent years (approximately in the last 15 years). The study of analytic aspects has been stimulated strongly by the progress in the theory of Pade approxima.tion and the asymptotic theory for orthogonal polynomials. Asymptotics of Hermite-Pade polynomials were investigated in ([Kal] , [Nutl] , [Nut2] , [NutTr], [Chu1], [Chu2], [Chu3], [Pi], [801], [802], [803], ... ) and convergence theorems for simultaneous rational approximants and C<m) (or p(m» -continued fractions have been obtained in ([Ap1], [Ap2], [ApKaU], [deBr4], [deBr5], [deBr6], ... ).

The general formal theory of Hermite-Pade polynomials was initiated and developed to a great degree of perfection by K. Mahler [Mah3]. He introduced the definition of polynomials of type I (Latin) and type II (German) and obtained the basic formula. for generating polynomials of one type by those of the other type.


The formal theory of Hermite-Pade polynomials is extremely rich in specific formulas and non-trivial identities. These results find applications in the construction of algorithms, and for this reason the research in formal relations continues in the theory of Hermite-Pade polynomials up to the present day (see [lsSa] , [LoPo], [deBr7], [deBrS], [Pas], [S04], ... ).

The general formal theory of c(m) (or p(m»-continued fractions and its connection with simultaneous rational approximants has been studied in the last several years by M. de Bruin and V. I. Parusnikov ([deBr10], [deBrll], [ParI], [par2], [par3], ... ). They also obtained a generalization of the theorems of " Poincare, Perron [Gelfj, Van Vleck [VVI] about ratio asymptotics for polynomials, which are given by recurrence relations with limit-periodic coefficients, and about the convergence of the corresponding C<m) (or p(m»-continued fractions ([par4], [deBr12], [deBrJa]).

The investigation of analogies of the de Montessus de Ballore Theorem for simultaneous rational approximants in the m-Pade table was started by J. Mall [Ma]. These investigations have been continued by ([deBrJa], [GrMSa], [BaLu]).

Rather few results are known from the analytic theory of Hermite-Pade polynomials for general classes of functions. Certain systems of Markov functions

{J dl';(Z)}m Z - Z ;=1

(1.15)

are an exception here. The functions in (1.15) are called Markov functions if 1';, j = 1, ... , m, are positive measures with compact supports in R. (We reserve the term Markov function for the case that the measure 1'; has a compact support, and speak of Stieltjes functions if the support is not bounded but contained in one of the two halfaxes of R, and of Hamburger functions if the support is not bounded and not contained in one of the halfaxes.)

In the present paper we will survey analytic aspects of the theory. Systems of Markov functions will.play a prominent role. We concentrate on Hermite-Pade polynomials of type II. The first author will survey results on strong asymptotics (1.14) and the second author on n-th root asymptotics as introduced in (1.12).

In concluding this short introduction we will merely enumerate results which are connected with our investigation and upon which we will touch at several places later on. More details will be given at these places. The description of general perfect systems belonging to the class (1.15) was obtained in ([Ang1], [Ang2], [Ang3], [Nil], [Ni2]). The analytic aspects of the theory of class (1.15) or even more general classes have been investigated (for the diagonal case) in ([Ni2], [Am), [Ni3], [Nutl], [BGNut], [GoR], [St4], [ApKal2], [Ap3], [Ap4]).


The historic survey given here is very short and schematic. For a more comprehensive survey we recommend the paper by M. de Bruin [deBrO] (mainly formal and algebraic aspects) and the comprehensive paper by J. Nuttall [N ut2] (analytic aspects), which is also very rich with respect to algebraic aspects. For a review of applications of Hermite-Pade polynomials in number theory we refer to F. Beukers' survey paper [Be] and ([Ni5], [Ni6], [S05]).

2 Hermite-Pade Polynomials for Systems of Markov Type Functions

In this section we assemble results and definitions which are related to systems of type (1.15). Special emphasis will be given to Angelesco and Nikishin systems. We start with some general remarks and results concerning Markov type functions and multiple orthogonality.

2.1 MULTIPLE ORTHOGONAL POLYNOMIALS

If we consider simultaneous rational approximants (1.8) to systems of Markov type functions

( h(Z) = it;(z) = J dl';(Z»)m , Z - Z ;=1

(2.1)

then this leads to multiple orthogonal polynomials. Since we have assumed that It, ... , 1m are Markov functions, the 1'; 's have to be positive measures with compact support in R, i.e.

1'; ~ 0 and supp(I';) ~ R compact. (2.2)

It follows from (1.2) that Pade polynomials in the diagonal case developed at z = 00 to a single Markov function it satisfy

Q(z)jt(z) - P(z) = O(z-n-1) as z -+ 00

with deg(Q) = n, deg(P) = n - 1. This implies the orthogonality

jzkQ(Z)dJJ(Z)=O for k=O, ... ,n-l (2.3)

of the denominator polynomial Q. The orthogonality (2.3) determines Q up to normalization.

Similarly, the definition of simultaneous rational approximants (developed at z = 00) for a vector of Markov type functions (2.1) leads to the multiple orthogonality

j ZkQn(z)dl';(z) = 0 for k = 0, ... , n; - 1, j = 1, ... , m, (2.4)


for the denominator polynomial Qn, where now

Qn E IIlnl ' n= (nl,"" nm ) E Nm , and Inl = nl + ... + nm. (2.5)

Note that in contrast to the terminology in the preceding section we now use a multi-index with only m components. The first component no has been dropped, and the modulus Inl is now equal to No (as defined in (1.9».

In the diagonal case, i.e., if

n=(k, ... ,k), kEN, (2.6)

the orthogonality (2.4) coincides with (1.11) if in the latter formula we substitute f(lf()d( by dl'(x) and Qo by Qn. The non-diagonal case will be discussed later.

In the notation for Qn we use the subscript n = (nl,"" nm ) E N m instead of Qo in order to emphasize the dependency on the multidegree n.

The m numerator polynomials of the simultaneous rational approximants (1.8) are now given by

J Qn(z) - Qn(x) Pjn(z) = dJ.'j(x) E IIlnl-l

x-z (2.7)

for j = 1, ... ,m. The multiple orthogonality (2.4) together with (2.7) implies that

(QnfJj - Pjn) (z) = O(z-n;-l) as z -+ 00 (2.8)

and for the simultaneous rational approximants we have the remainder formulas

(2.9)

= 1 J (QnHj) (x)dl'j(x)

(QnHj)(z) x - z

for j = 1, ... ,m , where Hj E lIn;, j = 1, ... ,m, are arbitrary polynomials not identical zero. The last equality in (2.9) is a consequence of the multiple orthogonality (2.4).

If the polynomial Qn has Inl simple zeros Xl, •• • , xlnl' then these zeros have the following quadrature property: For every j = 1, ... , m there exist Christoffel numbers

ejk E R, k = 1, ... , Inl, j = 1, ... , m

such that

Inl J p(x)dJ.'j(x) = ~p(xk)ejk for all PElIlnl+nj-1, j= 1, ... ,m k=l

(2.10)


It has already been mentioned that in the diagonal case, i.e. if we assume (2.6), the orthogonality relations (2.4) and (1.11) coincide after an exchange of variables in (1.11). In the same way there is a correspondence between the relations (2.8) and (1.7): In (1.7) we have to substitute z by 1/ z, and we have to multiply the whole relation by zNo• Note that fa has been assumed to be identically 1. This correspondence shows that in the diagonal case the polynomials Qn, PIn, ... , Pmn are the inverse Hermite-Pade polynomials of type II to the system of functions (2.1) together with fa == 1.

In the non-diagonal case the definitions (1.7), (1.8), (1.9), and (1.11) do not immediately correspond with (2.4), (2.7), and (2.8). However, if we replace the functions h, j = 1, ... ,m by

ij(z) := zn;-no h(z), j = 1, ... , m,

we also have a correspondence for this case. In the present section the starting point of our discussion is (2.4) together with (2.7) and (2.8).

In taking (2.4) as starting point, we give an explicit role to the polynomial Qn. The polynomials PIn, ... , Pmn are then only deduced from Qn and for them we have no characterization comparable with (2.4). This phenomenon stands in a certain contrast to the definition of the Hermite-Pade polynomials of type II, qo, ... qm, in (1.6), where all m + 1 polynomials play an equal role. The now existing asymmetry is a consequence of the assumption fa == 1. It has already been mentioned after (1.7) that a normalization h == 1 for j E {I, ... , m} can be achieved by dividing all elements of (1.1) by h, which leaves (1.6) unchanged if h(O) =P 0 (remember that in (1.6) we consider expansion at z = 0). However, in the present context it is important to observe that such a division would result in new functions ii := fi/ h, which for i =P 0 are no longer of the assumed Markov type. Hence, such a procedure is impractical, and the polynomials Qn play indeed a special role for the system (2.1).

In the case m = 1 the situation is different. If II is a function of Markov type, then 1/ II is a function of Markov type plus a linear term and the connection between the polynomials Qn and PIn then is the well-known connection between orthogonal polynomials of first and second kind.

2.2 ANGELESCO AND NIKISHIN SYSTEMS (DEFINITION

AND FORMAL ASPECTS)

Contrary to the case of ordinary orthogonal polynomials Q, as defined in (2.3), where we have uniqueness up to normalization, this desirable property does not hold in general for multiple orthogonal polynomials Qn defined by (2.4). We only have uniqueness under additional assumptions. This fact may already give a hint how much more difficult it is to obtain positive results about the asymptotic behavior of the polynomials Qn for Inl -+- 00.


There are two general classes of systems of Markov type functions (2.1) which are understood and discussed in the literature. They are Angelesco and Nikishin systems.

The first class has been introduced and its investigation was started in a series of papers by Angelesco in 1916 - 1923 [Angl-3]. He considered Markov functions (2.1) under the additional assumption

(i) supp (Pi) = Ai' is a compact interval, j = 1, ... , m, (2.11)

(ii) Ai n Ai = 0, i =F j , i, j = 1, ... , m.

Under this assumption it rather immediately follows from the multiple orthogonality (2.3) that the polynomials Qn , n = (nl. ... , nm) E Nm, can be represented as a product

m

Qn(z) = II Qin(z) , deg(Qin) = ni , (2.12) i=l

and each of the m polynomials Qin , j = 1, ... ,m, has exactly ni simple zeros in the interval Ai (see [An 2]). From (2.12) and some additional analysis it follows that all polynomials Qn are unique up to normalization for Angelesco systems.

The second class of systems of Markov type functions was introduced by E.M. Nikishin in [Nil]. He considered systems (Ii )j=l with functions Ii defined by

h(z)

l2(z)

(2.13)

Im(z) = f (f ( ... ) dtT2(X2») dtTl(Xl) , J J},.1 J J},.2 X2 - Xl Xl - z

Here, the tTi , j = 1, ... ,m, are positive measures with suPP(tTi) ~ Ai' the supports supp (tTi) are assumed to contain infinitely many points, the .AI' ... ' Am are real compact intervals, and the pairs (Ai> Ai +1), j = 1, ... ,m - 1, are assumed to be disjoint, i.e.

Ai n Ai+1 = 0 for j = 1, ... , m - 1. (2.14)

In contrast to Angelesco systems the functions of a Nikishin system are Markov functions with a defining measure on the same interval

A =.Al ~ R


for all m functions /j = pj , j = 1, ... , m. All m measures PI, ... , Pm are mutually absolutely continuous, and the density functions with respect to 0'1 again form a Nikishin system of m - 1 functions. We have

(2.15a)

and

(2.15b)

In the case of two functions and two intervals.6. = .6.1 = [a, b] and d2 = [c, d] the definition (2.13) reduces to

l1(z) = 16 dO'l(Z) II z-z

(2.16)

16. 9(Z2d_0'1z(Z) /2(z) = ...

with g(z) = ill d0'2(Z) .

e z-z

It has been shown in [Nil] that for Nikishin systems (2.13) all multiindices

. {k+l forj<q n = (n1, ... ,nm) Wlth nj = k r· - , 10r}>q

(2.17)

r E N and 0 ~ q ~ m, are normal. For such multi-indices the polynomials Qn are of exact degree

(2.18)

and all zeros are simple and contained in the interval .6. = .6.1 • Nikishin calls such systems weakly perfect.

A basic tool in Nikishin's investigations are algebraic Chebychev systems (AT-systems). A system (U1' ... , um) of functions continuous on an interval .6. = [a, b] ~ R is called an AT-system for the multidegree (n1,. ~., nm) E N m if any linear combination

P1(Z)Ul(Z) +.,. + Pm (z)um(z) , 1'; E lIn;, j = 1, ... , m (2.19)

has at most (nl + 1) + ... + (nm + 1) = Inl + m zeros in .6.. For nj = 0, j = 1, ... , m, this definition reduces to that of an ordinary T-system.


2.3 ANGELESCO AND NIKISHIN SYSTEMS (ANALYTIC

ASPECTS)

Two major results in the analytic theory of Angelesco and Nikishin systems were achieved by E.M. Nikishin in [Ni2] and by A.A. Gonchar and E.A. Rakhmanov in [GoRal]. We will review these results in the present section.

Let N ~ N m be an infinite sequence of multi-indices with the property that the m limits

nj I I j;;j -+ Cj as n -+ 00, n E N, (2.20)

exist. The constants Cj E R can be prescribed arbitrarily; the only restrictions are

Cj > 0, j = 1, ... , m, and Cl + ... + Cm = 1. (2.21)

Most important and, perhaps, of the greatest interest are close-to-diagonal sequences N ~ N m , which by definition are associated with the constants

1 Cl = ... = Cm =-.

m (2.22)

Besides the general assumption (2.2) and the special assumption (2.11) for Angelesco systems we now assume that the Radon-Nikodym derivatives of the measures I'j, j = 1, ... ,m, satisfy

dW(z) T >0 a.e.on aj (2.23)

for every j = 1, ... ,m. This condition is known in orthogonal polynomial theory as ErdOs' condition. Checking the proofs in [GoRal] shows that it seems possible to weaken (2.23), but some conditions are necessary in order to have a proper limit and equality in (2.26), (2.27) and (2.29) below.

With every polynomial Q we associate the zero-counting-measure vQ, which places a unit weight at every simple zero of Q and a weight equal to the multiplicity at every multiple zero of Q. A sequence of measures {Vn}~=l is said to converge weakly to v, written

• Vn--+V as n -+ 00,

if for every function f continuous in C we have

lim J f dVn = J f dv. n-oo


The logarithmic potential of a measure v is defined as

We recall that in (2.12) for Angelesco systems the m factors of Qn, which have their zeros in fl.j. have been denoted by Qjn, j = 1, ... , m. We assume that these factors (and thus the polynomial Qn itself) are monic.

Theorem 2.1. ([GoRal]) Let (2.11) and (2.23) be satisfied and let N = {n = (nl, ... , nm ) ~ Nm} be an infinite sequence of multi-indices such that (2.20) holds for given constants (Cl, ... , em). Then there exist m probability measures Vj, j = 1, ... , m, with

and m constants aj E R, j = 1, ... , m, such that:

(i) We have

1 • - vQ. ---+v,' n. ,ft , (ii) We have

as Inl- 00, n E N, for j = 1, ... , m.

locally uniformly for z E C \ fl.j, and

for j = 1, ... , m.

(iii) If we set

then we have

lim IIQjnQnll~l(n~.) = exp(aj) Inl-oo,nEN ,

Wj (z) := 2cj p( Vj , z) + L CiP( Vi, z) , i#j

I ( p.) Illlni lim ftj - Q,n (z) = exp(Wj(z) - aj)

Inl-oo,nEN n

locally uniformly for z E C \ (fl.l U ... U fl.m) and j = 1, ... , m.

(2.24)

(2.25)

(2.26)

(2.27)

(2.28)

(2.29)


Remark: Set

D+ := {z E C \ (~1 u ... U ~m)j W;(z) < aj} ,

D_ :={ZEC\(~lU ... U~m)j W;(z»aj},

j = 1, ... , m. Then (2.29) implies that p;n/Qn converges locally uniformly in D+ to pj for every j = 1, ... , m, and the approximant diverges to infinity for z E D_. The domain D_ may be empty, for instance, if all intervals ~1' ..• ,.6.m are of equal length. However, there are also examples with D_ 1:- 0. A detailed discussion of these cases is contained in [GoRal] for m=2.

By Theorem 2.1 the problem of determining the asymptotic behavior of multiple orthogonal polynomials Qn in the weak sense is solved and the convergence behavior of simultaneous rational approximants is completely described for Angelesco systems satisfying condition (2.23). It only remains to learn more about the measures IIj and the constants aj,j = 1, ... ,m, and to give a characterization of these quantities that is independent of the asymptotic problems considered in Theorem 2.1.

The measures IIj, j = 1, ... , m, can be described by a potential theoretic minimality problem: Let M = M(Cl' ... ' em) be the set of probability measures II with the properties

SUpp(lI) ~ .6.:=.6.1 U ... U ~m (2.30)

1I(~j) = Cj for j = 1, ... ,m.

Every measure II E M can be represented as

where 1

IIj := -111 4 ., j = 1, ... ,m, Cj J (2.31)

are probability measures with supp(lIj) ~ ~j.

Consider the kernel function

{ 2 log Iz ~ yl for (z,y) E ~j x ~j, j = 1, ... ,m

K(z,y) = 1 log-I--I for (z,y) E.6.j X ~i, j 1:- i

z-y

(2.32)

which is defined on .6. x d, and define the energy for II E M by

J(II):= J J K(z, y) dll(z)dll(Y)· (2.33)

In [GoRal] the following theorem was proved.


Theorem 2.2. For any vector of constants (Cj )j=l satisfying (2.21) there exists a unique measure Vo EM = M(ct, ... , em) such that

I(vo) = inf I(v). /lEM

(2.34)

The probability measures Vj, j = 1, ... ,m, in Theorem 2.1 are given by

1 V· = -volA' .J Cj J

and the constants aj, j = 1, ... ,m, by

a· = min W·(x) J xEAj J

with functions Wj defined as in (2.28).

(2.35)

(2.36)

Remark: We always have supp (Vj) ~ t1j , but in general it may happen that

t1j := supp(Vj), j = 1, ... ,m, (2.37)

is a proper subset of t1j. In any case t1j is an interval.

In [GoRal] and [GoRa2] some further extremality properties, especially properties related to the functions Wj , j = 1, ... ,m, defined in (2.28) are proved. The methods applied in the investigation belong to the theory of logarithmic potentials. The methods for studying the existence of the measure Vo in (2.34) and for deriving special properties of these measures are variants of Frostman 's technique. However the problem is more complicated than in the classical case here because of the interdependency between the potentials Wj for different indices j.

With Theorems 2.1 and 2.2 the main results concerning weak asymptotics for Hermite-Pade polynomials of type II associated with Angelesco systems have been presented. We will come back to this topic in Section 4, where strong asymptotics for the same problem will be discussed.

We close the present section by stating and discussing Nikishin's result on the convergence of simultaneous rational approximants. For the proofs of Theorems 2.1 and 2.2 it is essential that all functions II, ... ,fm of an Angelesco system have defining measures 1'1, ... ,I'm on m intervals t11 ... , t1m

that are all disjoint. In this respect Nikishin systems represent the opposite extreme since now all functions II, ... ,fm have defining measures 1'1, ... ,I'm on the same interval t11.

For the case m = 2 Nikishin was able to prove the convergence of the simultaneous rational approximants.


Theorem 2.3 ([Ni2], Theorem 4). Let the functions It and 12 be defined by (2.16), and assume that the measure 1'1 = 001 has infinitely many points of increase. Then the sequence of diagonal simultaneous rational approximants

n=(k,k), kEN (2.38)

converges to It, 12, locally uniformly in C \ [a, b] as n -+ 00.

A main tool in the proof of Theorem 2.3 is the use of AT-systems and a detailed investigation of the quadrature properties of the zeros of the polynomials Qn.

In [Ni2] it has been shown that the denominator polynomial Qn, n = (k, k), is of exact degree Inl = 2k and that allzeros are simple and contained in the interval [a, b].

It has been mentioned in Section 2.1 that if Qn has only simple zeros, then these zeros have the quadrature property (2.10). In case of Nikishin systems all Christoffel numbers Cjk, k = 1, ... , Inl, j = 1,2, introduced in (2.10) are positive. The proof of Theorem 2.3, in [Ni2] is based on this fact.

A similar strategy would be impossible for a proof of Theorem 2.1 since in case of Angelesco systems the Christoffel numbers Cj k are oscillating for all zeros Xk of Qn that do not belong to fl.j , j = 1, ... ,m. Therefore, the proof of part (iii) of Theorem 2.1 has been deduced from the weak asymptotics for Qn, that has been established in part (ii) of Theorem 2.1.

In case of a Nikishin system the deduction of weak asymptotics for Qn is more complicated than in case of Angelesco systems. We will come back to this in the next section.

3 The Role of Different Sheets

Now the investigation of Hermite-Pade polynomials to systems (1.15) will be continued from a different perspective. We no longer assume that the functions in (1.15) are of Markov type, instead we make assumptions about the possibility of analytic continuations of these functions across the interval on which the defining measures J.'j, j = 1, ... ,m, live. If the functions have sufficient analyticity, then we can describe the asymptotic behavior of Hermite-Pade polynomials. It is natural to consider polynomials of both types simultaneously, where in case of type II polynomials we only consider the multiple orthogonal polynomial Qn. With the new approach we can investigate systems that fill the gap between Angelesco and Nikishin systems. We restrict ourselves to the case m = 2.


3.1 DEFINITIONS AND ASSUMPTIONS

For a, b E R, a < b, let n = n[a.b] denote the lliemann surface of the equation y2 = (z - a)(z - b). The two sheets of n will be denoted by So and S1, and they are assumed to be copies of C cut along [a, b]. If [a, b] is approached from {1m (zo) < OJ, then the limit points belong to So otherwise to S1. The canonical projection is denoted by 11" : n --+ C, and Zo and Z1 are coordinate functions on So and S1, respectively, i.e. we have 11" 0 Zj = ide, j = 0, l.

On S1 we consider a second cut with endpoints c, d E R \ (a, b), c i: d. By [c,d] we denote the segment that connects c and din R \ (a,b). If both points c and d are in R and on the same side of (a,b), then [c,d] is the usual interval, otherwise it is an interval on the lliemann sphere C.

We say that a function 1 belongs to the class A = A(a, b, c, d) if

(i) 1 is analytic in n \ z1(R \ (a,b)), and (ii) if the jump function

1 9(Z):= -2 . (J(zI) - I(Z2)), Z E [a,b],

11"1 (3.1)

which has an analytic continuation to (C \ R) U (a, b) because of (i), is analytic in C \ [c, d], has continuous boundary values on [c, d] from both sides, where we denote the continuation from {Im(z) > O} by 9+ and the continuation from {Im(z) < O} by g_, and the second jump function

1 h(z) := -2 .(9+(Z) - 9-(Z)), z E [c,d]

11"1 (3.2)

has only isolated zeros in (c, d).

It follows from (i) and (ii) that every 1 E A has the representation

l b 9(x)dx I(zo) -- + 1(000), for Zo E So,

a X - Zo (3.3a)

I(Z1) = I(zo) + g(z), for Z1 E S1 \ Z1([C, d])

with Zo E So and '/r(zI) = '/r(zo) = z. The function 9 can be represented as

Jd h(x)dx {g(00)' g(z) = --+

c x - Z 0

if 00 ¢ [c, d] (3.3b)

else.

A formal comparison of (3.3) with (2.16) shows that two functions 11, 12 E A( a, b, c, d) form a Nikishin system if

11(000) = 12(000) = 91(00) = 92(00) = 0, and (3.4)

91(X) ~ 0, g2(X) ~ 0 for x E (a,b).


A comparison of (3.3a) alone with (2.1) and (2.11) further shows that two functions It E A(aI, 61 , C1, dt} and hE A(a2, 62 , C2, d2) form an Angelesco system if (3.4) holds and the two intervals [a1, 6tl and [a2'~] are disjoint.

We will now extend our investigation in such a way that intermediate cases between Nikishin and Angelesco systems can also be included.

Let I; E A(aj, 6j, Cj, dj), j = 1,2, we then distinguish three cases:

Case I: [aI, 61] n [a2, 62] = 0 or a single point, Case II: [aI, 61] n [a2, 62] =: [a3, 63], a3 < 63 ,

Case III: [a1,6d = [a2,62].

In order to keep the notation simple, we assume that a1 ::5 a2, 61 ::5 62 ,

aj < 6j ::5 Cj < dj ::5 00, j = 1,2.

Case I corresponds to Angelesco systems or to the case of two intervals meeting in one point, which has been investigated by Kalyagin [Kal]. Case III corresponds to Nikishin systems, and case II is the intermediate case, which will be the main object of investigation in the present section. Case II fully or partly covers the other two cases as limiting cases.

The analysis is based on a special Riemann surface which we define now. If we consider simultaneous analytic continuation of the system of functions (11,12) starting from 000 = 00, and follow this process over at least two sheets, then this results in a Riemann surface ii = ii(It,/2,00) with four sheets 80, ... ,83. The sheets 80 and 8 1 are connected across [ai, a2], 80 and 82 across [bI, b2], and the sheets 80 and 83 across [a3, b3]. A fifth sheet 84 is then added. By connecting 84 and 83 across [CI, d1] U [C2, d2]. The final Riemann surface with five sheets is denoted by'R, = 'R,(It, 12, 00). It is of genus 9 = lor 9 = 0 depending on whether [Cb d1] U [C2, d2] is connected or disconnected. By Zo, ••• , Z4 we denote the coordinate functions on the sheets 8 0 , ••• ,84 and by 1r : 'R,' _ C the canonical projection of'R, on c.

It follows from the assumptions made for the functions Ii in A( aj, 6j, Cj , dj ), j = 1,2, that each of the two functions It and 12 can be lifted to the first four sheets 80 , ••• ,83 of'R, and we have continuous boundary values over [C1, dtl U [C2, d2] if the sheet 84 is approached from 83. It follows from the structure of 'R, that It has identical values on the two sheets 80 and 82 and likewise on the two sheets 81 and 83. In the same way 12 has identical values on the two sheets 8 0 and 81 and on 82 and 83. The representations (3.3) are also valid on 'R,. They will be very useful in the calculation of the special determinants that now follows.

lt turns out that the determinants

(3.5)


are of fundamental importance, where Ie = 2,3, {FI , ... , FA.} ~{1, iI, f2}, {il, ... ,jA.} ~{O, ... ,4} with iI = 0, iI < ... < jle. Here Zo, ... , Z4 again denote the local coordinates on So, ... , S4. Since we have 1r(zo) = ... = 1r(Z4) = Z the determinants Die are locally analytic functions of z. We consider only determinants on adjacent sheets starting with So. Altogether there are 13 different determinants. We give explicit calculations for the first two of them, for the others only the results:

1 iI(zo) h(zo)

Da(l, iI, hi Zo, ZI, Z2) = 1 iI(Zl) h(Zl)

1 b(Z2) h(Z2)

1 b (zo) h(zo)

0 9l(Z) 0 9l(Z)92(Z),

0 0 92(Z)

1 b (zo) h(zo)

Da(l,fl,f2i Zo,Zl,za) = 1 b (Zl) h(zI)

1 b(Z3) h(za)

1 b(zo) h(zo)

0 9l(Z) 0 9l(Z)92(Z),

0 9l(Z) 92(Z)

Da(l,b,hi zo,z2,z3) = -9l(Z)92(Z)

Da(l, b, hi Zo, Za, Z4) = 9l(Z)hl(z) - 92(Z)hl (z), Z E [cl, dl] U [C2' d2],

D2(I,fj i Zo, Zj) = D2(1, f;; Zo, Z3) = 9j(Z), j = 1,2,

D2(I,biZo,Z2) =D2(I,h;zo,zl) == 0,

D2(b,hi zo,zI) = -h(Zo)9l(Z),

D2(b,hi Zo,Z2) = b(zo)92(Z),

Asymptotics of Hermite-Pa.d6 Polynomials 147

From the assumptions made with respect to It and 12 it follows that none of the determinants, except D2(1, It, zo, Z2) and D2(1, 12, zo, Z1), is identically zero.

3.2 WEAK ASYMPTOTICS FOR TYPE I AND TYPE II POLYNOMIALS

From the fact that 11 of the 13 determinants in the last subsection do not vanish identically, weak asymptotics can be deduced for Hermite-Pade polynomials of botIi types.

In order to exclude complicated geometric structures of the lines on which the zeros of the polynomials asymptotically cluster, we restrict the class of admissible functions (11,12). As before we assume that case II holds and in addition it is now assumed that

length [a1' 61] = 61 - a1 = 62 - a2 = length [a2' 62], (3.6)

For such systems of functions (It, 12) we can describe the asymptotic behavior of polynomials in the weak sense. However, the proof of these results is too long to be included here and will be published elsewhere. The method of proof has been described in [St3] for the case of Pade approximants.

Theorem 3.1. Let a1 < a2 < 61 < 62 satisfy (3.6), and let the two functions !;, j = 1,2, 6e defined by

/j(z) = l bj gj(z)dz, Z E C \ [aj, 6j], OJ z- Z

(3.7a)

and .( ) -14j hj(z)dz g, Z - , Cj Z - Z

(3.7b)

Cj, dj E R, Cj #- dj ,. [Cj, dj] n [aj, 6j] = 0 for j = 1,2, and the functions hI and h2 are assumed to be continuous on [Cj, dj ], j = 1,2, and have only isolated zeros in (Cj, dj ), j = 1, 2.

For the diagonal sequence

N = {n = (k,k,k)}~1 ~ N 3 (3.8)

and the system of functions (1,11,12) we consider type I Hermite-Pade polynomials

Pon(l/z), PIn(l/z), P2n(l/z), n E N,

expanded around z = 00, together with the associated inverse polynomials

Pjn(Z) = zJ:pjn (l/z), j = 0,1,2, n E N, (3.9)


and the sequence Qn(Z), n = (k,k), kEN, (3.10)

of multiple orthogonal polynomials with respect to the system of functions (It, 12).

There exist three non-empty, disjoint domains Bj, j = 1,2,3, for which the union is dense in C, i.e.

We have (a1' a2) ~ B1, (bt, b2) ~ B 2, (a2' bd ~ B 3, and there exist three probability measures IIj, j = 0,1,2, with

supp (liD) = (B1 n B2) U (B2 n B3) U {B3 n Bd U

(B3 n ([ct, d1] U [C2, d2]»,

supp (Ill) = [alo a2] U (B2 n B3) U (B3 n ([C1' d1] U [C2, d2]», (3.11)

supp (112) = [bt, b2] U (R1 n B3)(B3 n ([Clo d1] U [C2' d2]»

such that for n = (k, k, k), j E N, we have

1 • "k"P; .. --+ IIj, as k - 00. (3.12)

for j = 0,1,2. There exists further a probability measure .,p with

(3.13)

such that for n = (k, k), kEN, we have

1 • 2k "Q .. --+.,p, as k - 00. (3.14)

Remarks. (1) The main information in the Theorem 3.1 is the existence of the limits (3.12) and (3.14). However, with this result we still do not know much about the structure of the limit distributions liD, "l, 112, and .,p. In the next theorem a potential theoretic characterization will be given for the measures liD, "1, 112 and .,p which is independent of Hermite-Pade polynomials. In this characterization four of the five sheets of the Riemann surface 'R = 'R(It, 12, 00), introduced in the last subsection, have simultaneously to be taken into consideration. From (3.3) and (3.7) it readily follows how the functions It, 12 can be lifted to a sub domain of'R consisting of So U ... U S3


(2) The domains Bt, B 2 , B3 are symmetric with respect to R and to the line perpendicular to R through (1/2)(a2 + b1). In Figure 3.1 a typical situation is sketched.

Figure 3.1

(3) The measures Vo, v!, and V3 are not restricted to the real line, and therefore also the polynomials Pjn, j = 0,1,2, will have zeros outside of R with an asymptotically positive density.

(4) The convergence in weak· topology in (3.12) and (3.14) does not imply that asymptotically all zeros have to cluster on the support of the limit measures.

(5) As in Theorem 2.1, from the weak· limits in (3.12) and (3.14) one can deduce the convergence of the sequences {IPjnI1/k}k:l' j = 0,1,2, and {IQnI1/k}f=1' i.e. n-th root asymptotics for these polynomials, if the polynomials are appropriately normalized. However, since now not all zeros are restricted to supp (Vj), j = 0,1,2, or supp (1/J), the convergence in general holds only in capacity.

In the next theorem we present the potential theoretic characterization of the measures Vo, V!, V2 and 1/J that has been mentioned in Remarks (1). We use the notations introduced in Theorem 3.1.

Theorem 3.2. (a). There exists a unique pair (Go, vo) of a domain Go ~ R and a probability measure Vo in C such that

(i) 000 EGo (ii) We have supp(vo) = 7I"(oGo) =: Ko. and Ko consists of piecewise analytic arcs.


(iii) The projection 7r : 'R, - C is 2-valent on Go \ 7r-1(Ko), and 3-valent on Go n 7r-1(Ko). (iv) Let G ~ 'R, be a domain such that Go ~ G and assume that G possesses a Green function. By 7r-1(lIo) we denote the lifting of 110 onto'R, and define

h(z):= [ gG(z, z) d(3cooo - 7r-1(lIo»(z), ./aD

(3.15)

then the restriction of h to Go is independent of the choice of G.

(b). The domain Go and the measure 110 are symmetric with respect to R.

(c). The measure 110 in (i) is identical with the first measure that appears in (3.12).

Set Bj = 7r(Go n 8j), j = 1,2,3, where 81,82 ,8a are sheets of'R, as defined before, and set

(3.16)

The functions hj can be represented as

hj(z) = 2 [ 9Bj(Z, z) d1/;(z) for Z E B j and j = 1,2,3, (3.17) JBj

where 1/; is a probability measure on [aI, b2]. Then the BI, B2, Ba are the three domains appearing in Theorem 3.1,

(3.18)

112 .- 21/;1[61,621 + 1I01BsnB1'

are the second and third probability measure in (3.12), and1/; is the probability measure in (3.14).

Remarks. (1) It follows from assertion (iii) that Go covers C \ Ko two times and Ko only once. The boundary aGo lies doubly over Ko. This implies that aGo consists of piecewise analytic arcs and these arcs appear in pairs, the members of each pair have the same ground path, and one member has Go on its right and the other one on its left side.

If 110 is lifted to Go, then it appears on 'R, three times; two times on aGo and once inside of Go.

(2) The independence of function (3.15) from the choice of the domain G implies that h(z) = 0 for all z E G \ Go.

Let f1 and f2 be two arcs in aGo with the same ground path, and let 8~1 and 8~2 be the normal derivatives of h on f1 and f 2, respectively,


directed towards Go. The fact that on r1 and r2 the measure 1I"-1(vO) is identical implies that

l) l) -l) h(zt} = -l) h(Z2) for Zj E rj and 1I"(zt} = 1I"(Z2)' (3.19)

n1 n2

Further we know that there is a third arc rs in Go on which h has an edge, and the difference of the right and the left side normal derivative of h is equal to the value of (3.19). The identity (3.19) generalizes the symmetry property of the convergence domains in Pade approximation theory (see [Stl], [St2], [GoRa3]).

With Theorem 3.2 we have achieved a characterization of the asymptotic zero distributions lIQ, V1, 112, and 1/J that is independent of Hermite-Pade polynomials. The measures Vo, VlI 112, 1/J are determined exclusively by the structure of the concrete Riemann surface 'R.

In closing this subsection we shall shortly discuss how the domains B1, B2, and Bs vary with respect to the overlapping interval [as, 6s], and how much the assumptions made with respect to the system off unctions (1, It, h) can be weakened.

If the length of the two intervals [aj, 6j], j = 1,2, is kept fixed, then there are two extreme cases. One is given by a1 < 61 = a2 < 62 , i.e. the two intervals [all 61] and [a2,62] overlap in just one point. In this special case the domain Bs is empty, and the Riemann surface 'R has only three sheets. The sheets 8s and 84 are missing.

The other extreme case is given by [all 61] = [a2' ~]. Here the sheets 81 and 82 are missing and the domains B1 and B2 are empty. The overlapping interval [as, 6s] is equal to the two original intervals.

If the overlapping interval is slowly growing, starting from the single point 61 = a2, then the domain Bs grows at the same time, and the two domains B1 and B2 are shrinking. Figure 3.1 gives a typical picture of the constellation in the first growing period. Here all three domains are simply connected and Bs n (R.\ (as,~» = 0. Then after a critical length of [as,6s], Bs is doubly connected. Only in this later period l)Bs may contain parts of the set [ClI d1] U [C2' d2]. At the end of the variation, when [all 61] = [a2'~] = [as,6s], the two domains B1 and B2 are empty and Bs = C \ ([ct, d1] U [C2' d2]).

If the special assumption (3.6) is dropped then similar results hold, but it is more complicated to determine and to describe the sets Bt, B2 , B3.

For Theorem 3.1 we have made assumptions about It and 12 that involve analyticity on the four sheets 80, ... ,83 of 'R(It, 12, 00) and a certain boundary behavior on the common boundary of 83 and 84 , However, a check of the proof of Theorem 3.1 and 3.2 shows that it is only necessary to have the properties assumed on 80, on 83 n Go, on the common boundary between 80 and 8j, j = 1,2, and on the common boundary between 83 n Go and 84.


3.3 A COMPARISON WITH RESULTS FROM SECTION 2

There are fundamental differences between the type of assumptions made in Section 2 with respect to Angelesco and Nikishin systems and the assumptions made in the present section. While in Section 2 the positivity of the measures I-'j, j = 1, ... , m, in (2.1) was of basic importance, it is now the possibility to continue the functions II and h analytically across the intervals [a1' b1] and [a2' b2] into adjacent sheets of an appropriately defined Riemann surface 'R.

If, however, the two measures 1-'1 and 1-'2 in (2.1), m = 2, have an analytic positive density function on [aj, bj]' j = 1,2, then the assumptions of both sections can be satisfied. Hence, such functions are admissible for both approaches, and in their case we can compare the results from the Subsections 2.3 and 3.2.

We start by a comparison with Nikishin systems. As in (2.16) we assume m = 2. In the terminology of Subsection 3.1 we set

a=aj, b=bj, c=Cj, and d=dj for j=I,2. (3.20)

The measures 0"1 and 0"2 in (2.16) are now defined by

dO"l(Z) := gl(z)dz, z E [a,b], (3.21)

where we assume that gl is analytic in C \ [c, d], and

(3.22)

The function h2 is assumed to be continuous on [c, d] and to have only isolated zeros in (c,d). The functions II and h are then defined by (2.16) with

g2(Z) g(z):= -,

gl(Z) (3.23)

where g2 is now assumed to be analytic in C \ [c, d] and it has to have h2 as a jump function on [c, d] (compare (3.3b». It is easy to check that under these assumptions the definitions (2.16) and (3.7) coincide.

If we accept that Theorem 3.1 can be extended to the limiting case [a1,b1] = [a2,b2] ,then we learn from Theorem 3.1 that the Hermite-Pade polynomials of both types have weak asymptotics, and from Theorem 3.2 we can derive a unique characterization of the asymptotic distributions IIj, j = 0,1,2, and ,po We have

liD = III = 112 =: II,

(3.24) SUpp(lI) = [c, d], supp(,p) = [a,b],


and the two probability measures v and ,p satisfy the two equations

v = J WC\[e,dj,.,d,p(z)

(3.25)

,p = ~ [3wc\[a,bj ,(X) + J WC\[a,bj,.,dv( z)] where wG,z denotes the harmonic measure of the domain G ~ C with

respect to the point z E G. The pair (v,,p) is uniquely determined by (3.25) and the second line of (3.24).

A comparison of Theorem 3.1 and 3.2 with Theorem 2.1 and 2.2 is not possible in a strict sense since for Angelesco systems both intervals [aI, bl ] and [a2' b2] have to be disjoint (compare (2.5». However, the analysis for two disjoint intervals is more or less the same as that for two adjacent intervals, and the latter case is covered as a limiting case by Theorem 3.1. (We restrict ourselves to m = 2.)

As already mentioned earlier, in case of two adjacent intervals [aI, bl ] and [a2' b2] the Riemann surface n = n(il, 12, 00) has only the three sheets So, Sl, and S2. From the analyticity assumptions with respect to il and 12 we only need the following in this case: The functions il and 12 have to be analytic on So, and they must have continuous jump functions (3.1) on [aj, bj], j = 1,2. These jump fl,lnctioI).s must not have too large zero sets. If, for instance, the jump functions have only isolated zeros in each interval (aj,bj), j = 1,2, then the results of Theorem 3.1 and 3.2 will hold.

In Theorem 2.2 the measures Vj, j = 1, ... , m, were determined by a minimality problem. This is only possible if the measures Jl.j in (2.1) are positive. In this respect the approach in Subsection 3.2 is more general, and opens new possibilities. One of its main advantages is the possibility to study systems (1, il, 12) with overlapping intervals.

4 Strong Asymptotics for Angelesco Systems

Up to now only weak asymptotics have been investigated. In this last section we give asymptotic formulas in the strong sense of (1.14) for HermitePade polynomials of type II associated with Angelesco systems. The result is a generalization of the well-known Bernstein-Szego Theorem about the asymptotic behavior of orthogonal polynomials on an interval or the unit circle. Here, only the basic ideas of the proof ofthis result can be presented. For a full treatment see [Ap3] and [Ap4]. A complete proof is contained in [Ap4].


4.1 THE BERNSTEIN-SZEGO THEOREM AND ITS

EXTENSION TO m > 1

Let p be a weight function on A = [-1,1]' define the measure I' by dl'(z) = p(z) dz, z E A, and let Q" E II", k = 0,1, ... , be the k-th monic orthogonal polynomial with respect to 1', i.e. Q" satisfies relation (2.3).

Theorem 4.1 (Bernstein-Szeg8). Let p be a weight /unction on A = [-1,1], that satisfies the Szeg6 condition

11 lnp(z)dz > -00.

-1~ (4.1)

Then the orthogonal polynomials Q,,(z), satisfy the asymptotic formula

Q,,(z) Cl>"(z) -+ F(z) as k -+ 00 (4.2a)

locally uniformly in the domain 0 := C \ A, and on the interval A we have

II Qn(z) {( Cl>(z) )" ( Cl>(z) )" }II 1Cl>(z)I" - 1Cl>(z) I F(z) + 1Cl>(z) I F(z) = 0(1), L2,p(A)

where

Cl>(z) =

F(z) =

z+~ 2

D(z) D(oo)'

and

as k -+ 00,

Cl>(Z)1/2 {~11 lnp(t)dt } D(z) = [z2 _ 1]1/4 exp - 41r -1 (z - t)v'f'="'t2 .

(4.2b)

(4.3)

In (4.2b) L2.p(A) denotes the L2-norm with respect to the measure dl' = pdz on A, and 0 denotes the small Landau symbol.

It has been outlined earlier that the classical orthogonality corresponds to the case m = 1 in the multiple orthogonality (2.4).

Let now m > I, and let {Aj := [aj, bj]}j=1 be a system of m nonintersecting real, compact intervals Aj ~ R. Let further pj, j = 1, ... , m, and dl'j := pj(z)dz, z E AjJ j = 1, ... , m, be systems of positive weight functions and measures on {Aj}. Since the intervals All ... , Am have been


assumed to be disjoint, {I'j }J;l defines an Angelesco system of m Markov functions.

For the diagonal multi-indices n = (k, ... , k) E Nm, kEN, we consider the monic multiple orthogonal polynomials Qn E 'll"lnl, Inl = mk, defined by (2.4) with respect to the system of measures {I'j }j=l. With the weight functions Ph j = 1, ... , m, the orthogonality relations (2.4) have now the form

r Qn(Z)Zipi(z)dz=O, ;=0, ... ,k-1, j=l, ... ,m. (4.5) }I}.i

As a generalization of the Bernstein-Szego Theorem we have

Theorem 4.2 ([Ap3], [Ap4]). Let {aj}j=l be a system of non-intersecting intervals, i.e. ai n aj = 0 for; #:- j, i, j = 1, ... , m, and let {Pj }j=l be a system of weight functions, that satisfy the SzeglJ condition (4.1), i.e.

1 lnp(z)jdz . > -00 for J = 1, .. . ,m. I}.i Vlaj - zllbj - zl

Then there exists a system of subintervals {anj=l, aj ~ aj, j = 1, ... , m, such that

Qn(z) .Inl(z) -- F(z) as Inl- 00 (4.6a)

locally uniformly in. 0" := C \ ~ .. , a" := ai U ... U a~, and

Qn(z) {( .(z) )Inl ( .(z) )Inl } 1.(z)llnl - 1.(z)1 F(z) + 1.(z)1 F(z) = 0(1),

L 2 ,p(l}.j)

as Inl-oo. (4.6b)

Here. is a branch of an algebraic function of order m + 2 with branch points at the end points of the intervals aj, j = 1, ... , m, and F is an analytic function in 0", which is uniquely determined by the weight functions {Pj(Z)}j=l·

Remark. For m = 1, Theorem 4.2 reduces to Theorem 4.1. However, even on a geometrical level the behavior of the polynomials Qn is more complicated for m > 1. So for instance a new main point arises with the phenomenon that the zeros of Qn will fill up only a" = ai U ... u a~, which may be a proper subset of a = al U ... u am.

4.2 A SKETCH OF A PROOF FOR THE BERNSTEIN-SZEGO

THEOREM

In order to have an orientation for the proof of Theorem 4.2 and to get acquainted with the functions that will appear in that proof, we reproduce


the basic ideas of a proof for Theorem 4.1 in the spirit of Widom's approach in [Wi]. .

The orthogonal polynomial Q" is the solution of the extremal problem

[11 IQ,,(z)12p(Z)dZ = P.(";>=!~.+ ... [: IP,,(z)12p(z)dz, (4.7)

i.e. the polynomial Q" is minimal in the L2,p-norm among all monic polynomials of degree k.

Let us consider the function ~ which is defined by the properties:

(i) (ii)

(iii)

~ is analytic in C \ [-1,1], ~(z)=z+ ... nearz=oo, and 1~(z)1 = const. for z E [-1,1].

(4.8)

Since I~I is constant on [-1,1], we can divide the integrands on both sides of (4.7) by 1~(z)12" without changing the equality in (4.7). This leads to the extremality problem

t 1 r;(~~) 12 p(e)ldel = ~ t 1.r(eWp(e)ldel, (4.9)

in a finite-dimensional hyperplane .r" of analytic functions in C \ [-1,1]. The integrals in (4.9) extend over the entire boundary of the domain n = C \ ~. The minimum in (4.9) is taken over all functions .r E .r" that are defined by the properties

. P,,(z) (1) .r(z) = ~(z)'" P,,(z) = z" + ... E II", and

(4.10) (ii) .r( 00) = 1.

If now k (the dimension of the hyperplane) tends to 00, we can expect that the extremal element of the finite-dimensional extremal problem (4.9) approaches the extremal element of the corresponding infinite-dimensional extremal problem. With the H2,p-space on n = C \ [-1,1] we have the appropriate infinite-dimensional space of analytic functions for the solution of the infinite-dimensional extremal problem.

We expect that

Q,,(z) ~"(z) --+ F(z), zEn, as k - 00, (4.11)

if F is the solution of the infinite-dimensional extremal problem, i.e. if

(4.12)

The limit (4.11) implies the Bernstein-Szego strong asymptotic formulas (4.2a,b) for orthogonal polynomials Q".


If we want to obtain the explicit form of the function F, we have to reduce the extremal problem (4.12) to a boundary problem for analytic functions. (The reduction can be done by a conformal mapping of n onto the exterior of the unit disk and then using Bessel's inequality for the Fourier coefficients.)

The extremal function F is then uniquely characterized by the properties:

(i) F is analytic in n = c \ [-1,1], (ii) F( (0) = 1 (4.13)

(iii) IF(z)12p(z)[1 - z2]-1/2 = const. for z E [-1,1].

It finally follows that the Zukovski function (4.3) satisfies the defining properties (i)-(iii) in (4.8) for~, and this completes our sketch of a proof of Theorem 4.1.

4.3 THE SCHEME OF THE PROOF OF THEOREM 4.2 IN

CASE OF m = 2

Set m = 2 and n = (k, k), kEN. From (2.12) we know that Qn can be factorized as

where each of the two polynomials Qjn is supposed to be monic and to have all its zeros in ~j, j = 1,~. The zeros are all simple.

The system (2.4) of orthogonality relations can be reduced to

I Q1,n(z)Zi IQ2,n(z)IP1(Z)dz = 0 JAl i = 0, ... ,k-l

From these two separate relations it follows that the two polynomials Qjn, j = 1,2, satisfy the following system of extremal problems:

il IQ1,n(zWIQ2,n(z)lp1(Z)dZ = p=~~ ... il IP(zWIQ2,n(z)IP1(Z)dz

(4.15)

L2IQ2,n(z)12IQ1,n(z)lp2(Z)dZ = P~~ ... L2IP(zWIQ1,n(z)IP2(Z)dZ

The system (4.15) of extremal problems will be transformed into an equivalent system of boundary value problems for analytic functions analogously to (4.12). For this aim suppose that there exist functions ~1 and


~2 satisfying the conditions

(i) (ii)

(iii)

~j is analytic in OJ := C \ Aj, j = 1,2, ~j(z) = Z + ... near Z = 00, j = 1,2, and 1~1(Z)I2I~2(Z)1 = const. for Z E A1 1~2(Z)121~1(Z)1 = const. for Z E A2.

(4.16)

By using the same reasoning as in Subsection 4.2, we can expect that

as Ie: -+ 00 for Z E OJ, j = 1,2, (4.17)

where the two functions F1 and F2 are now determined by the system of extremality problems:

AB in (4.9), the integration extends in- (4.18) over both parts of the boundary of 0 1 and O2 • A careful study of (4.17) then leads to (4.6a) and (4.6b).

In order to support the heuristic reasoning given here, we have to study the question of existence for the functions ~j and Fj, j = 1,2, more closely. We start with a construction of the functions ~1 and ~2.

Let 'R, be a Riemann surface with three sheets So, S1, S2. The sheets So and S1 are connected across the interval A1 and the sheets So and S2 across A2. The surface is of genus 9 = 0, and it is practically the same one as that used in Section 3, only that now the sheets S3 and S4 are missing since A1 n A2 = 0.

On 'R, we consider a rational function ~, which is uniquely determined (since the genus of'R, is zero) by the divisor

~(Z)IOOl Z + .. . ~(Z)IOO2 = Z + .. . ( 4.19) ~(z)looo = az-2 + ...

By Zo, Z1, Z2 we denote the coordinate functions on So, S1, S2, respectively. Consider the branches

~j(Z)=~(Zj), ,z=1I"(Zj), j=0,1,2,

of~. The two branches ~1 and· ~2 satisfy the three conditions (i)-(iii) in (4.16). Indeed, (i) and (ii) are evident. In order to show (iii) we consider the

Asymptotics of Hermite-Pa.de Polynomials 159

product c)Oc)l c)2, which is a single-valued function analytic on C. Hence, by Liouville's Theorem it is equal to a constant a, i.e.

On the intervals aj, j = 1,2, we have Ic)j I = le)o I since c) is a realsymmetric function. This proves (iii) in (4.16).

Next, we study the existence of the functions F1 and F2, which determine the strong part of the asymptotics. The functions are solutions of the system (4.18) of extremal problems. As in the case of orthogonal polynomials, it can be shown that (4.18) is equivalent to the following system of boundary problems:

(i) Fj is analytic in OJ, and Fj(z) j 0 for all Z E OJ, j = 1,2, (ii) F1(oo) = F2(OO) = 1 (4.20)

(iii) IF1(z)I2IF2(z)lp1(Z)[(Z - at}(b1 - Z)]-1/2 = const. for ZEal, IF2(z)I2!F1(Z)lp2(Z)[(Z - a2)(b2 - z)]-1/2 = const. for Z E a 2,

where aj = [aj,bj]' j = 1,2.

For these boundary problems we can describe the solutions in case of the special weight functions

v'(Z - a·)(b· - z) Pj,.(Z) = ()J, j=I,2

p. Z (4.21)

where •

P.(z) = b. II(z - Ci), s = 1,2, ... i=l

is an arbitrary polynomial of degree s with real coefficients. Let us consider on the Riemann surface n the rational function which is

defined by the divisor:

F(cm) = 0, ,i = 1, ... , s,

where Cm is the point on So lying over Ci. The branches

Fj(z) := f(zj), j = 0, 1,3,

are algebraic functions, and it can be verified that F1 and F2 satisfy (4.20).

Indeed, (i) is satisfied, and (ii) is satisfied after normalization, which can be done for each branch separately. In order to prove (iii) we consider the product FoFlF2' which is a single-valued function and has its only pole at infinity. This pole is of order m and the highest coefficient of its principle

160 A.!, Aptekarev, H. Stahl

part is b •. Further there are s zeros at the points Ci, i = 1, ... , s. Hence the product is a polynomial equal to p., i.e.

Fo(z)Fl (z)F2(Z) == P.(z).

On the intervals Il.j we have IFj I = /Fol, j = 1,2. This proves (iii) in (4.20).

After the solution for special weights (4.21) is obtained the solution for general weights can be obtained by approximation.

What we have described here is only a sketch of the main step of the proof of Theorem 4.2. Major difficulties have been left out, among them the determination of the subintervals Il.j ~ Il.j, on which the asymptotic distribution ofthe zeros of Qjn, j = 1, 2 lies, and we have given no attention to the difficulties that arise by extending the investigations to more than 2 functions. A complete proof can be found in [Ap3] and [Ap4].

References [Am] Ambraladze, A.U., On convergence of simultaneous Pade ap

prozimants, Soobjenya ANGSSR (1985), 119(3), 467 - 480, (in Russian).

[Angl]

[Ang2]

[Ang3]

Angelesco, A., Sur'l'approximation simultanee de plusieurs integrales definies, C.R. Paris 167(1918), 629 - 631.

Angelesco, A., Sur deux extensions des fractions continues algebriques, C. R. Paris 168(1919), 262 - 265.

Angelesco, A., Sur certains polynomes orthogonaux, C. R. Paris 176(1923), 1282 - 1284.

[ApI] Aptekarev, A.I., Convergence of rational approximations to a set of exponents, Moscow Univ. Math. Bull. 36(1) (1981),81 -86.

[Ap2] Aptekarev, A.I., Pade approximation for the system {lFl (1, c, ..\iZ)j 1 ~ i ~ k}, Moscow Univ. Math. Bull. 36(2) (1981), 73 - 76.

[Ap3] Aptekarev, A.I., Asymptotic behavior of polynomials of simultaneous orthogonality and of a system of extremal problems for analytic functions, Preprint No. 168, Keldysh Inst. Appl. Math. Acad. Sci. USSR, Moscow (1987), (in Russian).

[Ap4] Aptekarev, A.I., Asymptotics of simultaneous orthogonal polynomials in the Angelesco case, Mat. Sb. 136(1), (1988), English transl. in Math. USSR Sb. 64(1) (1989), 57 - 84.


[ApKall] Aptekarev, A.I., Kaljagin, V.A., Analytic properties of twodimensional P-fraction expansions with periodical coefficients and their simultaneous Padt-Hermite approximants, In: Rational Approximaton and its Application in Mathematics and Physics, Lancut 1985, (J. Gilewicz, M. Pindor and W. Siemaszko, Eds.), Springer, Lecture Notes 1237, SpringerVerlag, Berlin (1987), 145 - 160.

[ApKal2] Aptekarev, A.I., Kaljagin, V.A., Analytic behavior of an n-th degree root of polynomials of simultaneous orthogonality, and algebraic junctions, Preprint No. 60, Keldysh Inst. Appl. Math. Acad. Sci. USSR, Moscow (1986), (in Russian), MR 88f:41051.

[Ba] Baker, A., A note on the Pade table, Proc. Kon. Akad. v. Wet. A'dam Ser. A 69 = Indag. Math. 28(1966), 596 - 601.

[B-GrM] Baker, Jr. G.A., Graves-Morris, P.R., Pade Approximants, Part II: Extension and Applications, Encycl. of Math. and its Applies. vol. 14, Addison-Wesley, London 1981.

[BaLu] Baker, Jr. G.A., Lubinsky, D.S., Convergence theorems for rows of differential and algebraic Hermiie-Padt approximants, J. Compo and Appl. Math. 18(1987),29 - 52.

[Be] Beukers, F., Padt approzimation in number theory, In: Pade approximation and its applications, Amsterdam 1980, (M.G. de Bruin and H. 'van Rossum, Eds.), Springer Lecture Notes 888, Springer-Verlag, Berlin (1981), 90 - 99.

[deBrO] de Bruin, M.G., Some aspects of simultaneous rational approximation, Banach International Math. Centrum, lecture (1987).

[deBrl] de Bruin, M.G., Three new examples of generalized Padt tables which are partly normal, Dept. of Math., Univ. of Amsterdam, report 76 - 11, Amsterdam (1976).

[deBr2] de Bruin, M.G., Some explicit formulas in simultaneous Padt approximation, Lin. Algebra and its Applies. 63(1984), 271 -281.

[deBr3] de Bruin, M.G., Simultaneous rational approximation to some q-hypergeometric jUnctions, to appear.


[deBr4] de Bruin, M.G., Convergence of some generalized continued fractions, Dept. of Math., Univ. of Amsterdam, report 79 - 05, Amsterdam (1979).

[deBr5] de Bruin, M.G., New convergence results for continued fractions generated by four-term recurrence relations. J. Compo and Appl. Math. 9(1983), 271 - 278.

[deBr6] de Bruin, M.G., Some convergence results in simultaneous rational approximation to the set of hypergeometric functions hFl(ljcjjz)j1:::; j:::; n}, In: Pade Approximation and its Applications, Bad Bonnef 1983, (H. Werner and B.-J. Bunger, Eds.), Springer Lecture Notes 1071, Springer-Verlag, Berlin, 1984, 12 - 33.

[deBr7] de Bruin, M.G., Generalized Pade tables and some algorithms therein, In: Proc. of the first French - Polish meeting on Pade approXimation and convergence acceleration techniques, Warsaw 1981, (J. Gilewicz, Ed.), CPT-81/PE 1354, CNRS Marseille, 1982.

[deBr8] de Bruin, M.G., Simultaneous Pade approximation and orthogonality, In: Polynomes Orthogonaux et Applications, BarLe-Duc 1984, (C. Brezinski, A. Draux, A.P. Magnus, P. Maroni and A. Ronveaux, Eds.), Springer Lecture Notes 1171, Springer-Verlag, Berlin, 1985.

[deBrlO]

[deBrll]

[deBr12]

[deBrJa]

de Bruin, M.G., Generalized C-fractions and a multidimensional Pade table, Thesis, Univ. of Amsterdam (1974).

de Bruin, M.G., Convergence along steplines in a generalized Pade table, In: Pade and Rational Approximation, (E. B. Saff and R. S. Varga, Eds.), Acad. Press, New York (1977), 15 - 22.

de Bruin, M.G., Co.nvergence of genemlized C-fractions, J. Approx. Theory 24(1978), 177 - 207.

de Bruin, M.G., Jacobsen, L., Modification of generalized continued fractions I: Definition and application to the limitperiodic case, In: Rational Approximation and its Application in Mathematics and Physics, LaIicut 1985, (J. Gilewicz, M. Pindor and W. Siemaszko, Eds.), Springer Lecture Notes 1237, Springer-Verlag, Berlin (1987), 161 - 176.


[Chu1] Chudnovsky, G.V, Pade approximants to the generalized hypergeometric functions I, J. de Math. Pures et Applies. 58(1979), 445 - 476.

[Chu2] Chudnovsky, G.V, Pade approximation and the Riemann monodromy problem, In: Bifurcation phenomena in mathematical physics and related topics, (C. Bardos and D. Bessis, Eds.), Reidel, Dordrecht (1980), 449 - 510.

[Chu3] Chudnovsky, G.V, Rational and Pade approximations to solutions of linear differential equations and the monodromy theory, In: Springer Lecture Notes in Physics 126, Springer-Verlag, Berlin (1980), 136 - 169.

[Co] Coates, J., On the algebraic approximation of functions I-III and IV, Proc. Kon. Akad. v. Wet. A'dam Ser. A 69 and 70 = Indag. Math 28(1966), 421 - 461, and 29(1967), 205 - 212.

[Gelf] Gelfond, A.O., Calculation of limit differences, Moscow, Nauka (1967), (in Russian)

[GoRal] Gonchar, A.A., Rakhmanov, E.A., On convergence of simultaneous Pade approximants for systems of functions of Markov type, Proc. Stekl. Math. Inst. Issue 3(1983), 31 - 50.

[GoRa2] Gonchar, A.A., Rakhmanov, E.A., Equilibrium measure and the distribution of zeros of extremal polynomials, Mat. Sh. 125(1984), 117 - 127, English transl. in Math. USSR Sh. 53(1986), 119 - 130.

[GoRa3] Gonchar, A.A., Rakhmanov, E.A., Equilibrium distributions and the rate of rational approach of analytic functions, Mat. Sh. 134(1987), 306 - 352, English transl. in Math. USSR Sh. 62(1989), 305 - 348.

[GrMSa] Graves-Morris, P.R., Saff, E.B., Vector-valued rational interpolants, In: Rational Approximation and Interpolation, (P.R. Graves-Morris, E.B. Saff and R.S. Varga, Eds.), Springer Lecture Notes 1105, Springer-Verlag, New York (1984), 227 - 242.

[He 1] Hermite, Ch., Sur la fonction exponentielle, Comptes rendus de l'Acad. Des Sciences t. LXXVII (1873), 18 - 24, 74 - 79, 226 - 233, 285 - 293 = Oevres t. III (1873), 150 - 181.


[He2] Hermite, Ch., Sur la generalisation des fractions continues alge-briques, Extrait d'une lettre a M. Pincherle, Annali di Matematica 2-ieme serie, t. XXI (1893), 289 - 308 = Oevres t. IV (1893), 357 - 377.

[Hi] Hilbert, D., tiber die 1hmszendenz der Zahl e und 11", Math Annal. 43(1883), 216 - 219.

[IsSa] !series, A., Saff, E.B., Bi-'-orthogonality in rational approximation, J. Compo and App!. Math 19(1987), 47 - 54.

[Ja] Jager, H., A multidimensional generalisation of the Pade table, Proc. Kon. Akad. V. Wet. A'dam Ser. A 67 = Indag. Math. 26(1964), 192 - 249.

[Kal] Kalyagin, V.A., On a class of polynomials defined by two orthogonality relations, Mat. Sb. 110(1979), 609 - 627, English trans!. in Math USSR Sb. 38(1981), 563 - 580.

[KI] Klein, F., Elemantarmathematik vomhoheren Standpunkt aus, Erster Band, Springer-Verlag, Berlin (1924).

[10] Lopes, G.L., On the asymptotics of the ratio of orthogonal polynomials and converyence of multipoint Pade appro:cimants, Mat. Sb. 128(1985), 216 - 229, English trans!. in Math. USSR Sb. 56(1987), 207 -219.

[1oPo] Loxton, J .H., van der Poorten, A.J., Multidimensional generalizations of the Pade table, Rocky Mountain J. Math 9(1979), 385 - 393.

[Mah1] Mahler, K., Zur Appro:cimation der E:cponentialfunktion aund des Logarithmus I, II, J. Reine Angew. Math. 166(1932), 118 -150.

[Mah2] Mahler, K., Applications of some formulas by Hermite to the appro:cimation of e:cponentials and logarithms, Math. Ann 168(1967), 200 - 227.

[Mah3] Mahler, K., Perfect Systems, Compositio Math. 19(1968),95-166.

[Ma] Mall, J., Grundlagen fur eine Theorie der mehrdimensionalen Padeschen Tafel, Inaugural Dissertation, Munchen (1934).

[Nil]

[Ni2]

[Ni3]

[Ni4]

[Ni5]

[Ni6]

[Nutl]

[Nut2]

[Nut3]

[NutTr]

[Pal]

[Pa2]


Nikishin, E.M., A System of Markov functions, Moscow Univ. Math. Bull. 34(4) (1979),63 - 66.

Nikishin, E.M., On simultaneous Pade approximants, Math. USSR Sb. 41(4) (1982),409 - 425.

Nikishin, E.M., On asymptotics of a linear form for simuUaneous Pade approximants, Izvestiya VUZ, Matematika, No.2 (1986), 33 - 46, (in Russian).

Nikishin, E.M., On a set of power series, Siberian Math. J. 22(4) (1981), 164 - 168 (in Russian)

Nikishin, E.M., On logarithms of natural numbers, Izv. Akad. Nauk USSR, ser. Nat. 43(6) (1979), 1319 - 1327, English transl. in Math. USSR Izvestiya 15(1980), 523 -530.

Nikishin, E.M., On irrationality of values of function F(x, s), Mat. Sb. 109(1979) : 410 -417, English transl. in Math USSR Sb., 37 (No.3, 1980), 381-388.

Nuttall, J., Hermite-Pade approximants to functions meromorphic on a Riemann·surface, J. Approx. Theory, 32(1981), 233 -240.

Nuttall, J., Asymtotics of diagonal Hermite-Pade approximants, J. Approx. Theory, 42(1984), 299 - 386.

Nuttall, J., Pade polynomial asymptotics from a singular integral equation, Constr. Approx., 6(1990), 157 - 166.

Nuttall, J., Trojan, G.M., Asymtotics of Hermite-Pade approximants for a set of functions with different branch points, Constr. Approx., 3(1987), 13 - 29.

Pade, H., Sur la genemlisation des fractions continues algebriques, C.R. Hebd. de l' Acad. de France a. Paris 118(1894), 848 - 850.

Pade, H., Sur la genemlisation des fractions continues algebriques, J. de Math. Pures et Appl. 4-ieme serie, t. X (1894), 291 - 329.

166 A.!, Aptekarev, H. Stahl

[Pa3] Pade, H., Mimoire sur les diveloppements en fractions continues de la fonction exponentielle pouvent servir d'introduction a la thiorie des fractions continues algibriques, Ann. Sci. Ec. Norm. Sup. (3) 16(1899), 395 - 426.

[Pad] Parusnikov, V.I., The Jacobi-Perron algorithm and simultaneous approximation, Math. USSR Sb. 42(1982), 287 - 296.

[Par2] Parusnikov, V.I., Coefficientwise convergence rate of approximations obtained by Jacobi-Perron algorithm, Siberian Math. J. 25(2) (1985),935 - 941.

[Par3] Parusnikov, V.I., Weakly perfect systems and multidimensional continued fractions, Moscow Univ. Math. Bull., 39(2) (1984), 16 - 21.

[Par4] Parusnikov, V.I., On the convergence of the multidimensional limit-periodic continued fractions, In: Rational approximation and its Application in Mathematics and Physics, Laitcut 1985, (J. Gilewicz, M. Pindor and W. Siemaszko, Eds.), Springer Lecture Notes 1273, Springer-Verlag, Berlin, 1987,217 - 227.

[Pas] Paszkowski, S., Hermite-Padi approximation (basic notions and.theorems), will appear in J. Compo and Appl. Math.

[Pi] Pineiro Dias, L.R., On simultaneous approximants for some set of Markov functions, Moscow Univ. Math. Bull. No.2 (1987), 67 - 70, (in Russian).

[Sol] Sorokin, V.N., Asymptotics of a linear functional form with two logarithms, Uspechi Mat. Nauk USSR, 38(1) (1983), 193 -194, (in Russian).

[S02] Sorokin, V.N., Generalization of classic orthogonal polynomials and convergence of simultaneous Padi approximants, Proceeding Petrovsky Seminar (MGU), 11(1986), 125 - 165, (in Russian).

[S03] Sorokin, V.N., Convergence of simultaneous Pade approximations to functions of Stieltjes type, Izvestiya VUZ, Matematika, No.7 (1987), 48 - 56, English transl. in Soviet Math. (Iz VUZ) 31, No 7 (1987), 63 - 73.

[So4] Sorokin, V.N., Simultaneous approximation of several linear forms, Moscow Univ. Math. Bull. 38(1983), 53 - 56.


[S05] Sorokin, V.N., On irrationality of values of hypergeometric functions, Mat. Sb. 127(1985), 245 - 258, English transl. in Math USSR Sb., 55(No. 1, 1986), 243-257.

[Stl] Stahl, H., Extremal domains associated with an analytic function I and II, Complex. Var. 4(1985), 311 - 324, 325 - 338.

[St2] Stahl, H., Orthogonal polynomials with complex-valued weight functions I and II, Constr. Approx. 2(1986), 225 - 240,241 -251.

[St3] Stahl, H., Three approaches to a proof of convergence of Pade approximants, In: Rational Approximation and its Applications in Matheniatics and Physics, La.ncut 1985, (J. Gilewicz, M. Pindor and W. Siemaszko, Eds.), Springer Lecture Notes 1273, Springer-Verlag, Berlin (1987), 79 - 124.

[St4] Stahl, H., Asymptotics of Hermite-Pade polynomials and related approximants - A summary of results, In: Non linear Numerical Methods and Rational Approximation, (A. Cuyt, Ed.), Reidel Publ. Corp., Dordrecht (1988), 23 - 53.

[Wi] Widom, H., Extremal polynomials associated with a system of curves in the complex plane, Advances in Math. 3(1969), 127 -232.

[VVI] Vleck van, E.B., On the convergence of continued fractions with complex elements, Trans. Amer. Math. Soc. 2(1901), 476 - 483.

A.1. Aptekarev Keldysh Institute, Moscow RUSSIA

Herbert Stahl TFHjFB2 Luxemburger Str. 10 D-1000 Berlin 65 GERMANY

On the Rate of Convergence of Pade Approximants of Orthogonal Expansions

A.A. Gonchar E.A. Rakhmanov S.P. Suetin

ABSTRACT A variety of constructions of rational approximations of orthogonal expansions has been discussed in the series of works of 1960-1970 (see [H], [F], [CL], [Gr], and also the monograph of G.A. Baker, Jr. and P. Graves-Morris [BG, Part 2, §1.6]). The greatest interest relates to the definitions of rational approximants which extend the basic definitions (in the sense of Pade-Baker and Frobenius) ofthe classical Pade approximants of power series to the case of series in orthogonal polynomials. In contrast to the classical case, these definitions lead to substantially different rational approximants of orthogonal expansions. The problems of convergence of the rows of the corresponding Pade tables have been investigated by S. Suetin [S2], [S3], and [SI]. The main results of the present article concern the diagonal Pade approximants of orthogonal expansions. Our purpose is to investigate the rate of convergence of these approximants for Markov type functions.

1 Introduction

We start with some definitions and notations. Let s be a (positive Borel) measure on the segment [a,b] of the real line R, Pk(X) = Pk(X;S), k = 0,1,2, ... , a sequence of orthonormalized polynomials generated by the measure s, and let f be a real-valued function belonging to the class L 1 (s) defined by its Fourier expansion in the polynomial system {Pk }:

00

f(x) = L CkPk(X), k = 0, 1, ... (1.1) k=O

(In other words, the initial data are the Fourier coefficients of the function f with respect to the system {Pd.) In the sequel, we assume that the following condition holds true for the measure s:

s' = ds/dx > ° almost everywhere on [a, b].


170 A.A. Gonchar, E.A. Rakhmanov, S.P. Suetin

In particular, we admit measures corresponding to arbitrary weight functions on the segment [a, b] : ds(z) = h(z) dz, where h is a positive (almost everywhere on [a, b]) integrable function. The approximants considered below are of special interest for Legendre series (ds(z) = dz) and Chebyshev series (ds(z) = (1 - z2)-1/2 dz and a = -1, b = 1).

Fix an arbitrary pair of nonnegative integers L, M. Denote by R(L,M) the class of all rational functions of the form r = p/q, where the polynomials p and q have real coefficients, deg(p) $ L, deg(q) $ M, and q i: ° on [a, b]. Note that the number of free parameters of functions in the class R(L, M) equals L + M + 1. The following definitions extend the concept of ordinary Pade approximants (in the sense of Pade-Baker and Frobenius) from the association with power series to the case of orthogonal expansions. As we have already remarked, this concept leads to essentially different approximants.

The rational function FL,M of the class R(L, M), whose Fourier expansion with respect to the system {PI:} has the form

where CI: = CI:(f), k = 0,1, ... , L + M, is called the nonlinear Pade approximant of type (L, M) of series (1.1) (or of the function I). In other words, the rational function FL,M = P/Q is determined by the system of (nonlinear) equations

CI:(FL,M) = CI:(f), k = 0,1, ... ,L + M. (1.2)

We need to find the coefficients of the polynomials P and Q from this system. A method for solving system (1.2) (in the case of Legendre polynomials) is presented in [F]. System (1.2) is not always solvable. Thus it is possible that there exists no nonlinear Pade approximant of the given type. However, the uniqueness property is always valid: there exists at most one nonlinear Pade approximant of the given type (L, M). Indeed if PdQ1 and P2/Q2 are two approximants of such a kind, then CI:(P1/Qt} = CI:(P2/Q2), k = 0,1, .. . ,L + M, and thus

Consequently the polynomial P1 Q2 - P2Q1 posseses at least L + M + 1 zeros (on the interval (a, b». Since deg(P1Q2 - P2Qt} $ L + M, the polynomial is identically zero, and the rational functions PdQ1 and P2/Q2 coincide.

The rational function C)L,M of class R(L,M) is called the FrobeniusPade approximant oftype (L, M) ofseries (1.1) (or ofthe function I) if it is represented by the ratio p/q, where p and q are arbitrary polynomials (deg(p) $ L, deg(q) $ M, q i: ° on [a, b)) and the following relations hold

cl:(qf - p) =.0, k.= 0,1, ... ,L + M. (1.3)

On the Rate of Convergence of Pade Approximants 171

System (1.3), which determines the approximant ~L.M = p/q, is a homogeneous system of linear equations for the coefficients of the polynomials p and q. The number of equations of the system is equal to L + M + 1, and the number of unknown parameters is equal to L + M + 2. Thus there always exists a non-trivial solution to (1.3). It is easy to see that in this case, the polynomial q t: 0, and moreover, if q :I 0 on [a, b], then the ratio p/q determines the Frobenius-Pade approximant of the given type (L, M). The uniqueness of such approximants is not guaranteed. Here we only mention that if every polynomial q t: 0 determined by system (1.3) has degree M exactly and does not vanish on [a, b], then there exists a unique approximant ~L.M of series (1.1). In addition, the polynomials p and q are uniquely determined by system (1.3) (up to a constant multiplier). For proof, if we had two pairs of polynomials (Pl, ql) and (P2, q2) satisfying (1.3), then after equating the leading coefficients of the polynomials ql and q2, we see that for the pair Pl - P2 and ql - q2 relation (1.3) holds and deg(ql - q2) < M.

From the point of view of applications, it is important to rewrite system (1.3) in terms ofthe coefficients Ck of series (1.1) (i.e. the Fourier coefficients Ck(f) of the function I). Thus, we rewrite the polynomials P and q in the form

L M

p(z) = EakPk(Z), q(z) = E bkPk(Z), o o

and use the relation IHil

PiPi = E AkPk, li-il

which holds for any orthonormalized system {Pk} (the coefficients Ak depend on system {Pk})' In this way we obtain a system of M linear homogeneous equations for the coefficients bk, k = 0,1, ... , M, (the coefficients {ak} are then calculated on the basis of {bk}). The coefficients of this system are represented in terms of the coefficients Ck, k = 0, 1, ... , L + 2M, of series (1.1). For more details in the case of Chebyshev polynomials see [BG, Part 2, §1.6].

The fact that for the construction of the Frobenius-Pade approximants of type (L, M) we need the L + 2M + 1 coefficients of series (1.1) lowers the efficiency of the linearized method for the construction of Pade approximants from orthogonal expansions. In this connection, we emphasize that for the nonlinear Pade approximant of type (L, M) we use only L+ M + 1 coefficients of series (1.1). As we shall see in the sequel, the nonlinear approximants have substantial advantages over the Frobenius-Pade approximants with respect to the degree of approximation of the function f (at least this can be proved for functions of Markov type ). The corresponding numerical examples are presented in [F] (there one compares the following possible constructions of approximants based on the seven initial coefficients of Legendre series: partial sums of the series, nonlinear


Pade approximants oftype (3,3), and Frobenius-Pade approximants oftype (2,2».

In the present paper we deal with Pade approximants of orthogonal expansions (on the segment [a, b]) for functions I of the Markov type; that is, a function which can be represented in the form:

I(z) = ld (t - z)-1 duet), (1.4)

where u is a measure whose support belongs to a segment [c, d] of the real line R. In the sequel we shall suppose that the segments [a, b] and [c, d] are non-overlapping. For definiteness let us suppose that b < c. The case d = +00 is also included; in this case we shall consider only the measures u

for which the condition J Itl- 1 duet) < +00 holds. The function I of type

(1.4) is holomorphic in the domain G = C \ [c,d], and, in particular, it is holomorphic on the segment [a, b] (C denotes the extended complex plane). On [a, b] the function I is a real-valued.

Sequences of type (1.5)

(where j is a fixed integer) are called diagonal (in accordance with their position in the Pade tables {FL,M}, {~L,M}' L,M == 0, 1,2, ... ). The main theorem of the present article is valid for arbitrary sequences of type (1.5) under the condition j ~ -1. To simplify the notations, we shall consider in the sequel the case j = -1. In addition, we shall write

Fn = Fn-1,n, ~n = ~n-1,n, n = 1,2, ... (1.6)

It will be shown below that for functions I of Markov type, each of the considered approximants (Fn and ~n) exists and is uniquely defined for any n. To make a comparison, we shall also consider the best rational approximants of the function I (in Chebyshev metric on the segment [a, b]) together with the Pade approximants Fn and ~n. Denote by RL,M the rational function attaining the best approximation to I on [a, b] (in the Chebyshev metric) in the class R(L, M):

III - RL,MII[II'''l = reArl.M) III - rll[II,"1'

where 11·11[11,"1 is the sup norm on [a, b]. Set Rn = Rn-1,n. See [A, Chapter 2] about the best uniform approximants. All of the results on the approximative properties of the sequences Rn for Markov type functions I cited below are presented in essence in [G2] and [GL].

In Section 2, we introduce the potential-theoretic notations, formulate the main theorem of the article (Theorem 1), and. discuss briefly the results related to the theorem. Section 3 has an auxiliary character: here we


cite Theorems 2 and 3, which were proved in [GL] and [GR2]. They are related to the multipoint Pade approximants and asymptotics of polynomials orthogonal with respect to a varying (depending on the degree of the polynomials) weight function. These theorems are used substantially for the proof of Theorem 1. Sections 4 and 5 are devoted to the proof of the assertions in Theorem 1 that are related to the approximants Fn and C)n' In Section 6, functions characterizing the speed of convergence of the approximants are discussed, and the assertions relating to them, which will be stated in Section 2, are proved.

2 Main Theorem

It is convenient to characterize the convergence of Pade approximants Fn and C)n by notions that are connected with the equilibrium measures for mixed (Green-logarithmic) potentials. Let us now introduce these notations.

The logarithmic potential of a measure I' is denoted by VI' and defined by

VI'(z) = J log Iz - tl-1 dl'(t), z E C.

As above, let [a, b] and [c, dJ be non-overlapping segments of the real line R, and let G be the complement of the segment [c, dJ (in the extended complex plane C). By g(z,t) we denote the Green function ofthe domain G and by GI' the corresponding Green potential of the measure I' defined as

GI'(z) = J g(z, t) dl'(t), z E G.

Fix an arbitrary 0 > O. There exists a unique measure ~ = ~(O) minimizing the energy functional

J(I'; 0):= J J (g(z,t) + o log Iz - tl-1) dl'(z) dl'(t)

= J (GI'(t) + OVI'(t» dl'(t)

in the class of all unit measures I' whose supports are contained in [a, b). The measure ~(O) (and only this measure) is the equilibrium measure with respect to segment [a, b) and potential

WI'(z; 0) = GI'(z) + OVI'(z).

In other words, the measure ~(O) is the unique (unit) measure on the segment [a, b] satisfying the equilibrium condition

W>'(II)(Z; 0) = w(O) = const, z E [a, b]. (2.1)


Let I be a function of form (1.4) corresponding to a measure u with support lying in [e, d]. Let Fn and ()n represent the nonlinear Pade approximant and the Frobenius-Pade approximant oftype (n-l,n), respectively, of the orthogonal expansion (1.1). In Sections 4 and 5 we shall prove the existence and uniqueness of the Pade approximants Fn and ()n (for every n). The key element in the proof of existence and the analysis of the convergence is the connection with multipoint Pade approximants. Taking into account this connection, we obtain immediately from the results of [GL] the uniform convergence of the sequences Fn and ()n (n = 1,2, ... ) to the function I inside the domain G (for an arbitrary measure u whose support lies in [e, d]); see Theorem 2 below. Here we formulate a theorem characterizing the rate of convergence of the approximants Fn and ()n for measures u satisfying the following condition:

u' = du /dx > 0 almost everywhere on [e, d]. (2.2)

The assertions related to the best rational approximants Rn for the function I are also cited in the theorem.

Theorem 1. If the measure u in (1.4) satisfies condition (2.2), then uniformly on the compact subsets of the domain fl = G \ [a, b], the following limit holds:

lim I/(z) - In(z)11/ 2n = exp( -G>'(z)), z E fl, (2.3) n

where A = A(O) when In = Rn, A = A(I) when In = Fn, and A = A(3) when fn = ()n.

In each of the three cases In = Rn, Fn, ()n we have

lim sup I/(x) - In(x)ll/2n ::; exp( -G>'(x)) for all x E [a, b], n

where A = A(O), 0 = 0,1,3, respectively; in addition, the upper regularization of the function in the left-hand side 01 the inequality is equal to the right-hand side for all x E [a, b].

We have characterized the speed of convergence in a general form for the considered approximants. The form is also convenient for the proof of relation (2.3). Let

v(z; 0) = exp( _G'~(9)(z));

and notice that the Green's potential G>'(x) is identically zero on the segment [c, cl]; thus v(x; 0) = 1, x E [e, cl].

The function v(z; 0), characterizing the speed of convergence of the best approximants Rn, is easily described in geometric terms and is represented by elliptic integrals. The measure A(O) is the equilibrium measure on the segment [a, b] for the Green's potential ofthe domain G. The value w(O) = h of the potential G>'(O)(x) on [a, b] is called the modulus of the capacitor


([a,b], [e,d]) (or the annular domain 0). We have v(ZjO) = Iw(z)l, where w is the function conformally mapping the domain 0 onto a circular annulus e-h < 1(1 < 1 (the circumference 1(1 = e-h corresponds to the cut [a, b] and the circumference 1(1 = 1 corresponds to the cut [e, d]). The formulas representing function w(z) and the constant h by the elliptic integrals are well known. We emphasize that from (2.3) there follows the relation

1~1I1 - Rnlltf.i] = e- h

(the main result of [GIl). The function v(Zj 1), related to the nonlinear approximants Fn , also has

a simple geometric description and can be calculated in an explicit form. Here we recall the corresponding formulas. If t::. is a segment on the real line, then denote by <PI:!.. a function conformally mapping the complement of t::. (in C) onto the interior of the unit circle, so that <P1:!..(00) = O. For t::. = [-1,1],

(we choose the branch of the square root having positive values for Z > 1). We have

We remark that both functions v(Zj 0), 0 = 1,3 (unlike the function v(Zj 0» are nonconstant on [a, b]. Under the above assumption on the relationship of the segments [a, b] and [e, d] (b < e), the functions increase monotonically on [a, b]. Thus the speed of convergence of Fn and C)n to I on [a,b] is substantially improved (the common rate of a progression becomes smaller) when moving off [e, d]. In particular,

l~ III - In lit!,!] = v(bj 0)

(0 = 1,3 for In = Fn, C)n, respectively). Note that v(bjO) increases on o E [0, +00). The inequalities

e-h = v(bj 0) < v(bj 1) < v(bj 3)

show the relation between the speeds of approximation of the function I in the uniform metric on [a, b] by the rational functions Rn, Fn , and C)n. From the point of view of the speed of approximation of the function I in the uniform metric on [a, b], it is clear that the approximants Fn are worse than the best ones (v(bj 1) > e-h ). However they have a substantial advantage. On a part of the segment [a, b] adjoining the point a, Fn approximates I faster than Rn (v(Zj 1) < e-h , a ~ Z < b' < b). If e -+ b (as d is fixed), then e-h -+ 1 while v(z; 1) tends to a function corresponding to the pair of segments [a, b], [b, d]. This function is less than one for all Z E [a, b),

176 A.A. Gonchar, E.A.Rakhmanov, S.P. Suetin

and thus the speed of convergence of Fn to f in every point Z E [a, h) is majorized by the geometric progression v(z)2n with a fixed (independent of c) common ratio v = v(z) < 1 (in particular, hi _ h as c - h). The same is valid for the approximants C)n'

From the point ofview of applications of the approximant, it is interesting to compare those which are constructed directly on N initial coefficients Co, Ct, ... , CN_l of the expansion (1.1) (let us mention that it is impossible to construct the best approximants Rn on the basis of a finite number of Fourier coefficients Cj: of function f). Denote the corresponding partial sum of series (1.1) by CN = CN(Z). The rational function Fn is constructed on the basis of N = 2n initial coefficients Cj (see (1.2», and at first it is interesting to compare the approximative properties of functions Fn and C2n. For the function f, appearing in Theorem 1 of series (1.1), the sequence CN(Z) converges uniformly inside a domain E bounded by an ellipse with foci at the points a and h and passing through the point C (and diverges in the exterior of the domain E). The speed of convergence of the sequence CN is characterized by the following relation:

lim I/(z) - CN(z)11/ N = 1¢(z)1 = 1¢.o.(c)/¢.o.(z)l, z E F, (2.4) n

where 6. = [a, h] and F = E \ 6.; the function ¢ conformally maps the domain F onto the circular annulus ¢.o.(c) < 1'1 < 1. The analog of the foregoing remarks related to Theorem 1 is also valid for the character of convergence of the sequence CN on [a, h]. In particular, we have

All that has been mentioned above for CN also holds for the best polynomial approximants of f in the Chebyshev metric (on [a, h]). The reason for this is the fact that the partial sums CN of series (1.1) minimize the distance from / in L2( s) metric (in the class of all polynomials of degree not exceeding N). For the rational approximants in which we are interested, the situation is different. The character of convergence of Pade approximants and the best rational approximants is substantially different (compare with the facts that have been mentioned above on the functions v(z; 0) and v(z; 9), 9 = 1,3). The approximants Fn have an undoubted advantage over the Fourier approximants C2n. Firstly, the sequence Fn converges to / in the whole domain G (in which / is defined by formula (1.4» not only in thesubdomain E. Secondly in the domain E (and in particular on the segment [a, b]), the sequence converges to / substantially more rapidly than the sequence C2n :

v(z; 1) < 1¢(z)l, z E E.

For the Frobenius-Pade approximants C)n, the situation is more complicated. The approximants C)n converge to / in the whole domain E. Like the approximants Fn , these approximants are constructed on the basis of


the coefficients of series (1.1); however, to construct the function ~n, we need the 3n coefficients CJ:, k = 0, 1, ... , 3n - 1. Vice versa, on the basis of 2n coefficients CJ:, k = 0,1, ... ,2n -1, we can construct the approximants ~m only for m ::; 2n/3, and from the constructive point of view, it is necessary to compare the approximants ~m' m = [2n/3], with C2n and Fn.

To characterize the speed of convergence of the approximants ~m, we must use the function v( z; 3)2/3. From the description of the function given in Section 6, the next inequality follows:

v(z; 1) < v(z; 3)2/3, Z E G.

Thus in the whole domain G (and in particular on [a, b]), the nonlinear Pade approximants Fn approximate the function f substantially more rapidly than the Frobenius-Pade approximants ~[2n/31'

Everywhere in the sequel, we follow the notations introduced in Sections 1 and 2.

3 Multipoint Pade Approximants

Here we shall discuss the notations and the related results mainly for the purposes of the present article. For more details, see the monograph ([BG, Part 2, §1.1D, which contains an extensive bibliography; see also the works [G2], [GRl], and [GR2], in which the method of multipoint Pade approximants is applied to the problems of the best rational approximations of real functions as well as complex functions.

Consider an arbitrary table X of the form

where xJ:; are the points belonging to the segment [a,b]. Set

wJ:(z) = (z - xJ:I)(z - XJ:2) ... (z - xu), k = 1,2, ....

Let f be a function holomorphic on [a, b] (f E H[a, bD and L, M be a fixed pair of nonnegative integers. Let us consider an arbitrary pair of polynomials p = p(z), q = q(z) (q t 0) satisfying the conditions

deg(p) ::; L, deg(q)::; M, (qf - P)/WL+M+1 E H([a, b)). (3.1)

The last condition in (3.1) signifies that the difference qf - p vanishes in the points of the (L + M + l)th row of the table X (taking into account


the multiplicities). It is clear that such polynomials exist (compare with the definition of the Frobenius-Pade approximants). The ratio p/q of all polynomials satisfying (3.1) defines the same rational function rL,M of class R(L, M). This fraction is called the multipoint Padt approximant of type (L, M) of the function J corresponding to the table X. If for polynomials p and q, condition (3.1) holds and the polynomial q does not vanish in the nodes of the (L + M + 1 )th row of the table X, then the rational function rL,M interpolates J in the nodes.

Let us consider the case in which we are interested: J is a Markov type function (1.4), and u is a measure whose support lies in the segment [c, dj (c > b). To simplify matters here (and also in Sections 4 and 5), we shall suppose that [c, dj is a bounded segment and that the support of the measure u is an infinite set (if supp(u) is a finite set then all the considered rational approximants are identically equal to J if n is sufficiently large). Let rn = r n -1,n.

Writing rn = p/q, we obtain the following conditions on the polynomials p and q:

deg(p) ~ n - 1, deg(q) ~ n, (qJ - P)/W2n E H(G) (3.2)

(here we took into account that J E H(G), G = c \ [c, dj). It follows that for any contour "'( surrounding segment [c, dj and every j = 0,1, ... , n - 1

1 (qJ /W2n)(t)ti dt = o.

From this we obtain (taking into account (1.4» the following orthogonal relations J q(t)tiw;,;(t) duet) = 0, j = 0, 1, ... , n - 1. (3.3)

From (3.3) it follows that any polynomial q satisfying (3.2) has degree n, and therefore is uniquely defined, provided that the normalizing condition q(z) = zn + ... holds. Denote this polynomial by qn. Moreover it follows from (3.3) that all zeros of polynomial qn lie in the interval (c, d) (and all of them are simple). Consequently, the rational function rn = Pn/qn interpolates J in all of the nodes of the 2nth row of the table X.

From (3.2), (3.3), and the Cauchy integral formula, the corresponding Hermite formula follows:

(f - rn)(z) = (W2n/q~)(z) J (q~/W2n)(t)(t - z)-l duet), Z E G. (3.4)

The next theorem is an analog of Markov's theorem on the convergence of the classical Pade approximants (Chebyshev continued fractions) for Markov type functions. The theorem has been proved in the work [GL].

Theorem 2. For any table X and any measure u, the sequence of multipoint Pade approzimants rn (for a function f of form (1.4» converges uniformly to J inside the domain G.

On the Ra.te of Convergence of Pa.de Approximants 179

Under condition (2.2) for measures u and provided that the interpolation nodes of the table N have a limit distribution, it is possible to characterize the speed of the convergence of rn to f inside G.

Let l:!. be a segment on the real line. Denote by M(l:!.) the set of all unit measures whose supports belong to f).. The notation I-'n ~ I-' will mean that the sequence of measures I-'n weakly converges to a measure 1-'. From any sequence of measures I-'n E M(f).), it is possible to extract a subsequence of measures weakly converging to I-' E M(l:!.). For the polynomial p(z) = (z - at} ... (z - an), we let

n

I'(p) = L:c5oj ,

j=1

where c50 is a unit measure whose support consists of only one point, a. We have

vp(p) = log Ipl-l.

If deg(Pn) = n and all zeros of Pn lie in l:!., then

Let v = vex) be a continuous function on the segment l:!. (an external field). There exists a unique measure v E M(l:!.) minimizing the energy

in the class M(l:!.). The measure v = v(v) and only this measure (in the class M(l:!.», has the following equilibrium property:

VII (x) + vex) = W, x E supp(v),

> W, x E l:!., (3.5)

where W = w( v) = const. The following arises from the main result of [GR2].

Theorem 3. Let <Pn be a sequence of positive continuous functions on the segment l:!., I' be a measure on l:!. satisfying the condition 1" > ° a.e. on l:!., and Sn be a sequence of monic polynomials (deg(Sn) = n) satisfying the following orthogonal relations:

J Sn(t)ti<pn(t) dl'(t) = 0, j = 0,1, ... , n - 1, (n E A eN).

If the sequence


converges uniformly to a function v(x) > -00 on (.6.), then

n-1,,(Sn)-"'V (nEA),

!Wl (/ S~ <Pn d" ) 1/2n = e-w ,

where v = v(v), w = w(v).

(3.6)

(3.7)

It is easy to see that for the external field v = -vA, where A is a unit measure with compact support lying in R \ .6., the support of the equilibrium measure v = v( - V A) coincides with .6., and therefore it is possible to rewrite the equilibrium relation (3.5) in the form

VV(x) = V\x) + w, x E.6. (w = const).

In other words, the measure v = v( - V A) is the balayage of the measure A from the domain C \ .6. onto the segment .6.. In the sequel we shall often deal with a measure v which will be the balayage of a measure" E M([a, bD onto the segment [c, d]. In the remainder of the article, we shall denote the measure v by fJ(,,) and the associated constant w by b(,,). The relation

V.8(I')(x) = VI'(x) + b(,,), x E [c, d], (3.8)

uniquely determines the measure fJ(,,) and the constant b(,,). The next theorem follows directly from relations (3.3), (3.4), and Theo

rem 3 (for A = A(O) the theorem has been proved in [GL)).

Theorem 4. Let f be a function of form (1.4) with a measure (J' satisfying condition (2.2) and X be a table of interpolating nodes on the segment [a, b] having a limit distribution A (for even n):

(3.9)

Then the speed of convergence of the sequence of multipoint Pade approximants rn (associated with the table X) in the domain n is characterized by the relation

lim If(z) - rn (z)11/ 2n = exp(-GA(z», zEn, n

(3.10)

where GA is the Green's potential of the measure A for the domain G = C \ [c, d] (passage to the limit in (3.10) is uniform inside n).

Proof. Using orthogonal relations (3.3) (for q = qn) we apply Theorem 3 (for .6. = [c, d], Sn = qn, and <Pn = w2nl). From (3.9) it follows that v(x) = - VA(X), x E [c, d]. Thus from (3.6) and (3.7) we get

n-1,,(qn) -... V(_VA) = fJ(A) (n -... 00),


( )1/2n

lim f q2w-1 du = e-lI(~). n_oo n 2n

Using these relations and the Hermite formula (3.4), we obtain (uniformly inside 0)

II - rnI1/2n -+ exp(V), (3.11)

where V = V"(~)- V~-b(A).1t follows from (3.8) that V(z) = 0, z E [e, d]. Clearly, the function V + G~ is harmonic in the domain G and vanishes identically on [e, d]. Thus V = -G>' and (3.10) follows from (3.11).

It also follows from (3.10) that for z E [a, b] (and thus everywhere in G) the following inequality holds:

limsup II - rnll/2n ~ exp(-G>').

Here the upper regularization of the function of the left-hand side of the inequality is identical to the right-hand side function (everywhere in G)j see [L, Chapter 3, §3].

From Theorem 4 and the above remarks, the assertions concerning the best approximants Rn, which have been discussed in Section 2, follow. The fact that function Rn interpolates I in 2n nodes on segment [a, b] is a consequence of Chebyshev's theorem (see [A, Chapter 2] and also [G2]). We need to prove that a sequence of polynomials W2n, constructed from these nodes, satisfies the relation

(3.12)

where A(O) E M([a, b]) is the equilibrium measure on the segment [a, b] for the considered Green's potential G>'(O)(z) = w(O) = h, z E [a, b] (see Section 2). If X is an arbitrary table for which (3.12) holds, then

(3.13)

Thus the same inequality holds for the sequence Rn. If for the sequence W2n, corresponding to the best approximants Rn, relation (3.12) does not hold, then for a sequence A eN, the relation

(2n)-11'(W2n) -+ A"I A(O), n E A,

is valid. Then we should obtain from Theorem 4 (taking into account that the equilibrium measure A{O), and only this measure, minimizes the maximum of GP on [a, b] in the class M([a, b])) that

lim sup III - Rnllt!~] > e-h • neA '

This relation contradicts (3.13), and thus for the points Z2n,t. .•• ' Z2n,2n,

in which Rn interpolates I, the relation (3.12) holds. Now the statement of Theorem 1 for the case In = Rn follows from Theorem 4.

182 A.A. Gonchar, E.A. Rakhma.nov, S.P. Suetin

In the above reasoning, we essentially used the fact that A(O) is the equilibrium measure on the segment [a, b) for the Green's potential of the domain G. The same potential arises in relation (2.3) for Pade approximants Fn and ~n, but now the measures A(1) and A(3) are not an equilibrium for it. However, in these cases, it is also possible to deduce Theorem 1 from Theorem 4. We shall prove below that the approximants Fn and ~n are the multipoint Pade approximants associated with the tables X, having the limit distributions A(1) and A(3) respectively. For this purpose, we shall also apply Theorem 3 for b. = [a,b].

4 Nonlinear Pade Approximants

First we need to prove that for Markov type functions f of the form (1.4) (where (1' is an arbitrary measure on [c, d] whose support consists of infinitely many points) there exists a unique Pade approximant Fn (for every n EN). Fix an arbitrary n and suppose that Fn exists. We rewrite relations (1.2) determining Fn in the form:

Ck(f - Fn) = 0, k = 0,1, ... ,2n-1.

These relations are equivalent to the following ones:

J (f - Fn)(t)ti ds(t) = 0, j = 0,1, ... ,2n - 1. (4.1)

Consider an arbitrary set of 2n points of the segment [a, b):

Suppose that the points Z2n,k are enumerated in such a way that Z2n,k :::; Z2n,k+l' The corresponding simplex from the space R2n is denoted by K2n.

Set W2n(Z) = (z - Z2n,1)'" (z - Z2n,2n),

and consider a multipoint Pade approximant rn of the function f, corresponding to W2n. In accordance to what has been said in Section 3, we have rn = Pn/qn, where polynomial qn(z) = zn + ... is uniquely determined by orthogonal relations (cf. (3.3))

J qn(t)tiw;,;(t) d(1'(t) = 0, j = 0,1, ... ,n - 1.

Fix the polynomial qn and define a polynomial 02n, deg(02n) = 2n (as a monic polynomial) by the following orthogonal relations:


j = 0, 1, ... ,2n - 1 (recall that all zeros of qn lie on [c,dJ). Relations (4.2) determine a unique polynomial 02n with the properties described above. The polynomial has 2n simple zeros on the interval (a, b); arranging them in increasing order, we obtain a point

Y2n = (Y2n.1, Y2n.2, ... , Y2n.2n) E K2n.

The constructed correspondence X 2n -+ Y2n defines a mapping of the simplex K 2n onto itself. It is easy to see that the mapping is continuous, and consequently it has a fixed point (by the Brouwer theorem). Keeping the same designation X 2n for the fixed point, we get 02n = W2n and can rewrite relations (4.2) as follows:

J W2n(t)ti (q;;-2(t) J (q~/W2n) (r)( r - t)-1 du( r») ds(t) = 0,

j = 0, 1, ... , 2n - 1. (4.3)

Comparing the inner integral expression in (4.3) with the formula (3.4) associated with the table X 2n multipoint Pade approximant rn ofthe function I, we get

J (f - rn)(t)ti ds(t) = 0, j = 0,1, ... , 2n - 1.

Setting Fn = rn, we obtain that for this function, relations (4.1) hold. Since all zeros of Qn = qn lie on [c, dJ, Fn is a nonlinear Pade approximant of the function I.

We emphasize that together with the existence of Fn, we have proved the fact that the sequence Fn (n = 1,2, ... ) is a sequence of multipoint Pade approximants of the function I corresponding to the table X whose even rows coincide with the X 2n constructed above (the odd rows of the table are not used in the construction of Fn). In particular, from here it follows that the denominators Qn of the rational functions Fn . satisfy the orthogonal relations

J Qn(t)ti w2;(t) du(t) = 0, j = 0,1, ... , n - 1 (4.4)

(cf. (3.3»; all zeros of Qn (poles of Fn) lie on the segment [c, dJ. Taking into account what has been said above, we get from Theorem 2

the uniform convergence of the sequence Fn to the function I inside the domain G.

Now we shall prove relation (2.3) of Theorem 1 for In = Fn (assuming that the measure u satisfies condition (2.2». Let W2n be a sequence of polynomials whose zeros are the nodes at which Fn interpolates the function I. Let us show that the following relation holds:

(4.5)


Consider a limit point ~ of the sequence I'n:

(4.6)

First we apply Theorem 3 for fj. = [e, d], Sn = Qn, tPn = l/w2n' Using (4.4) and (4.6) we get

n-11'(Qn) -II(-V>') = ,B(~) (n E A) (4.7)

(/ )1/2n

lim Q2w-1 du - e- lI(>') neA n 2n - , (4.8)

where ,B(~) is the balayage of the measure ~ onto the segment [e, d], and b(~) is the corresponding equilibrium constant (see (3.8». Next we use relation (4.3) for qn = Qn and n E A. Here it is possible to once more apply Theorem 3 for fj. = [a, b], S2n = W2n and

We assumed that as n - 00, n E A, there is a limit distribution ~ of zeros of the polynomials W2n' To get the equilibrium relation that we need, apply Theorem 3. With the aid of (4.7) and (4.8), we calculate the limit function ofthe sequence V2n = (4n)-ttog tP2"; (as n - 00, n E A):

v = - VP(>')/2 + b(~)/2.

By Theorem 3, we have

It is easy to see that for the external field v = _va + const, where a is a measure whose compact support lies outside fj., lal ~ 1, the support of measure lIe v) coincides with fj.. Thus in the case under consideration, the equilibrium relation (3.5) takes the form

VII(tI)(z) - VP(>.)/2(z) = const, z E [a,b]. (4.9)

On the other hand, it follows from (4.6) that lIe v) = ~, and one can rewrite (4.9) in the form

(4.10)

Now using the representation of the Green's potential 0>' over logarithmic potentials of measures ~ and ,B( ~),

O>'(z) = V>'(z) - VP(>')(z) + b(~), z E 0,


we get from (4.10)

G"'(z) + V~(z) = w = const, z E [a, b].

Measure..\ = ,,$1) is the only measure satisfying this relation (cf. (2.1». Therefore relation (4.5) is proved because..\ is an arbitrary limit point of the sequence Pn. Now the assertion of Theorem 1 related to the approximants Fn follows from Theorem 4. •

From what has been proved, it also follows that the poles of Pade approximants Fn have a limit distribution characterized by measure {3(..$, ..\ = ,,\(1). Clearly, the zeros of approximants Fn have the same limit distribution.

5 Frobenius-Pade Approximants

Let us rewrite relation (1.3) determining (at L = n-1, M = n) polynomials p and q in the ratio C)n = p/q in the following equivalent form:

j(qf - p)(t)ti ds(t) = 0, j = 0, 1, ... , 2n - 1. (5.1)

System (5.1) (for the coefficients of polynomials p and q) always has a non-trivial solution. The polynomial q ~ 0 corresponds to this solution. Fix an arbitrary pair of polynomials p, q (q ~ 0), satisfying (5.1). We shall suppose that the leading coefficient of the polynomial q is equal to one. It follows from (5.1) that function (qf - p) (analytic on [a, b]) has at least 2n zeros on (a, b). Choose an arbitrary set

ofzeros of (qf-p) on [a,b]. Setw2n(Z) = (Z-Z2n,t} ... (Z-Z2n,2n). Function Tn = p/q is a multipoint Pade approximant off unction f, corresponding to the row X 2n . Consequently (see Section 3), polynomial q satisfies orthogonal relations (3.3), has degree equal to n, and all its zeros lie in (c, d) (in particular q i= 0 on [a, b]). From this it follows (see Section 1) that there exists a unique approximant C)n = p/q of function f. Moreover, polynomials p = Pn and q = qn are also uniquely defined (under the normalizing condition on qn mentioned above). Also it is easy to show that the number of zeros of function (qnf - Pn) on [a, b] is precisely equal to 2n. Otherwise the polynomial qn should satisfy orthogonal relations of the form

and that is impossible. Thus the polynomial W2n is also uniquely defined.


From Theorem 2 it follows the uniform convergence inside domain G of the sequence n to the function f (corresponding to an arbitrary measure (T with a support lying in [e, d]).

The proof of the limit relation

(5.2)

(for measures (T satisfying condition (2.2)) is carried out by the same scheme as the proof of relation (4.5) in Section 4. The orthogonal relations for qn are the same ones as for Qn (see (4.4)). Therefore formulas (4.7) and (4.8) remain valid after substituting qn in place of Qn. Instead of (4.3), we now have

j = 0,1, ... , 2n - 1,

with the only difference being q;l(t) in place of q;2(t). The difference leads to substituting the measure /3(>.)/4 for measure/3(>.)/2 in relation (4.9) for the measure v(v) = >.. So instead of (4.10), we get

4V\x) - VP('>")(x) = const, x E [a, b].

Using the Green's potential of measure >., we obtain an equilibrium relation

G'>"(x) + 3V'>"(x) = const, x E [a, b],

determining the measure>. = >'(3). Thus relation (5.2) is proved, and from Theorem 4 there follows the assertion of Theorem 1 related to the approximants n. The limit distribution of poles (and zeros) ofthese approximants is characterized by the measure /3(>'), >. = >'(3). •

6 Functions v(z; 1) and w(z; 3) = V(Z; 3)2/3

These functions characterize the speed of convergence of the approximants Fn and m, m = [2n/3] (Pade approximants of the function f which can be constructed by the 2n initial coefficients of the decomposition of function f into series (1.1); see Section 2). Here we shall give the description of these functions in terms connected with Green's functions on the corresponding Riemann surfaces.

We shall consider the compact Riemann surfaces 'R, which have been realized as a finite sheet covering of the extended complex plane (Riemann sphere) C. Denote by z(j) a point ofthe Riemann surface lying on the jth sheet of the surface over a point z E C. We use an analogous notation for the sets on 'R, univalently covering the corresponding sets on C.

On the Rate of Convergence of Pad~ Approximants 187

First consider the Riemann surface 'R, = 'R,2, arising as a result of a standard pasting of the two planes with the cuts over the segment [a, 6] (we paste the upper shore of the cut on the ''first sheet" to the lower shore on the "second sheet" and inversely). Let D be a domain on 'R, supplementary to [e, dj(l). In the domain, we consider a function V defined by the formulas

V(z) = G"'(l)(z), Z = z<1) E 0(1),

V(z) = -V"'(l)(z) + w(1), Z = z(2) E (C \ [a,6])(2).

From the equilibrium relation

G"'(l)(Z) + V"'(l)(z) = w(1), Z E [a, b],

(cf. (2.1) with () = 1) it follows that the two branches ofthe function V are harmonically "pasted" together (over [a, b]). Thus V is a harmonic function in the domain D \ {00(2)} and V(z) ~ log Izi as z = Z(2) -+ 00(2). Now it is clear that

V(z) = 9D(Z, 00(2», zED C 'R"

where 9D is the Green's function of the domain D. Taking into account that on the first sheet of Riemann surface 'R" the function V is identical to G"'(l), we obtain

v(Zj1) = exp(-V(z(l») = ItP(i1»I, (z E 0, z(l) E 0(1», (6.1)

where tP(z), zED, is a function conformally mapping domain D onto the unit circle 1'1 < 1 in such a way that 00(2) -+ o. The last equality also holds on [a, b]. From (6.1), the representation of the function v(Zj 1) easily follows in the form (cited in Section 2) of superposition of the mappings of the exterior of a segment onto the interior of the unit circle.

To describe the function w(zj3), we construct the three sheet Riemann surface 'R, = 'R,s by the following method. Three copies of the plane (the first copy with a cut over the segment [a, b], the second one with cuts over [a, b] and [e, dj, and the third one with a cut over [e, dj), are pasted together in the following way: the first sheet is pasted together with the second one over the segment [a, b] and the second is pasted together with the third one over the segment [e, dj. On this Riemann surface, we consider a double connected domain

D = 'R, \ ([e, dj(l) U [a, b](S».

Define a function V on the sheets of the surface 'R, by the following formulas:

V(z) = 2G"'(S)(z), Z = z(l) E 0(1),

V(z) = G"'(S)(z) - 3V"'(S)(z) + w(3),

V(z) = _G"'(S)(z) - 3V"'(S)(z) + w(3),

z = z(2) E 0(2),

z = z<S) E o(S).


From the equilibrium relation

G>'(3)(Z) + 3V>'(3)(z) = w(3), z E [a, 6]

(see (2.1) with () = 3) and self-evident properties of the Green's potential G>', it follows that the given branches are "pasted" together (over [a, 6] and [c, d] respectively) and define a harmonic function

V(z), zED \ {00(2),00(3)},

with singularities of the form

V(z) ~ 3 log Izl, z = z(~) -10 oo(~), k = 2,3.

In addition, the function V(z), zED, is continuously extended as identically zero onto the boundary segments [c, d](l) and [a, 6](3). From what has been said, the following formulas follow:

V(z) = 3W(z), W(z) = gD(Z, 00(2» + gD(Z, 00(3», zED C 'R,

where gD is a Green's function of the domain D. Thus (see the formula for V in the domain 0(1» we have

w(z; 3) = v(z; 3)2/3 = exp( - W(z(l»), (z E 0, z(l) E 0(1». (6.2)

Formula (5.2) is also valid for z E [a, 6]. To compare the functions v(z; 1) and w(z;3) in the domain G, we in

troduce more precise notations connected with the description of v(z; 1). Denote by Dl the domain D C 'R2 constructed above, and by Vi the corresponding function V:

Vl(Z) = 9D1 (Z,00(2», z E Dl .

We do not change the notations connected with w(z; 3) (the domain D C 'R3 and functions V and W). We shall consider the lliemann surface 'R3 as a result of the standard pasting of the third sheet (the plane cut over the segment [c, d](2» with the second sheet of the surface 'R2. Thus the function VI is defined in the domain Di = Dl \ [c, d](2) C 'R3, harmonic in Di \ {00(2)}, and VI(Z) ~ loglzl as z = Z(2) -10 00(2). It is possible to extend the function onto the third sheet (doubling its values on the second sheet). As a result, we get a harmonic function V1(z), zED \ {oo(~)}, such that Vi(z) ~ log Izl, z = z(~) -10 oo(~) (k = 2,3). The difference VI - W is harmonic in domain D. The boundary values of function VI - W == 0 on the segment [c, d](l) and are greater than zero on the segment [a, 6](3). Thus VI (z) > W (z) for all zED, and in particular, on the first sheet (outside the segment [c, d](l». From this the inequality we need the following inequality:

v(z;1),< w(z;3), zE G

(cf. Section 2).


It is possible to characterize the functions v(z; 1) and w(z; 3) in terms connected with Abelian's integrals on the compact Riemann surfaces (with the duplicate numbers of sheets in comparison to the surfaces 112 and 1ls considered above).

References [A] Akhiezer, N.I., Lectures on Theory of Approximations, 2nd ed.,

"Nauka", Moscow, 1965; English transl. of 1st ed.: Ungar, New York, 1956.

[BG] Baker, George A. Jr., and Graves-Morris, Peter, Pade Approximants, Part I: Basic Theory. Part II: Extensions and Applications. Encyclopedia of Mathematics and its Applications, 13 & 14, Cambridge University Press, Cambridge, 1981.

[CL] Clenshaw, C.W., and Lord, K., Rational approximations from Chebyshev series, In: Studies in Numerical Analysis, Academic Press, London, (1974), 95-113.

[F] Fleischer, J., Nonlinear Padl approximants for Legendre series, J. of Math. and Physics, 14, No.2, (1973), 246-248.

[G1] Gonchar, A.A., On the degree of rational approximation of analytic functions, Trudy Math. Inst. Steklov, 166 (1984), 52-60; English transl. in Proc. Steklov Math. Inst., 166, No.1, (1986), 53-6l.

[G2] Gonchar, A.A., On the speed of rational approximation of some analytic functions, Math. Sb., 105 (147) (1978), 147-163; English transl. in Math. USSR Sb., 34, No.2, (1978), 131-146.

[GL] Gonchar, A.A., and Lopez-Lagomasino, Guillermo, On Markov's theorem for multipoint PaM approximants, Mat. Sb. 105 (147) (1978), 511-524; English transl.· in Math. USSR Sb., 34, No.4, (1978), 449-460.

[GR1] Gonchar, A.A., and Rakhmanov, E.A., Equilibrium distributions and degree of rational approximation of analytic functions, Mat. Sb., 134 (176) (1987), 306-352; English transl. in Math. USSR Sb., 62, No.2, (1989), 305-348.

[GR2] Gonchar, A.A., and Rakhmanov, E.A., The equilibrium measure and the distribution of zeros of extremal polynomials, Mat. Sb., 125 (167) (1984), 117-127; English transl. in Math. USSR Sb., 53, No. 1, (1986), 119-130.


[Gr] Gragg, W.B., Laurent, Fourier and Chebyshev Padl tables, In: Pade and Rational Approximation, (Eds. E.B. Saff and R.S. Varga), Aca.demic Press, New York (1977), 61-70.

[H] Holdeman, J.T. Jr., A method for approximation offunctions defined by formal series expansion in orthogonal polynomials, Math. Comp., 23, No. 106, (1969), 275-287.

[L] Landkof, N.S., Foundations of Modern Potential Theory, "Nauka", Moscow, 1966; English trans!., Springer-Verlag, Berlin, 1972.

[Sl] Suetin, S.P., Inverse theorem on generalized Padl approximants, Math. Sb., 109 (151) (1979),629-646; English trans!. in Math. USSR Sb., 37, No.4, (1980), 581-597.

[S2] Suet in , S.P., On the convergence of rational approximants to the polynomial expansions in the domains of meromorphy of a given function, Math. Sb., 105 (147) (1978), 413-430; English trans!. in Math. USSR Sb., 34, No.3, (1978), 367-381.

[S3] Suetin, S.P., On Montessus de Ballore's theorem for rational approximants of orthogonal expansions, Math. Sb., 114 (156) (1981), 451-464; English transl. in Math. USSR. Sb., 42 (1982).

A.A. Gonchar, E.A. Rakhmanov, S.P. Suetin Steklov Math. Institute Vavilova 42 117966 Moscow GSP-l RUSSIA

Spurious Poles in Diagonal Rational Approximation

D.S. Lubinsky

ABSTRACT Any function f meromorphic in C admits fast rational approximation. That is, if K is a compact set in which f is analytic, there exist rational functions Rn of type (n, n), n ;::: 1, such that

More generally, any function f defined on an open set U, and admitting such approximation on a compact K C U with positive logarithmic capacity, is said to belong to the Gonchar- Walsh Class on U. We discuss at an introductory, non-technical, level, the problem of spurious poles for diagonal and sectorial sequences of rational approximants to functions in the Gonchar-Walsh class. In particular, we concentrate on some recent positive results on the distribution of poles, and some of their consequences.

1 Introduction

For n ;::: 1, let Rnn be a rational function of type (n, n) (both numerator and denominator degrees are at most n). Such a sequence {Rnn}~=l is often called a diagonal sequence.lt has radically different convergence properties from sequences of rational functions in which the numerator degree, or denominator degree, is bounded. Similar in nature to diagonal sequences, are sectorial sequences: These are of the form {Rm jn;}b::l' where Rmjnj has type (mj, nj) (numerator and denominator degrees at most mj, nj respectively) and the ratio mj / nj is bounded above and below by positive constants independent of j.

We survey, at an introductory level, results on the poles of such sequences, formed by approximating functions admitting sufficiently rapid rational approximation. In particular, spurious poles, the Nuttall-Pommerenke theorem, Stahl's theorem, new results on the distribution of poles and their consequences, and the Baker-Gammel-Wills Conjecture are covered. Some new results are also established. Although there is overlap, our focus is different from Stahl's excellent recent survey [38], which concentrated on convergence in capacity, and included functions with branchpoints, which are mostly omitted here. Converse theorems, which deduce information on f from the behaviour of the poles of its approximants, are also omitted.


192 D.S. Lubinsky

2 The Gonchax-Walsh Class

It is an elementary exercise in complex analysis that the partial sums {Pn}~=l of the Maclauren series of an entire function I satisfy

lim III - Pn lIi'n(K) = 0, n~oo 00

on every compact K C C. Here, as elsewhere in this paper, the norm is the sup norm on K. It is a little harder to show that if I is meromorphic in C (that is, analytic except for poles), there exist rational functions Rn of type (n,n), n ~ 1, such that

lim III - RnIli'n(K) = 0, n~oo co

(2.1)

on every compact K in which I is analytic. Both these classes of functions admit last rational approximation, that

is, may be approximated by rational functions faster than any geometric sequence. But many more types of functions admit such rapid approximation, and it becomes interesting to give them a name and to study their properties. In fact, this theme goes back to one of the founding fathers of approximation theory in the United States, Joseph Leonard Walsh, and was developed by A.A. Gonchar, a most prominent Soviet protagonist of the art of approximation. So its presence is not inappropriate at a joint U .S.-U .S.S.R. symposium.

It is clear that if K consists of, say, a single point, then (2.1) will have no implications for I. SO for a meaningful study, K should be "large enough" , and it is at this stage that logarithmic capacity invariably rears its head: Complex approximation without capacity is like analysis without measures. There are almost an infinite number of ways to present definitions, but the simplest is the following:

Throughout 1rn denotes the set of polynomials of degree at most n with complex coefficients, and 1rmn is the set of rational functions R of type (m, n), that is R = P/Q, with P E 1rm , Q E 1rn , and Q not identically zero.

Definition 2.1 IS

Let K be compact. Then the logarithmic capacity of K

cap(K):= lim ( min IIPIIL (K»)l/n. n-+oo PE ... n. 00

Pmonic

For arbitrary L C C, its (inner) logarithmic capacity is

cap(L) := sup{cap(K) : K C L, Kcompact}.

As simple examples, cap([a,bD = (b - a)/4;

cap({z: Izl = r}) = cap({z: Izl::5 r}) = cap({z: Izl < r}) = r.

Spurious Poles in Diagonal Rational Approximation 193

A good introduction, despite an error in a proof, is given by Hille [13]. From the point of view of rational approximation, it is mostly whether or not a set has positive capacity that is important: Sets of capacity zero are thin and often ignorable. But an estimate on a set of positive capacity leads to some sort of estimate on any bounded set.

From a measure theoretic point of view, a set K of capacity 0 is thin indeed: It has planar Lebesgue measure zero, and its intersection with every line has linear Lebesgue measure zero. Even more, it has what is called Hausdorff dimension zero. By contrast, the usual Cantor ternary set in [0,1], (obtained by repeatedly removing open middle thirds), has positive logarithmic capacity. Still, any countable set has capacity zero.

We shall write I E Ho(K) if I is analytic at each point of K, and I E M e'(K) if each point of K is either a point of analyticity or an isolated pole of I.

Definition 2.2 The Gonchar-Walsh Class Let U be open and I E Ho(U \ S), where cap(S) = O. We write I E Ro(U) and say that I belongs to the Gonchar- Walsh Class on U, if for each compact K C U with IE Ho(K), there exists R,. E 1rnn, n ~ 1, such that

lim III - RnIli'n(K) = O. n~oo 00

(2.2)

Thus I is required to admit fast rational approximation on each compact subset of U in which I is holomorphic. One of the main features of the Gonchar-Walsh class is what is called quasi-analyticity, a type of uniqueness property generalizing that of ordinary analytic functions. If I, 9 E Ro(U), and there is a compact set K C U with cap( K) > 0 and I == 9 on K, then I == 9 throughout U, except possibly on a closed set of capacity zero. Another feature is that in each component of U, I is single valued and in particular, cannot have branch points [9].

Because ofthe quasi-analyticity, it is quite common in defining Ro(U) to omit any reference to U and simply to demand that (2.2) holds on a single compact K with positive capacity. However, the above definition is more convenient for our purposes.

It is clear from the introduction to this section that entire and meromorphic functions belong to Ro(C). In fact, any function I that is single valued and analytic in C except on a set S of capacity 0, automatically' satisfies (2.2) on each compact K C C \ S [9,11]. This means that Ro(C) admits all functions whose singularities form a countable set, so allowing essential singularities rather than just poles.

For further orientation on Ro(U), see [38].

194 D.S. Lubinsky

3 Rational Approximants

Perhaps the simplest, most explicit, and most constructive method of approximation is that of interpolation: Though any m+ 1 points (z j, Yj) in the plane, with all Zj distinct, one can explicitly exhibit the unique Lagrange interpolation polynomial P E 1I"m satisfying P(Zj) = Yj, 1 ~ j ~ m + 1.

In contrast to the polynomial case, interpolation by rational functions R = P / Q E 11" mn is more complex. P and Q have respectively m + 1 and n+I coefficients, so R has (m+I)+(n+I)-I = m+n+I free parameters: we subtract one to account for the division. This leads one to expect that R E 1I"mn might satisfy m+n+I interpolation conditions. Unfortunately, the non-linearity caused by division by Q inherently complicates the situation, so this is not always possible.

For example, suppose /(0) = 0, /(1) :F 0, and we try to find R E 11"01

satisfying the 0 + 1 + 1 = 2 interpolation conditions

R(O) = /(0) = OJ R(I) = /(1) :F O.

Since a R(z) = bz+c' some a,b,cE C,

the condition R(O) = 0 forces a = 0 and so R == O. Then we cannot satisfy R(I) = /(1).

To avoid this difficulty, one linearizes the interpolation conditions by multiplying by the denominator Q:

P(O) = /(O)Q(O)j P(I) = /(I)Q(I). (3.1)

More compactly, we can require that

(fQ - P)(z)/{(z - O)(z - I)} is holomorphic at z = 0 and z = 1.

This forces (fQ - P)(z) = 0 for z = 0,1, and hence (3.1). The advantage of this linearized problem is that it always has a solution R = P/Q, and moreover, this is unique. In the above example, the solution is

This motivates:

Definition 3.1 Let

o R(z)=-=O. z-I

au

A ·-.-


be a triangular array of (not necessarily distinct) complex numbers akj

contained in a compact set K. Let

k

Wk(Z):= II(z - akj),k ~ 1. j=1

(3.2)

Given m,n ~ 0, and IE Ho(K), the (m,n) multipoint Padl approximant to I lor the interpolation scheme A is

Rrrm(z) := Rmn(f;A;z):= P/Q E 1I"m,n,

such that (fQ - P)/wm+n+1 E H o(K). (3.3)

An important special case is that where all interpolation points are at 0, that is, all akj = O. Then

and (3.3) requires that

(fQ - P)(z)/zm+n+1 is analytic at O.

In this case, we obtain the familiar Pade approximant, denoted [m/n](z):

Rmn(f; {OJ; z) = [m/n](z).

The multipoint Pade approximant Rmn(f; A; z) exhibits an explicit representation in terms of values of I or its derivatives at the interpolation points {akj} [2]. At the opposite end of the approximation spectrum is the rather implicit best approximant, the "closest" rational function to a given function:

Definition 3.2 Let K be compact, I E H o(K) and m, n ~ O. Then we define the error in best approximation to I on K Irom 1I"m,n,

Emn(f; K):= inf III - RIILoo(K), RE'lI"m ...

(3.4)

and a best approximant on K to I from 1I"m,n is any rational function

satisfying (3.5)

We also define the normalized errors

(3.6)

196 D.S. Lubinsky

It is well known that R';;..n exists, but need not be unique. Of course, one might consider more generally best approximants Rmn E '1rmn minimizing some weighted Lp norm involving some p > 0 and a finite positive Borel measure 1':

where

IIgllLp , .. = (J IglPdl') lip

Despite the contrast between the explicit and implicit nature of multipoint Pade approximants and best uniform or best Lp approximants, they are all "near-best" approximants to functions in the Gonchar-Walsh Class Ro(U). The phrase "near-best" is used in widely different senses in the literature, but of course means that a given approximant(s) is (are) close to best possible.

In our context, we use "near-best" only in connection with sequences of rational functions. Which sequences? The convergence behaviour of such a sequence depends inherently on the relative size of the numerator and denominator degrees as n -+ 00. We shall concentrate on a few important cases: A diagonal sequence {rj }};l has the property

rj E'1rjj, j = 1,2,3, ....

A sectorial sequence {rj }};l has the property

rj E'1rmjnj , j = 1,2,3, ... ,

where the upper bounds {mj}};l on the numerator degrees, and {nj}};l on the denominator degrees, satisfy for some A > 1,

I/A ~ mj/nj ~ A,j ~ 1; .lim mj = 00. J ..... OO

(3.7)

That is, the ratio of numerator and denominator degrees is bounded away from :tero and infinity.

Of course, diagonals are a special case of sectorials, but already exhibit most of the difficulties of sectorial sequences.

Definition 3.3 Let U C C be open and f E Ro(U). Let {mj }};l' {nj }};l be sequences of non-negative integers satisfying (3.7) and let

rj := pj/qj E 7rmj,nj,j ~ 1.

We say {rj }};l is a sequence of near-best approximants to f if there exists compact K C U, with cap(K) > 0, such that for each 0 < {) < 1, we have for j large enough,

(3.8)


Clearly, (3.8) is a linearized form of the condition

"' - rj IILoo(K) ::s; Em;n; (I; K)1-6 . (3.9)

We have multiplied by (}j, thereby linearising (3.9) and circumventing the nasty (and not uncommon) situation that rj has poles in K. In turn, since (2.2) implies

.lim Em ·n .(1; K)l/(m;+n;) = 0, )_00 J J

(3.9) and hence (3.8) is certainly true if for some C > 0,

II' - rjIlLoo(K) ::s; cm;+n; Em;n;(I; K), j ~ 1.

So the above admits a fairly wide class of sequences of rational functions. For further orientation on arrays of rational approximants, and different

sequences ofrational functions (sectorial, rows, columns, ... ), see [4,37,38,41].

4 Spurious Poles

To the newcomer to rational approximation, a somewhat surprising and disconcerting feature is that the poles do not always behave as they ought to: The poles of a sequence {rj}~l of rational approximations may not reflect the analytic properties of the underlying function ,. Thus , may be analytic in a compact set K, but no matter how large is j or how large are the numerator and denominator degrees of rj, rj may have poles in K. This phenomenon arises irrespective of whether {rj} ~1 is formed by multipoint Pade, or best, approximation. The neat phrase "spurious poles" that describes this, was coined by George Baker in the early 1960's.

It is difficult to determine who first noticed the problem. In his landmark 1821 essay on rational interpolation, Cauchy [7] did not seem aware of the need to linearize in Definition 3.1, never mind of spurious poles. In his 1908 thesis [8], Dumas investigated a class of functions for which the phenomenon occurs, in a relatively mild form. Almost certainly, de Montessus de Ballore and Wilson, who worked on convergence of Pade approximants with fixed denominator degree in the period 1900-1930, must have confronted it.

However, it probably was first given real prominence by O. Perron in his book on continued fractions [28]. He constructed an entire function such that each point in C is a limit point of poles of {[m/l](z)}:=l. From then on, examples illustrating various divergence features have appeared. Perhaps the most influential one is due to Hans Wallin [40].

Theorem 4.1 There exists an entire function f with the following properties: (a) Each point in C is a limit point of poles of {[n/n](z)}~=l.

198 D.S. Lubinsky

(b) At each z E C \ {O}, {[n/n](z)}~::::l diverges. More precisely,

lim sup l[n/n](z)1 = 00, Vz E C \ {O}. n ..... oo

Thus no point in the plane is free of spurious poles! Furthermore, initial hopes that the diagonal sequence of Pade approximants might converge uniformly, or pointwise, or even just almost everywhere, are dashed once and for all.

At this stage, one might be tempted to throw up one's hands in despair and deduce that nothing positive can be said about convergence in general. But there is a veritable sea of different convergence concepts available to the modern analyst. In 1970, John Nuttall discovered the relevant one -convergence in measure.

John Nuttall [25] proved that the diagonal Pade sequence ([n/n](z)}~::::l to a function meromorphic in C, converges in planar Lebesgue measure. In 1973 [29], Pommerenke showed that one could replace meromorphic functions by functions in Ro(C), and planar measure by logarithmic capacity. In retrospect, this is not surprising, as cap has a long association with rational approximation. It was not long before Gonchar [11] replaced Ro(C) by Ro(U) and Wallin [41] replaced Pade by multipoint Pade approximants. Evidently the Nuttall-Pommerenke theorem is a fundamental theorem, which should be valid for any reasonable sequence of rational approximants:

Theorem 4.2 Let U be open and f E Ro(U). Let {mj}~l and {nj}~l satisfy (9.7), and let

rj E 1f'm;nj' j = 1,2,3, ... ,

be a sequence of near-best approximants to f. Then

rj -+ f in capacity as j -+ 00.

More precisely, given compact K C U and (. > 0,

cap{z E K : If - rj I(z) > (.m;+n;} -+ 0 as j -+ 00.

In particular, we may choose {rj }~1 to be a sequence {Rm;n;(f; A; Z)}~l of multipoint Pade approximarl.is, where A is any array of interpolation points contained in a compact set L with FE Ho(L). Or, we may choose {rj}~l to be a sequence {~;n;<f;L;z)}~l of best approximants to f, where L is compact and f E Ho(L).

The above N uttall-Pommerenke theorem is implicit in works of Gonchar, Wallin, and Karlsson [11,15,41]: essentially a proof is contained in Lemma 3.2 in [23]. For a more detailed discussion of convergence in capacity, the reader is encouraged to consult [38].


5 Distribution of Poles

The Nuttall-Pommerenke theorem is a positive response to the problem of spurious poles, which shows that they disrupt only sets of small measure or capacity. Nevertheless, it does not say much about the distribution of poles,for little can be said under only the Nuttall-Pommerenke hypotheses.

What is meant by distribution of poles? In a sense, this is how the poles behave on average. Individual poles are not so important as how positive proportions of the poles behave. This is best studied with the aid of pole counting measures:

Definition 5.1 Let R = P/Q E 1I"mn where P and Q have no common factors. The pole counting measure limn for R is

limn := n-1 ~ 6z ,

Q(z)=O

(5.1)

where 6z denotes a unit mass (Dirac delta) at z, and poles are repeated according to multiplicity.

Note that limn will have total mass one only if R has poles of total multiplicity n. It is a remarkable feature that for some entire functions, an infinite subsequence of their diagonal Pade approximants may concentrate a positive proportion of their poles at a single point:

Theorem 5.2 Fix bEe \ {OJ. Let {nj }b=l be a sequence of positive integers such that for some f > 0,

(5.2)

Then there exists an entire function f and 6 > 0, such that for j ~ 1, [nj/nj](z) has a pole of multiplicity at least 6nj at z = b.

Equivalently, if lin jn j denotes the pole counting measure for [nj / nj] (z), we have

lInjnj({b}) ~6, j ~ 1.

Although not explicitly stated in Wallin's 1974 paper, Theorem 5.2 may be extracted directly from his construction. This example is, in some ways, even more disturbing than that in Theorem 4.1: f is entire, and so analytic at b, but b still attracts a positive proportion of the poles!

Perhaps the message of this example is that the only singularity of f -its essential singularity at infinity - is not strong enough to tell the poles where to go. From the point of view of rational approximation, essential singularities are weak.

In remarkable contrast, branchpoints are strong singularities and always determine a specific asymptotic behaviour of the pole counting measures. This discovery, due to H. Stahl [34,35], (with earlier work by J. Nuttall),

200 D.S. Lubinsky

was the major breakthrough in diagonal Pade approximation during the 1980's.

Theorem 5.3 Let I be analytic in C \ S, where.o E Sand cap(S) = O. Furthermore, let I have branchpoints. Let {mj}~l and {nj}~l satisfy

.lim mj/nj = 1; .lim mj = 00, J-+OO J-+OO

(5.3)

and let vm;n; denote the pole counting measure lor [mj/nj],j ~ 1. There exists a positive unit measure 1', depending only on the location and nature 01 the branchpoints 01 I, such that

(5.4)

Furthermore, there e:cists a domain D, depending only on the location and nature 01 the branchpoints 01 I, such that in D,

[mj /nj] - I in capacity, j - 00.

Of course, ~ denotes weak convergence. It is more customary to formulate the theorem for Pade approximants formed from expansions at infinity, in which case I' is the equilibrium measure for D, which is itself Stahl's e:ctremal domain. For further orientation, see Stahl's survey [38].

Recall that functions in Ro(U) are single valued in their Weierstrass domains of analytic continuation, so Stahl's results cannot be applied to Ro(U). Further, Wallin's example in Theorem 5.2 plainly indicates that no complete analogue is possible for Ro(U). However, there is a ray of hope: Wallin's example applies only to a very thin subsequence, for example ([ai /ai]}~l' of the diagonal sequence.

Can we say something positive about other subsequences? Yes [23]:

Theorem 5.4 Let U be open and f E Ro(U). Let {mj}~l and {nj}~l satisfy (3.7) and

(5.5)

Then there e:cists an infinite sequence :J of positive integers with the following property: Let rj E 'lrm;n;, j = 1,2,3, ... , be a sequence of near-best appro:cimants to I. Let vm;n;, j = 1,2,3, ... , denote the corresponding pole counting measures. Then for each compact K C U in which I is meromorphic,

Jim Vm 'n .(K) = O. '-"00 'J ;E3


In particular, the result applies to multipoint Pade approximants {Rmjnj(l; Aj Z)}~l associated with an array A such that I E H 0(.4), or to best approximants {w,;.jnj(l;KjZ)}~l' whenever I E Ho(K), for both are sequences of near-best approximants. More generally, so too are best approximants in some possibly weighted Lp norm.

Note that .:J above is independent of the particular sequence of nearbest approximants. It may be chosen, for example, as follows: Let T C U be compact with cap(T) > 0 and I E Ho(T). Because I E Ro(U), the normalized errors {'7mjnj(l;T)}~l' defined by (3.6), satisfy

.lim TJm;n;(I; T) = o. )_00

Choose j E .:J iff

This suggests that when the normalized errors of approximation decrease monotonically, one may treat full sequences, rather than subsequences: [23]

Theorem 5.5 Let U be open and IE Ro(U). Assume that T C U has cap(T) > 0 and I E Ho(T), and lor some k ~ 1,

TJk1c(l; T) ~ TJ1c+l,1c+l(1; T) ~ TJ1c+2,1c+2(1; T) ~ .. . . (5.6)

Then il rn E 'll"nn, n = 1,2,3, ... , is any sequence 01 near-best approximants to I, and Vnn , n = 1,2,3, ... , are the corresponding pole counting measures,

lor every compact K C U in which I is meromorphic.

Note that (5.6) can be weakened to the requirement that

le~f TJ1c-(1c£),1c-(1cf) (I; T)/TJk1c(l; T)l+6 > 0,

for every (,6 E (0,1). Here {x} denotes the greatest integer ~ x. To the Pade practitioner, the idea that regularity of decrease of errors

of approximation implies good behaviour of the poles, is no surprise: It is an old observation that Pade approximants tend to behave well when the underlying Maclauren series has smooth coefficients. And, Maclauren series coefficients decay much like errors in best approximation.

Of course, there is one function whose Maclauren series is a model of regularity: I(z) = eZ • The behaviour of the poles of Pade approximants to eZ has been thoroughly investigated by Saff and Varga [32,33]' and the rate at which all poles approach infinity obtained. For example, [32] all poles of [m/n](z) lie in the annulus

{z: (m+n)p< Izl < m+n+H,

202 D.S. Lubinsky

where p = 0.278 ... is the unique positive root of pe1+P = 1. This is sharp. Such a detailed analysis has not been performed for any other entire

function, and perhaps is not possible. Still, less detailed information is available in several cases, [1,18,19,21], and the Saff-Varga theorems and results in [18] suggest that for entire I, and for a suitable subsequence of {[n/n](z)}~=1' most spurious poles sit in an annulus

(5.7)

with C1 and C2 independent of n, and rn the root of

logmax{[/(z)l: Izl = rn} = n.

More precise conjectures based on heuristic considerations appear in [26]. For meromorphic functions, one must replace the log of the maximum

modulus by the Nevanlinna characteristic, and must also take account that poles of I in Izl < C1rn could attract a positive proportion of poles of rational approximants. Of course, for such I, Theorem 5.4 states simply that as j -+ 00, j E :J, all but o( nj) poles of rj leave each bounded set in C. The method of proof in [23] yields only weak rates at which most poles approach infinity, and nothing like (5.7). There is certainly scope for further work in this direction.

6 Consequences of a Reasonable Distribution of Poles

It is well known that when poles of the Pade approximants [min] leave each bounded set as m + n -+ 00, then in each ball centre zero, the error (f- [m/n])(z) traces a nearly circular curve as z traverses any circle centre zero. This has the consequence that the error in approximation by [m/n](z) on Dr := {z: Izl $ r} is a good estimate of Emn(fjDr). It was this type of observation that helped in proving Saft"s result [31] that for fixed n, and m -+ 00,

and the much later difficult extension to m + n -+ 00 [5,39]. It is less well known that information on asymptotic distribution of the

poles (rather than all of the poles), also has many interesting consequences. This type of result played a crucial role in the Gonchar- Rahmanov resolution of the "1/9"th conjecture [12,24]

lim Enn(e-.i j [o,oo»l/n = 1/9.289 .... n_co


Possibly the most impressive application of results on pole distribution is Herbert Stahl's "Thue-Siegel-Roth" Theorem [36]. Let us write u(r) = l, if

I(z) - r(z) = blzl + bl+lZl+1 + ... ,bl :I O.

Thus u( r) describes the order of contact of r with I. It is an elementary consequence of the definition of the Pade approximant that

u([mJn)) = max u(r). re1l'm.,..

In the same way as one can describe the degree of approximation of irrational numbers by rationals, u measures the approximability of I, in a certain sense, by rational functions. Following is Stahl's 1987 result:

Theorem 6.1 Under the hypotheses 01 Theorem 5.3,

.lim u([mjJnj))f(mj + nj) = 1. J-OO

(6.2)

Furthermore, il Tnj and hj denote the actual numerator and denominator degrees in [mjfnj]' j ~ 1, then

(6.3)

For functions 1 E Ro(U), analogous results are possible only if there is some regularity of decrease in the errors of approximation. Nevertheless, Theorems 5.4 and 5.5 on Ro(U) do permit comparison of the rates of polynomial and rational approximation, and rational approximation on different sets. We discuss only the latter here; the former will be presented elsewhere. For simplicity, we do not present results in the greatest possible generality.

A useful tool is the following lemma of Gonchar and Grigorjan [10]:

Theorem 6.2 Let U be open, bounded and consist of k simply connected components. Let 1 : U -+ C be meromorphic, with poles of total multiplicity l and let f be analytic on the boundary aU of U. Then if j denotes the analytic part of 1 in U, obtained by subtracting from 1 the sum of the principal parts of 1 in U,

(6.4)

Proof. For the case of one component, this is the lemma of GoncharGrigorjan, with 9 replaced by 7. The general case follows easily by considering the principal parts in each of the components, and by using the maximum modulus principle. •

204 D.S. Lubinsky

We shall need the generalized Green's function gK(Z,oo) with pole at infinity for a compact set K with connected complement. This is a function harmonic in C\K, and with gK(Z, oo)-log Izi harmonic at infinity, and with boundary value 0 at all points of K, except possibly in a set of logarithmic capacity zero. The simplest example is K := {z : Izi $ s}, for which

gK(Z, OO) = log(lzl/s).

We can now prove:

Theorem 6.3 Let f be meromorphic in C. There ezists an infinite sequence :J of positive integers with the following property: Let K be any compact set with cap(K) > 0, C \ K connected, and f E Ho(K). Given s> 1, let

K. := K U {z : gK(Z,OO) $ logs}.

Then if f E Ho(K),

limsuPf'Jnn(f;K)/1/nn(f;K.) $1/s. .. --E3

(6.5)

(6.6)

If, in addition (5.6) is satisfied on some compact set T with cap(T) > 0 and f E Ho(T), then

(6.7)

Proof. Let A be the array of interpolation points whose nth row Clnl, Cln2, ... ann are nth Fekete points for K: That is,

II JanA: - anj I ;:: J;na.X II IUA: - Uj I· I<O<A:<n ul.u:z •... u .. EK 1<·<A:<n J _ J _

It is well known that the associated polynomials {WA:}r=1 satisfy

lim IWA:(z)11/A: = cap(K) exp(gK(Z, 00», A:_oo

uniformly in compact subsets ofC\K [13,Ch. 16]. Let rj(z) := Rjj(f; A; z), j ~ 1. From Corollary 2.8 in [23], there is a sequence :J, independent of K, such that for any £ > 0, and 1 < (T < s,

cap{z E KtI : If - rj I(Z)I/(2H l) ~ (1 + £);1/jj(f; K.)} --+ 0,

as j --+ 00, j E :J. The same proof shows that this remains valid if we replace cap by one-dimensional Hausdorff measure mi. The latter is defined as follows: For SeC,

m,(S) ,= inf {t, diam(B;) , S c ;9. B; } .


The inf is taken over all covers of S by balls {Bj}~l with diameters {diam(Bj)}~l . Let 1 < u' < u < s. A standard argument involving the transformation properties of ml under projections of sets onto circles or lines, enables us to construct a contour rj in Kq \ Kql, such that for large j E .:1,

rj will consist of finitely many non-intersecting simple closed curves, which themselves may be constructed (for example), from finitely many vertical and horizontal line segments. (See for example, the proof of Step 3 of Lemma 3.2 in [23] or [9]). We may assume that there is at most one simple closed curve enclosing each component of K. Let rj denote the analytic part ofrj inside rj. Applying Theorem 6.2, we have for large enough j E.:1,

and hence 1Jjj(f; K) ~ (1 + 2t);·1Jjj(f; K.).

Since t> 0 and u > 1 are arbitrary, (6.6) follows. Now assume (5.6). Then we may take.:1 = {oo, E, 3, ... } above, (Corollary 2.7 in [23]), so

limsuP1Jnn(f;K)/1Jnn(f: K.) ~ l/s. (6.9) n_oo

Now fix 6 E (0,1/2), and take

l := lU) := j +(26j), j ~ 1,

where (z) denotes the greatest integer ~ z. Let T > s. Now by Lemma 3.3 in [23], we have for large j,

1Jll(f; K.,.) ~ 1Jll(f; T)1-6/8

~ 1Jjj(f; T)1-6/8(by (5.6»

< n .. (/. K)1-6/4 -- ·Q1' ,

by Lemma 3.3 in [23] again. For j ~ 1, write

(6.10)

rj(z) := pj(z)/qj(z) := R}j(f; K; z); rf(z) := pf(z)/qf(z) := Rt.(f; K.,.; z).

From the best approximation properties of these approximants, we deduce

206 D.S. Lubinsky

By the Bernstein-Walsh lemma [42],

IIPjqf - pfqjIlLoo(Kr) ~ ,-L+illpjqf - pfqjIlLoo(K)'

We deduce that for z E K.,.,

If - rjl(z) ~ If - rfl(z) + Ipfqj - Pj&qfl(z)/Iqjqfl(z)

< E (f' K ) + ,-L+i IIqjqfIlLoo(K) [E- ·(f· K) + E (f' K )]. _ u , .,. I '* # I( ) JJ' U,.,. qiql Z

Now in each compact subset of C, qiq; has o(j) zeros as j - 00 (Cor. 2.7 in [23]), so for each £ > 0, one deduces as in [23] that,

IIq~qfIlLoo(K) . . mI{zEK.,.: J # >(I+£)'}-OasJ-oo.

Iqjql I(z)

By choosing a suitable contour 1j in K.,. \ K., we obtain that for z E 1j,

If - rjl(z) ~ rL+j(1 + 6)L+j[Ejj(fj K) + Eu(fj K.,.)].

By forming the analytic part of rj in 1j and then applying Theorem 6.2 as above, we deduce that for large enough j,

L+j Ejj(fj K.) ~ (r(1 + 26») [Ejj(fj K) + Eu(fj K.,. )].

Hence, using the choice of i = i(j),

T/jj(fi K.) ~ (r(1 + 26»)1+6+0 (1) max{"1jj(fi K), "1u(fj K.,. )1+26+0 (1)}

= (r(1 + 26»1+6+0(1)"1jj(fjK),

for large j, by (6.10) and as (1.- 6/4)(1 + 26) > 1. Since r> s and 6 > 0 are arbitrary, we deduce that

lim sup "1jj(fi K.)/"1jj(fj K) ~ s, j_oo

which together with (6.9) yields (67). • As examples for which (5.6) holds, we mention f(z) = eZ and the slowly

growing entire functions treated in [19].

7 The Possibility of Uniform Convergence

While research in the 1970's and 1980's has yielded a deep and sophisticated understanding of diagonal sequences of rational approximants, the


possibility of uniformly convergent subsequences has still not been confirmed or refuted. Indeed, there is an often quoted conjecture, originally formulated on numerical evidence, that predates these developments [2], and which might well in 1991 celebrate its thirtieth birthday:

Conjecture 7.1 The Baker-Gammel-Wills Conjecture Let f be meromorphic in Izl < 1, and analytic at O. Then there exists an infinite sequence of:l of positive integers such that

lim, [n/n](z) = f(z), .. E.7

uniformly in compact subsets of Izl < 1 omitting poles of f.

The conjecture reduces to finding a sequence:l such that as n -+ 00, n E :I, the poles eventually leave every compact set in which f is analytic. Even to demonstrate that the poles avoid some neighbourhood of zero, even for entire functions, is a difficult unsolved problem.

It is widely believed that the conjecture is false in the form above, but true for functions meromorphic throughout in C, in analogy to the situation for rows of the Pade table [6].

As positive, but severely limited, progress, we note that the conjecture has been proven true for "most" entire functions in the sense of category, and true [20] for

00

f(z) = La;zi, ;=0

if limsup la; 11/;2 < 1/3. j-oo

(7.1)

The latter condition forces f to have order zero. As a first step towards obtaining uniform convergence, one might weaken

the requirement that all interpolation conditions are placed at 0, so replacing the Pade approximant by a multipoint Pade approximant. In this situation classical results of A.L. Levin [16,17] imply a positive answer.

Let f E H2(1z1 $ 1) : That is, f is analytic in Izl < 1, and

f(e i '):= lim f(re i ') r-l-

exists a.e., with

Levin proved that if Rnn E 1I'nn n Ho(lzl $ 1) satisfies

II! - Rnnll2 = min II! - r1l2, rE1I' ... nHo(l"I:S1)

208 D.S. Lubinsky

then there exists an 1, ... ann such that

(f - Rnn){k}(anj) = 0, k = 0,1, j = 1,2,3, ... nj

(f - Rnn)(O) = O.

In particular, Rnn is a multipoint Pade approximant for f. By approximating fin H2(lzl $ sn), where Sn increases to 1 sufficiently

slowly as n -+ 00, one may readily deduce

Theorem. 7.2 Let fez) be analytic in Izl < 1. There exists an array A of interpolation points in Izl < 1 such that

lim Rnn(fj Aj z) = fez), n ..... oo

uniformly in compact subsets of Izl < 1.

As a next step, one might demand that the interpolation points at the nth stage are independent of n. That is, the array A of interpolation points in Definition 3.1 has the form

A·- (7.2)

This type of array leads to what is called Newton-Pade interpolation. In this framework, one can prove [22]:

Theorem 7.3 Let f be meromorphic in C. There exists a distinct sequence {aj }~1 of interpolation points such that for the array A of (7.2),

lim Rnn(fjAjz) = fez), n ..... oo

uniformly in compact subsets ofC omitting poles of f.

The array A necessarily depends on f. For further orientation, see [22]. Can one replace Newton-Pade approximants in Theorem 7.3 by a se

quence of Pade approximants formed from the Taylor series of f at different points? This seems very doubtful for full diagonal sequences, but if true for subsequences would yield a weaker form of the Baker-Gammel-Wills Conjecture. There is certainly scope for further work in this direction!

In the presence of sufficiently rapid and regular decrease of errors of best approximation, one can prove uniform convergence of full sequences, and as a new result in this direction, we present:


Theorem 7.4 Let f be entire and let D. := {z : Izl $ s}, s> O. Suppose that for some (T > 0,

limsuP1JH1J+1(fj Da)/1Jjj(fj Da) < 1. j-oo

(7.3)

Let A be an array of interpolation points contained in a compact set. Assume that the associated polynomials {WI: }r=1 of (3.2) satisfy, for some p> 0,

(7.4)

Then if Tj(Z) := Rjj(fj Aj z), j ~ 1, we have for T > p,

,lim IIf - Tjlli'(2(~1» /1Jjj(fj D.,.) = 1. J-+OO 00 ".

(7.5)

In particular, {Tj}T=1 converges uniformly in compact subsets of C, and we may take Tj = Ii 131, j ~ 1.

Proof. We note that the Arzela-Ascoli theorem forces (7.4) to hold uniformly on compact subsets of Izl > p. Furthermore, (7.3) implies that (5.6) holds with T = Da.

Now suppose that for some a > 0, and for some infinite sequence of integers:l,VI has a pole in Izl $ a, for j E:I. Choose s > b:= max:{p, a, (T}. Applying Corollaries 2.7 and 2.8 in [23], we have for f > 0 and b < t < s,

cap{z : Izl $ t and If - Tj l(z)1/(2H 1) ~ (1 + fH1Jjj(fj D.)} --+ 0,

as j --+ 00, and the same result holds with cap replaced by m1. As in Theorem 6.3, we can for large enough j, choose tj E (b, b(1 + f» such that

If - Tj l(z)1/(2H 1) $ (1 + f)¥1Jjj(fj D.), Izl = tj.

Letting rj denote the analytic part of Tj in D" we have Tj E 1I'j-1J-1 as Tj has a pole there. We deduce that for large j E :I,

1Jj-1,j-1(fj Db) $ (1 + f)3~1Jjj(fj D,).

Using (6.7) in Theorem 6.3 with K = Da there (recall b ~ (T, S > (T), we obtain for large j E :I,

1Jj-1J-1(fj Da) $ (1 + f)41Jjj(fj Da).

This contradicts (7.3). So {Tj }~1 has no finite limit points of poles. Then Corollary 2.8 in [23] shows that given t > T > P and f > 0,

If - Tj l(z)1/(2H 1) $ (1 + 2f)f1Jjj(fj D,), Izl = T,

210 D.S. Lubinsky

j large enough, whence

< IIf - r·11 1/(2j+l) J Loo(Dr)

j large enough, by Theorem 6.3 again. • As examples of functions for which (7.3) is known to hold, we mention

those in [19]: 00

f(z) = Lajzj , j=O

with aj =I 0, j large enough, and for some Iql < 1,

Here, .lim 1]j+l,j+1 (f; Da )/1]jj (f; Da) = IqI3/2, J-OO

and the conclusion of Theorem 7.4 was known for Pade approximants, but not for multipoint Pade approximants. Theorem 7.4 admits an obvious generalization to compact K with cap( K) > 0 and C \ K connected.

References [1] R.J. Arms, and A. Edrei, The Pade Table and Continued Fractions

generated by Totally Positive Sequences, In: Mathematical Essays Dedicated to A.J. MacIntyre, Ohio Press, Athens, Ohio, 1970, 1-21.

[2] G.A. Baker, Jr., J.L. Gammel, and J.G. Wills, An Investigation of the Applicability of the Pade Approximant Method, J. Math. Anal. Appl., 2(1961), 405-418.

[3] G.A. Baker, Jr., Essentials of Pade Approximants, Academic Press, New York, 1975.

[4] G.A. Baker, Jr. and P.R. Graves-Morris, Pade Approximants, Part I: Basic Theory, Encyclopedia of Mathematics and its Applications, Vol. 13., Addison-Wesley, Reading, 1981.

[5] D. Braess, On the Conjecture of Meinardus to Rational Approximation to eX, II, J. Approx. Theory, 40(1984), 375-379.

[6] V.I. Buslaev, A.A. Gonchar and S.P. Suetin, On the Convergence of Subsequences of the mth Row of the Pade table, Math. USSR-Sb., 48(1984), 535-540.


[7] M.A.L. Cauchy, Sur la Formulae de Lagrange Relative a l'interpolation, Analyse, algebraique, Paris, 182l.

[8] S. Dumas, Sur Ie developpement des fonctions elliptiques en fractions continues, Thesis, Zurich, 1908.

[9] A.A. Gonchar, A Local Condition of Single- Valuedness of Analytic Functions, Math. USSR-Sb., 18(1972), 151-167.

[10] A.A. Gonchar and L.D. Grigorjan, On Estimates of the Nonn of the Holomorphic Component of a Meromorphic Function, Math. USSRSb., 28(1976), 571-575.

[11] A.A. Gonchar, On the Speed of Rational Approximation of Some Analytic Functions, Math. USSR-Sb., 34(1978), 131-145.

[12] A.A. Gonchar and E.A. Rakhmanov, Equilibrium Distributions and Degree of Rational Approximation of Analytic Functions, Math. USSR-Sb., 62(1989), 305-348.

[13] E. Hille, Analytic Function Theory, Volume II, Chelsea, New York, 1987.

[14] L. Jacobsen and H. Waadeland, When does f(z) have a Regular Cfraction or a Nonnal Pade Table?, J. Comput. Appl. Math., 28(1989), 199-206.

[15] J. Karlsson, Rational Interpolation and Best Rational Approximation, J. Math. Anal. Appl., 53(1976), 38-52.

[16] A.L. Levin, The Distribution of Poles of Rational Functions of Best Approximation and Related Questions, Math. USSR-Sb., 9(1969), 267-274.

[17] A.L. Levin, The Distribution of the Poles of the Best Approximating Rational Functions and the Analytical Properties of the Approximated Function, Israel J. Math., 24(1976), 139-144.

[18] A.L. Levin and D.S. Lubinsky, Best Rational Approximation of Entire Functions whose Maclauren Series Coefficients decrease rapidly and smoothly, Trans. Amer. Math. Soc., 293(1986), 533-545.

[19] A.L. Levin and D.S. Lubinsky, Rows and Diagonals of the Walsh Array for Entire Functions with Smooth Maclauren Series Coefficients, Constr. Approx., 6(1990), 257-286.

[20] D.S. Lubinsky, Pade Tables of a Class of Entire Functions, Proc. Amer. Math. Soc., 94(1985), 399-405.

212 D.S. Lubinsky

[21] D.S. Lubinsky, Pade Tables of Entire Functions of Very Slow and Smooth Growth II, Constr. Approx., 4(1988), 321-339.

[22] D.S. Lubinsky, On Uniform Convergence of Rational, Newton-Padl Interpolants of Type (n, n) with Free Poles as n -+ 00, Numer. Math., 55(1989), 247-264.

[23] D.S. Lubinsky, Distribution of Poles of Diagonal Rational Approximants to Functions of Fast Rational Approximability, Constr. Approx., 7(1991), 501-519.

[24] AI. Magnus, Caratheodory-Fejer-Gutknecht- Trefethen Determination of Varga's Constant "1/9", Preprint B-1348, Inst. Math., Katholieke Universiteit Leuven, Louvain, 1986.

[25] J. Nuttall, The Convergence of Padl Approximants of Meromorphic Functions, J. Math. Anal. Appl., 31(1970), 147-153.

[26] J. Nuttall, Location of Poles of Padl Appronmants to Entire Functions, In: Rational Approximation and Interpolation, (Eds.: P.R. Graves-Morris, E.B. Saff and R.S. Varga), Springer Lecture Notes in Math., Vol. 1105, Springer, Berlin, 1984, pp. 354-363.

[27] J. Nuttall, Asymptotics of Diagonal Hermite-Pade Polynomials, J. Approx. Theory, 42(1984),299-386.

[28] O. Perron, Die Lehre von den Kettenbriichen, Chelsea, New York, 1929.

[29] Ch. Pommerenke, Padl Approximants and Convergence in Capacity, J. Math. Anal. Appl., 41(1973), 775-780.

[30] E.B. Saff, The Convergence of Rational Functions of Best Approximation to the Exponential Function II, Proc. Amer. Math. Soc., 32(1972), 187-194.

[31] E.B. Saff, On the Degree of Best Rational Approximation to the Exponential Function, J. Approx. Theory, 9(1973), 97-101.

[32] E.B. Saff and R.S. Varga, On the Zeros and Poles of Padl Approximants to eZ , II, In: Pade and Rational Approximations: Theory and Applications, (Eds.: E.B. Saff, R.S. Varga), Academic Press, New York, 1977, pp. 195-213.

[33] E.B. Saff and R.S. Varga, On the Zeros and Poles of Padl Approximants to eZ , III, Numer. Math., 30(1978), 241-266.

[34] H. Stahl, Orthogonal Polynomials with Complex Valued Weight Functions I, Constr. Approx., 2(1986),225-240.


[35] H. Stahl, Orthogonal Polynomials with Complex Valued Weight Functions II, Constr. Approx., 2(1986), 241-251.

[36] H. Stahl, A Note on Three Conjectures by Gonchar on Rational Approximation, J. Approx. Theory, 50(1987),3-7.

[37] H. Stahl, Existence and Uniquess of Rational Interpolants with Free and Prescribed Poles, In: Approximation Theory, (Ed.: E.B. Saff), Springer Lecture Notes in Math., Vol. 1287, Springer, Berlin, 1987, pp. 180-208.

[38] H. Stahl, General Convergence Results for Pade Approximants, In: Approximation Theory VI, (Eds.: C.K. Chui, L.L. Schumaker, J.D. Ward), Academic Press, San Diego, 1989, pp. 605-634.

[39] L.N. Trefethen, The Asymptotic Accuracy of Rational Best Approximations to eZ on a Disc, J. Approx. Theory, 40(1984), 380-383.

[40] H. Wallin, The Convergence of Pade Approximants and the Size of the Power Series Coefficients, Appl. AnaL, 4(1974), 235-251.

[41] H. Wallin, Potential Theory and Approximation of Analytic Functions by Rational Interpolation, In: Proc. of the Colloquium on Complex Analysis at Joensuu, Springer Lecture Notes in Math., Vol. 747, Springer, Berlin, 1979, pp. 434-450.

[42] J .L. Walsh, Interpolation and Approximation in the Complex Domain, 5th edn., Amer. Math. Soc. Colloq. Publns., Vol. 20, Amer. Math. Soc., Providence, 1969.

D.S. Lubinsky Dept. of Mathematics Witwatersrand University WITS 2050 Johannes burg REP. OF SOUTH AFRICA

Expansions for Integrals Relative to Invariant Measures Determined by Contractive Affine Maps

C.A. Micchelli

ABSTRACT We discuss expansions for integrals to invariant measures of certain stationary Markov chains determined by contractive affine maps. In the homogeneous case, Appell polynomials generated by the Fourier transform of the invariant measure determines the expansion. Some facts about the spectral radius of a stationary subdivision operator and the Lipshitz class of refinable functions are also included.

1 Introduction

Our principal goal in this paper is to develop further ideas from Demko [5] on expansions for integrals relative to invariant measures of certain stationary Markov chains determined by contractive affine maps. At the same time we take this opportunity to discuss the relationship of Iterated Function Systems [1] and Stationary Subdivision Schemes as studied in [3].

As for subdivision, we focus on the problem of determining the spectral radius of the subdivision operator and show how it is useful in proving the convergence theorems. Upper and lower bounds are given for the spectral radius and its relationship to the Lipschitz class of the limit function of the subdivision operator is established.

To facilitate a comparison of the results obtained here with those of [5] we begin by reviewing the setting described in [5]. Let {Wi : 1:5 i :5 N} be a family of contractive affine maps on R m and {Pi : 1:5 i :5 N} probabilities, that is,

N

Pi> 0, i = 1, ... ,N, LPj = l. j=1

(1.1)

W := {W1, ... , WN; P1, ... , PN} is an example of an iterated function system (IFS). The attractor of W, denoted by Aw, is the unique fixed point of the set-valued mapping

N

A U (A) A~Rm. -+ Wi ,

i=1

PROGRESS IN APPROXIMATION THEORY (A.A. Gonchar and E.B. Saff, eds.), @Springer-Verlag (1992) 215-239.

(1.2)

215

216 C.A. Micchelli

Consequently, if co (A) := convex hull of A then

N

U wi(co(Aw» i=1 i=1

There exists a unique probability measure I' supported on Aw so that for all functions IE C(co(Aw»

N J Idl' = ~Pi J 1 0 Widl' 1=1

(1.3)

[6], see also [1]. Here log is the standard notation for function composition (f 0 g)(z) := I(g(z».

The measure is attractive in the sense that

lim (T" I)(z) = J Idl', z E R m

" ..... 00

(1.4)

for all IE C(Rm ), where T is the linear operator

N

TI= ~pdOWi' (1.5) i=1

cf. [2]. The theorem of Demko which is of interest to us here is the following

result.

Theorem 1.1 (Demko) Let W be a one dimensional homogeneous IFS system, that is, Wi(Z) = az + bi , i = 1, ... , N, a, bi , z E Rl. Suppose I is analytic on a sufficiently large neighborhood 01 co (Aw). Then, lor any integer k ~ 1 and any z E co(Aw)

J Idl' = (T" I)(z) + f: dn(z)an" J JCn)dl' (1.6) n=l

where

(1.7) it,.·· ,in ~ 1,

Expansions for Integrals Relative to Invariant Measures 217

and

j ~ 0, (1.8)

otherwise.

Moreover,

J dndll = 0, n = 0, 1,2, .... (1.9)

We will provide various extensions of this result which neither require the IFS be homogeneous, one dimensional, nor that {Pl, ... ,PN} are probabilities.

Our interest in the above theorem, comes from the close connection between a measure Il satisfying (1.3) and a refinable function associated with a stationary subdivision scheme (SSS), [3]. Therefore, because of [4], some of what we say has applications to Multiresolution A nalysis and Orthogonal Wave/ets. These connections will be pointed out later as we improve upon Demko's interesting formula (1.6). Furthermore, in the course of our analysis we will observe that the polynomials appearing in (1.6) form an Appell sequence, a fact that seems to have been known to Professor Demko (personal conversations) although not contained in [5]. This connection to Appell sequences and equation (1.9) also occurs in [3].

We begin with:

2 One Dimensional IFS

We suppose throughout this section that

(2.1)

for a" b" x E Rl with lail < 1 and that

(2.2)

However, we do not generally suppose that each Pj is positive; nevertheless, we still refer to Was an IFS.

Set

and

N

>'A: := >'A:(W) := LPjaJ, k = 0, 1,2, ... , j=l

(2.3)

(2.4)

218 C.A. Micchelli

Note that by (2.2) 1 = ~o E A. We also use lI'A:(Rl) to denote the linear space of all univariate polynomials of degree $ k.

Proposition 2.1. Suppose A consists of distinct elements. There exists a unique sequence of polynomials {cA:(x) : k E Z+} such that

CO(Z) = 1,

CA:+1(z) = ZA:+1 + rA:(z), rA: E lI'A:(Rl), k E Z+, (2.5)

and (2.6)

Proof. We let vi (z) := xi, j E Z+ and observe that

TVj = ~jVj + g;, j E Z+,

for some fJ; E lI'j_l(R1), j E Z+ \ {OJ, and qo = O. Thus, there is a lower triangular matrix M = (Mij), i,j E Z+ such that

and

00

TVi = EMijVj . j=O

Mii = ~i, i E Z+.

Specifically, the matrix M is given by

(2.7)

(2.8)

(2.9)

For each k E Z+, we determine a vector (wf), i E Z+ so that w% = 1 and w; = 0, j > k. The remaining components wf, 0 $ i $ k - 1 are determined inductively by

i: (~i: - ~l)wf = E wf Mil, 0 $ i $ k - 1, (2.10)

i=l+l

so that 00

Ewf Mij = ~i:w:, j E Z+. (2.11) i=O

If we set 00

Ci:(z) = Ewfvi(Z) (2.12) i=O


then equation (2.5) holds by construction of (wf), i E Z+, and from (2.7), (2.11), and definition (2.12) we get

TCk = T (twfVi) 1=0

t, (t,WtM+, 00

Ak Lwjvi ;=0

= Akck.

This computation proves the proposition.

ReIDark 2.1. Note that for each k there is a unique polynomial which satisfies (2.5) and (2.6). Also observe that even if A does not consist of distinct elements, there is a Ck E 'Irk (Rl) \ {O} which satisfies (2.6) because the matrix M is lower triangular. However, Ck may not be of exact degree k.

Next, we introduce the generating function for the sequence of polynomials

00 k

C(x,Z)=LCk(X)~!, XERl, zECl . (2.13) k=O

This (formal) power series satisfies a functional equation which we describe In

Proposition 2.2. Suppose (2.13) converges in the disk ~R = {z : Izl ~ R}, R> O. Then for z E ~R we have

Proof.

N N

LPtC(atx + bt , z) = LPtC(x, atz ). t=l t=l

N 00 k N

LPtC(atx + bt, z) L ~! LPtCk(atX + bt) t=l k=O t=l

00 k

L(Akck(x» ~! k=O

t,PI (~a~c.(%)~;) N

= LPtC(x, atz ). t=l

(2.14)

220 C.A. Micchelli

In the homogeneous case

al = a, l = 1, ... , N (2.15)

we see that C(x,z) = e"'ZF(z) (2.16)

satisfies (2.14) when

F(az) = (t,PleblZ) F(z), F(O) = 1. (2.17)

The generating function (2.16) has the characteristic form for Appell sequences. More on this connection later.

To proceed further we need to construct linear functionals which are biorthogonal relative to the sequence of polynomials {cn : n;:::: O} constructed in Proposition 2.1. As we have pointed out, when we are dealing with a homogeneous IFS, the sequence {cn : n;:::: O} forms an Appell sequence, so that

C~ = nCn_l, n;:::: 1.

This also follows from the fact that (Tf)' = aTf' where aj a, j 1, ... ,N. Consequently, when in addition, Pj ;:::: 0, j = 1, ... , N, the invariant measure p completely determines a biorthogonal system, by the prescription

J fdl'k := ;! J f(l<)dl'.

This is easy to see, since in particular

J cldl'k = ~! J c~l<)dl'. Thus, for l < k we clearly have

J cldp" = O.

When l > k

J cldpk = (l~! k)! J Cl_kdl'.

However, by specializing (1.3), (1.5), and (2.3), (2.5) to a homogeneous IFS we get

J Ckdp = J TCkdp = ak J Ckdl'

and so, recalling that lal < 1, we obtain

J Ckdp = 601<.


In summary, we have

The general nonhomogeneous case introduces some complications. We view the situation as follows. Suppose X is a Banach space of functions on co (Aw) which includes all polynomials 11' = U:EZ+ 1I'k(Rl). Let Co be the closed linear space spanned by {en : n ~ I}. Since

dist(l,Co) = inf 111- cll = max IL11, cECo ULUSt

LEe;'

(2.18)

we see, as is well known, that a necessary and sufficient condition for the existence of an L EX· such that

(2.19)

is that M-1 $111- cll, c E Co. (2.20)

A simple sufficient condition which insures for some M, that these requirements are met is the following result.

Proposition 2.3. Let X be as above and A be a set of distinct points. Suppose further there is an M > 0 and a p, 0 < p < 1, such that

(i) IITk fll $ Mllfll, f E Co, k E Z+, (ii) IAtl $ p, k =1,2, ....

There is an LEX· such that

Proof. For any finite sum c = E~=l dtCk we have

Hence letting m -+ 00 gives (2.20) for any c E Co.

Remark 2.1. When W is an IFS, so that Pi ~ 0, i = 1, ... , N, then, according to the set-up in the introduction, we can choose X = C( co Aw) which means that M = 1 works in (i) of Proposition 2.3 since IITtll = 1, k E Z+. Moreover, for k E Z+ \ {OJ, IAtl $ p := maxl<l<N lall < 1 so that (ii) also holds. Consequently, in this case the linear Junctional in Proposition 2.3 corresponds to a measure and so (1.4) follows when A consists of distinct elements because then Co = C(coAw ).

222 C.A. Micchelli

Proposition 2.3 gives us the first linear functional in our biorthogonal family. The remaining ones come from

Proposition 2.4. Let X be a Banach space of functions on co (Aw) which includes all polynomials. Suppose there are real numbers Mj > 0, j E Z+, o < p < 1, such that

(i) II(Tkf)(j)1I ~ (Aj)kMjllf(j)II, f U) E Co, j,k E Z+, (ii) 0 < IAk+11 ~ plAkl, k E Z+. Then there are linear functionals Lj EX", j E Z+, such that

LjC~i) = bjl' j,l E Z+,

Proof. First we define

and

Then

and (Tw f)(n = Aj(W)Twjfj.

Consequently, (i) implies

IIT~jfll ~ Mjllfll, f E Co, j,k E Z+,

and also (ii), (iii) give

and Ao(Wj) = 1.

Thus by Proposition 2.3 there is an Lj E X" such that

Li(cj,k) = bOlo,

(2.21)

(2.22)

(2.23)

(2.24)

(2.25)

(2.26)

where {Ck,j : k E Z+} are the unique sequences of polynomials such that

c~jl(z) = k! and ),

Tw jCj,k = Ak(Wi)Cj,k.

However, differentiating the equation


j-times and using (2.23) and (2.24) gives

Twjc~2j = ('\Hj(W)/'\j(W»cnj = '\A:(wj)cnr

Hence we have proved that

(j). _ (1:.+ j)! . cA:+j - j! CJ,A:

and therefore by (2.23), the linear functionals {Lj : j E Z+} have the desired properties described in the proposition.

Theorem 2.1. Let the hypothesis 01 Proposition 2.4 hold. Then lor any I: E Z+, Z E R, and polynomial IE 1f,

00

(TA: I)(z) = I>n(Z)(.\A:)n Ln/(n). (2.27) n=O

Proof. Since TA:c,. = (.\A:t c,. and LnJn) = 6rn both sides of (2.27) agree for I = cr , r E Z+ and hence also for any polynomial I. Corollary 2.1. Let W be a homogeneous IFS, with at = a, I = 1, ... ,N, invariant measure 1', and attractor Aw ~ R. Suppose

(2.28)

is the Fourier translorm 01 the invariant measure 1'. Define the Appell sequence {cn(z) : n E Z+} by

(2.29)

Then lor any Z E R, any I: E Z+, and any entire function I,

J OanA: f Idl' = (TA: I)(z) - E Cn(Z)~ fn)dl'.

n=1 n. (2.30)

Proof. Most of this result follows from Theorem 2.1. What remains to prove is the identification of the polynomials {cn : n ~ O} with the generating function,

Since I' has a compact support on co(Aw), it is clear that pew) is an entire function wE C of finite type. Moreover, p(O) = 1 which guarantees that

00

( A(. »-1 ""' Vn n I' az = L...J ,% , n=O n.

Vo =0,

224 C.A. Micchelli

in some disk {z : Izl::::; r}, r > O. Consequently, (2.29) defines polynomials {en : n ~ O} given explicitly as

n

cn(:r:) = L (~) vn_;xi. ;=0 J

(2.31)

Moreover, it is clear by the Cauchy integral formula that the series on the right hand side of (2.30) converges and absolutely for any :r: E R. Next we verify that

(2.32)

For this purpose, we use the functional equation (1.3) for the invariant measure to obtain the equation

fi(z) = (tpteiZbl) fi(za). t=l

Hence we obtain, for Izl < r,

Noon

LPt L cn(a:r: + btl; t=l n=O n. N

= L Pt(fi( _iz))-l e(ax+bl )z t=l

( A( . ))-1 axz jt(-iz) J.t -IZ e A ( • ) J.t -laZ

eaxz 00 zn A( • ) = Lancn(:r:)_,

J.t -laZ n=O n.

which proves (2.32) and hence finished the proof of the theorem.

3 Stationary Subdivision Schemes. and Refinable Functions

In this section we relate some of the concepts and results of the previous section to the study of refinable functions of [3]. Here we focus on the univariate case and recall the basic equation studied in [3]:

<p(:r:) = L aj<p(2:r: - j), :r: E R. (3.1) jEZ

This equation is fundamental in the study of iterates of the stationary subdivision operator

(BaAj) := L aj-2k Ak, A = (Ak)kEZ E J!OO(Z). (3.2) kEZ


The set a := {aj i e Z} is called the mask of the scheme and the Laurent series

a(z) = Eajzi (3.3) jEZ

is called the symbol of a. The iterates of So act as a "filling in" rule to create a curve defined on R. Therefore for practical reasons we assumed the mask is finitely supported. Thus

suppa = {j : aj i= O} (3.4)

has at most a finite number of elements. We said that the stationary subdivision scheme, SSS, converges, provided for every ~ = F(Z) there is an I>. e C(R) such that

Continuous limit curves are desirable in practice, and therefore most of what is given in [3] pertains to this situation. It was observed that a necessary condition for convergence to a nontrivial limit 1>.., ~ e F(Z), is that

E a2j = Ea2j+1 = 1 (3.5) jEZ jEZ

and that I>.(z) = E ~jlP(z - i), z e R, (3.6)

jEZ

where II' is the unique continuous solution to the refinement equation (3.1) which satisfies

EIP(z-i) = 1, z e R. jEZ

(3.7)

Note that II' necessarily has compact support in co (supp a) and we refer to it as the (unique) refinable function associated with So.

Choose any continuous function I(z), multiply both sides of (3.1) by I(z) and integrate with respect to Lebesgue measure on R. Upon simplification we get

where

and

J Idprp = 4 E·aj flo Rjdprp jEZ

Rjz = ~(z + bj), j = I, ... ,N,

{aj : iesuppa}={bt. ... ,bN }.

(3.8)

(3.9)

(3.10)

226 C.A. Micchelli

Thus we obtain the following identification with our previous notation:

and

1 Pj = -aj

2 (3.11)

(3.12)

A refinable function gives rise to a (signed) invariant measure with "probabilities" (1/2)aj and conversely. Note also the attractor of the set of contractive mappings Rl, ... , RN is [bl, b N], if b1 < b2 < ... < b N. Thus conditions on the mask which insure that the refinement equation (3.1) has a continuous solution, for instance, gives conditions on Pj = (1/2)aj, j E supp a, so that there exists an invariant (signed) measure dl' satisfying (1.3) which has a continuous Radon-Nikodym derivative relative to Lebesgue measure.

There are two approaches followed in [3] for the study of the refinement equation. First, and principally, we study the convergence question for SSS. This leads us to a "constructive" method for obtaining a solution of (3.1) by means of iterates of Sa. The other approach is through Fourier Analysis. We try to construct (j; as an infinite product and then apply the Paley-Wiener Theorem to invert the Fourier transform. Let us review these methods here and at the same time improve upon the discussions in [3].

For the convergence question a useful observation about the stationary subdivision operator is the following fact from [3] which we describe in somewhat different terminology.

Lenuna 3.1. [3] Let Sa be any SSS scheme with a finite mask a. Then limk_oo S! = 0 in the uniform operator topology on ,eOO(Z) if and only if it goes to zero in the strong operator topology.

Proof. From [3]

(S!A)j = L:a:_2• t At (3.13) tez

where 2 2.-1 '"' k j ak(z) := a(z)a(z ) ... a(z ) = L..J aj z . (3.14)

jez

From (3.13) it easily follows that

where IIAlioo = SUPjez IAj I. Hence S! -+ 0 in the uniform operator topology if and only if

Expansions for Integrals Relative to Invariant Meunres 227

while if S! -+ 0 on the strong operator topology then

lim sup la; I = O. ~_oo jez

Now, it was shown in [3] that there is a constant d > 0, which is independent of k, and only dependent on co (supp a) such that

IIS!lIoo = sup I: la;_2.tl $ dsup la;l. jez tez jez

(3.15)

Hence if S! -+ 0 in the strong operator topology then S! -+ 0 in the uniform operator topology. The converse is trivial.

This leads us to the following fact, most of which is contained in [3].

Theorem 3.1. Lefa = {aj : j E Z} be a given mask of compact support. Then the SSS Sa converges (nontrivially) if and only if

a(z) = (1 + z-l )q(z), q(l) = 1,

where the spectral radius of S9 satisfies

r(q):= lim 1I~II~n < 1. n-oo Moreover, when (3.16) and (3.17) hold,

/>.(z) = I:~jCP(z - i), jeZ

I: cp(z - j) = 1, jez

(3.16)

(3.17)

(3.18)

(3.19)

suppcP ~ co (supp a). (3.20)

For any p, 0 r(q) will do)

cp E LipT (3.21)

where T E (0, 1] and 2-1" = p. (3.22)

Finally, for the Appell sequence {cn : n ~ O} defined by

00 n

L cn(z); = e#:z /tjJ(-iz) n=O n.

we have for any z E R, any k E Z+, and any entire function f,

(3.23)

228 C.A. Micchelli

where

(Tf)(z) = ~ :E ajl (~(z + j») . iEZ

The part of this theorem which is concerned with the convergence of the SSS appears in [3]. The last claim, (3.23), is a special case of Corollary 2.1, applied to the refinable function cp for the convergent subdivision scheme Sa. We will prove (3.21) below, and at the same time review the convergence argument used in [3]. All of this will follow our discussion of several useful conclusions which can be drawn from Theorem 3.1.

Some information is available for estimating the special radius Sa. But this central question remains unresolved. We mention the following facts. Define

( )l/n

M(a) = lim max lan(z)1 , n_oo Izl:51 (3.24)

where, as before an(z) = a(z)a(z2) ... a(z2"- 1).

It is easy to see that the limit in (3.23) exists, and, moreover, we have the inequalities

1 '2M(a) :5 rea) :5 M(a). (3.25)

The proof of (3.25) is obtained from arguments used in [3]. Specifically, if supp a ~ [-N, N] then (3.14) implies that supp a i ~ 2i[-N, N] and so for Izl:5 1,

lai (z)l:5 2t+1Nrn:a:la:l:5 2t+1NIIS!lIoo.

Therefore, first maximizing over Izl :5 1, and then taking k-th roots of both sides of the resulting inequality, and letting k - 00 proves the lower bound in (3.25) for the spectral radius. For the upper inequality we have from (3.15)

consequently IIS!lIoo < dmax lai(z)l·

- Izl~l

Taking k-th roots of both sides of this inequality and letting k - 00 proves the upper bound for the spectrhl radius given in (3.25).

Note that, since M(a) 2:: la(I)1 we see that for q defined in (3.16) we have

1 r(q) 2:: 2' (3.26)

a fact already implicit in Theorem 3.1. If q(-I) = 0 and lJi 2:: 0, j E Z we also have

and so r(q) = 1/2 in this case.


Thus, whenever

a(z)

q(z)

(1 + Z)2 = q(Z), q(l) = 1 2

= E~rf, ~ ~o, jeZ+, jEZ

the SSS Sa converges and its refinable function is in Lip 1. As for the case of positive masks, we have the following stronger result

from [8] (see [3] for several multivariate versions).

Theorem 3.2. [8]. Suppose {aj : j E Z} is a non-negative mask, with support a = {j t :5 j :5 m} (a set of consecutive integers) such that m-t~ 2 and

(3.27)

Then the SSS Sa converges.

Proof. A proof of this result can be based on Theorem 3.1. We require the following quantities, introduced in [3]:

and

1 p:= -2 . ma,x E laj-2k - ar_ul,

IJ-rl<m-t kEZ

D("\) = max{l..\j - ..\kl : Ij - kl < m - t}.

(3.28)

It was shown in [3] that under the hypothesis of the theorem p < 1 and

D(Sz..\) :5 pD("\).

Using (3.27) we can factor the symbol a(z) as in (3.16)

a(z) = (1 + z-1)q(z), q(l) = 1.

Equivalently,

where (~..\j) := ..\H1 -..\j, j E Z,

is the forward difference operator. Hence from (3.29) we get

(3.29)

(3.30)

(3.31)

which implies that r(q) :5 p < 1 and so Theorem 3.1 proves the convergence of Sa. Note that we have also established that its refinable function in Lip T where 2-'1" = p.

230 C.A. Micchelli

The next theorem from [3] is useful for the construction of compactly supported orthonormal wavelets, [4]. We require two definitions. The first, a familiar concept from Function Theory, is the geometric mean of the Laurent polynomial q

(3.32)

and the other is the definition of refinably bounded. We say q is refinably bounded provided that there is a constant M > 0 such that

where, as before, q1:(z) = q(z)q(Z2) ..• q(Z21o

-1

).

Theorem 3.3. [3]. Let a = {aj : j E Z} be a finite mask such that

(i) a(z) = (1 + z-l )q(z), q(l) = 1 (ii) G(q) < 1, and (iii) q is refinably bounded.

Then Sa converges.

(3.33)

Proof. By the mean ergodic theorem for the doubling map, see [3], we have

lim .!.log I q1: (e2lrit)I = t log Iq(e2lria)I dO" 1:-+00 k Jo

a.e. t E [0,1]. Therefore (ii) implies

lim Iq1:(e2lrif)I = 0, a.e. t E [0,1]. 1:-+00 However, using (iii) and the bounded convergence theorem we get

lim 11 Iq1:(e2lrit)I dt = O. 1:-+ooJo

Consequently, using (3.15) we conclude r(q) < 1, because

IIS:lIoo ~ d~a:lqfl ~ d 11Iq1:(e2lrif)ldt J • Jo

and so Theorem 3.1 again implies that Sa converges. This proves Theorem 3.3.

Next we turn to the

Proof of Theorem 3.1. For the necessity, we observe that for the sequence

' .. _ { -1/2, A J .-

1/2,

j<O

j~O


we have by (3.31) that

(S!A)j+1 - (S!A)j = q;, i E Z.

Hence, using the triangle inequality twice, we obtain

IIqklloo = ~~~ Iq;1 ~ 2;~~ II>. (1k) -(S!A)jl

(3.34)

The function f>..(z) = EjEz AjCP(Z- i) is uniformly continuous. To see this, suppose that Iz - YI ~ 6, and consider the sum

f>..(z) - I>.(Y) = E Aj(cp(Z - i) - cp(y - i». (3.35) jEZ

If supp cP ~ [l, m], then there are at most m - l + r nonzero summands in (3.29) where r is the least integer ~ 6. Hence

If>..(z) - I>.(Y) 1 ~ (m -l + 1 + 6)IIAlloow(cpj 6).

Since cP is continuous and of compact support, its modulus of continuity, w(cp, 6), goes to zero as 6 -+ 0+. Recalling that Sa was assumed to converge, we conclude from (3.34) that limk-+oo IIqklloo = 0 which proves, by using (3.15), that r(q) < 1.

For the sufficiency we use the function

{ 1-11- zl,

g(z) = 0,

o ~ z ~ 2,

otherwise.

We suppose that co (supp a) = [l, m] so that m -l ~ 2, since r(q) < 1, and we set

fo(z) = g(z -l).

Therefore, supp fo ~ [l, m], and so it follows, inductively on n that the functions

fn(z) = L: a;!n_l(Z - i), n = 1,2, ... , (3.36) jEZ

are continuous and have support in [l, m] as well. Note that

and

Efo(z - i) = 1, fo(z) ~ 0, z E R, jEZ

fo(z) = L: b;!0(2z - j) jEZ

(3.37)

(3.38)

232 C.A. Micchelli

where ! 1/2,

b; = 1,

0,

j =l, l+2

j=l+1

otherwise.

This implies the operator Sa - S" is zero on the constant sequence. Hence there is a constant M > 0 such that

Iterating equation (3.36) gives

In(z) = L:(S:-lc5);/o(2n - 1z - j) ;EZ

(3.39)

(3.40)

where (15); = 1, if j = 0 and zero for j # O. Moreover, using (3.36) and (3.38) we get

In(z) = L:(S,,(S:-lc5»;/o(2nz - j). ;EZ

(3.41)

Hence (3.40) (with n replaced by n + 1), (3.41), (3.39) and (3.37) give the inequality

(3.42)

From (3.16) we have as:: = s; a and so if p satisfies the hypothesis of the theorem there is an no E Z+ such that for n ~ no

(3.43)

Furthermore, differentiating (3.40) we see that

I/~l)(z)1 ~ 2n- 1IlaS:-1c5 l1oo ~ 2n- 1pn-l,

a.e., z E R, which provides us with the inequality

Now, it follows from (3.43) that {In(z) : n ~ O} is a Cauchy sequence, uniformly for z E R. Hence there is a continuous function 11', with support in (l, m) such that

(3.45)

uniformly in z E R. Clearly, from (3.36) II' necessarily satisfies the functional equation

ip(z) = L: a;ip(2z - j). ;EZ

(3.46)


Moreover, using the equations

L:a2i = L:a2i+1 = 1 iEZ iEZ

(see (3.16)), (3.37) and (3.36) imply inductively on n that

L:fn(z - j) = 1, Z E R. iEZ

Hence, letting n -+ 00 in this equation, we also have

If we set

L: <p(z - j) = 1, Z E R. iEZ

J>..(z) = L: Aj<p(Z - j) iEZ

then, the refinement equation (3.46) gives for all n E Z+

J>..(Z) = ~)S~ A)i<P(2n z - j) jEZ

so that using (3.47) we have for each i E Z

1>.. (2~) - (S~ A)i = L:«S~ A)j - (S~ A)i)<p(i - j). jEZ

(3.47)

(3.48)

Observe that, independent of i, there are at most m - f nonzero summands in this sum and therefore we have

If>. (2~) - (S~)A)il ~ (m -f)lIaS~Alloo. This inequality implies that

lim suplJ>.. (2i) -(S~A)il =0, n_oo iEZ n

that is, Sa converges and <p is its refinable function. It remains to prove that <p E Lip T, where T is defined by equation (3.22).

For this purpose, first we observe that by (3.43)

M Ifn(z) - <p(z)1 ~ -1-pn- 1 , n ~ no, Z E R.

-p

Therefore, combining this inequality with (3.44) we get

2M l<p(z) - <p(y) I ~ -1-pn + 2npnlz - yl, n ~ no,

-p

234 C.A. Micchelli

I.e.

where

Since

Icp(z) ~ cp(y)1 ~ v(lz - yl)

v(t):= inf (2M p" + 2" P"t) . "~"o 1- P

pv(2t) =

=

we have v(t) ~ pv(2t), t > O.

Define the constant

then for t E [1/2,1]

We claim that

(3.49)

(3.50)

(3.51)

for all t E [0,1]. To see this, we suppose that (3.51) is valid for 2-t ~ t ~ 1. Then for the larger interval2-t - 1 ~ t ~ 1 we have 2-t ~ 2t ~ 1 and so by (3.22) and (3.50)

v(t) ~ pv(2t) ~ pT(2tY = p2TTtT = TtT.

Thus (3.51) follows by induction on i E N. Hence, combining (3.51) with (3.49) establishes that cp E Lip'T, which finishes the proof ofthe theorem.

Next we turn to the study of series expansions of the type (2.30) by Fourier transform techniques. This can be done in the generality of

4 Multivariate Homogeneous IFS

Our purpose in this section is to extend some of the results of Section 2 to multivariate homogeneous IFS. Thus, in this case we have

(4.1)


and N

LP;=1 (4.2) ;=1

where A is an m x m matrix and b1 , •. . , bN are m-vectors. Our first goal is to obtain the polynomial eigenfunctions of the operator

N

(Tf)(z) = LP;/(Az + b;). (4.3) ;=1

We do this under the hypothesis that A has eigenvalues A = (A!, ... , Am) such that p := max19~m IA; I < 1 and A is diagonalizable, that is, there is a matrix V, such that A = V- 1AV, where A = diag(AI, ... ,Am). We denote the standard inner product between Z,Y E R m as (z,y) = E~l ZiYi and define

N

g(z) = LP;e(Vbj,z), z E em. (4.4) ;=1

Then g(O) = 1 and g(z) is an entire function on em. Therefore it has an everywhere convergent power series expansion

g(z) = L ga za . aEZ+,

Let us consider the functional equation

I(z) = g(z)/(Az), z E em

(4.5)

(4.6)

for a function I. We claim that this equation has a unique solution which is continuous in a neighborhood of the origin normalized so that 1(0) = 1. Moreover, this solution is in fact also entire. To see this, we observe that the sequence of entire functions

converges uniformly and absolutely because

00 00

L 11- g(Alcz)1 < L L 1:=1 1:=1 aEZ+,\{O}

00

= L Igazal L(IAla)1: aEZ+,\{O} 1:=1

< (1- p)-1 L Iga(Az)al < 00.

aEZ+,\{O}

(4.7)

236 C.A. Micchelli

We let 00

goo(z) = II g(A1: z) (4.8) 1:=0

so that goo (0) = 1. Now it is an easy matter to see that J = goo, since (4.6) implies that

J(z) = g1:(z)J(A1:+1 z), k E Z+,

and so in the limit we get J = goo. Moreover, it is apparent that

goo(z) = g(z)goo(Az). (4.9)

We now consider the Appell sequence {qIJ : I' E Z+} associated with the function h(z) = 1/goo(z) which is analytic in some neighborhood ofthe origin;

It follows that :cIJ

qIJ(:C) = -, + v(:c) 1'.

(4.10)

where v E M; := span of all the monomials :cti , v E Z+, with v ::; 1', v =1= 1'.

We introduce the polynomials

CIJ(:C) := qIJ(V:c), I' E Z+ (4.11)

and proceed to prove that

(4.12)

Observe that

which, by (4.10), becomes e(Vz-.Az)h(Az). Recalling that h- 1 = goo, equation (4.9) shows that this is the same as

N

h(z)g(z)e(Vz-.Az) = h(z) :~:::>je(AVZ-+Vb;.z) j=l

N zIJ L:Pj L: qIJ(AV:c + Vbj ),

j=l IJEZ+ 1'.


where we have again used (4.10). Invoking definition (4.11) gives

which proves (4.12), by identifying powers of z. Our next step is to identify Ij fez) with the Fourier transform of a tem

pered distribution of compact support. We require, therefore, an estimate for the growth of goo in the complex plane. For this purpose, we say as in [3], that 9 is refinably stable ifthere is an integer k, such that

Ig(iz)g(iAz) .. . g(iAkz)1 ~ 1, z E R.

Note that, if Pi ~ 0, i = 1,2, ... , N, then

N N Ig(iz)1 = Epjei(V6i'~) ~ Epj = 1,

j=1 j=1

so that 9 is refinably stable (with I: = 0). In general, if 9 is refinably stable then gk(iz) is bounded by one for z E Rm and is of exponential type ~ Uk := B(1 - pH1)j(1 - p), where p = max1~j~m IAjl and B = max1~j~N IIVbj1l2, that is, for some constant I: > 0

m

Igk(iz)1 ~ l:eullllzll2, IIzll~:= ~)ZjI2, j=1

z = {ZlJ"" zm}. Therefore by the Pharagmen-Lindelof theorem, we have

19k(iz-yH~euIIII!lIl, z,yERm •

Furthermore since 00

goo(z) = II gk(Al (H1)z) l=O

we get the inequality

00

Igoo(iz - y)1 ~ eUII L: pl(k+l)IIYIl2 = e(1-p)-lulllb,U2 •

l=O

Thus by the Paley-Wiener Theorem, cf. [9] for distributions, goo(iz) is the Fourier transform of a tempered distribution of compact support. If we call this distribution v, and denote its Fourier transform by

238 C.A. Micchelli

then goo(t) = v( -it) and so equation (4.9) implies that

v(z) = g(iz)v(Az).

We set I'f = v(J(V- 1.». Then it follows that

I'f = I'Tf

for all Coo functions f. In summary, we have

Theorem 4.1. Let A be an m x m matrix such that A = V-1AV, A = diag(Al, ... ,Am) with p:= maxl5j5mlAji < 1. Suppose bj E Rm, j = 1, ... , Nand Pl, ... ,PN E R with Ef=l Pj = 1. Let g(z) = Ef:l pje(Vbj,z). Then the function

00

goo(z) = II g(Alz), g(O) = 1 l=O

is entire and the Appell sequence {q/J : I' E Z+} defined by

(4.13)

satisfies

where

and N

(TJ)(z) = L:pjf(Az + bj). j=l

If 9 is refinably stable, then there is a tempered distribution I' of compact support such that

f Tf f E Coo(Rm). I' =1' , (4.15)

Moreover, for every k E Z+, every z E R m , and every entire function f

I'f = (T" I)(z)- (4.16)

Proof. We have proved all these facts except formula (4.16) which follows along arguments similar to those used in Section 2 by using formulas (4.13), (4.14) and (4.15). We omit the details.


Acknowledgement. We wish to thank Professor Demko for providing us with a preprint of his paper "Euler MacLauren Type Expansions for Some Fractal Measures" as well as for several illuminating discussions concerning his results.

References

[1] Bamsley, M.F. and Demko, S. Iterated Function Systems and the Global Construction of Fractals, Proc. of the Royal Society of London A, 399(1985), 245-275.

[2] Bamsley, M.F., Demko, S., Elton, J., and Geronimo, J., Markov Processes Arising from Function Iteration with Place Dependent Probabilities, Annales de l'Institute Henri Poincare, 24, No.3, (1988),367-394.

[3] Cavaretta, A.S., Dahmen, W., and Micchelli, C.A., Stationary Subdivision, IBM Research Report No. 15194, 1989, to appear in Memoirs of AMS.

[4] Dahmen, W. and Micchelli, C.A., Stationary Subdivision and the Construction of Orthonormal Wavelets, in "Multivariate Approximation and Interpolation," ISNM 94, N. Haussmann and K. Jetter (eds.), Birkhauser Verlag, Basel, (1991), 69-90.

[5] Demko, Stephen, Euler, Maclauren Type Expansions for Some Fractal Measures, preprint.

[6] Hutchinson, J. Fractals and Self-Similarity, Indiana J. Math, 30(1981),713-747.

[7] Micchelli, C.A. and Prautzsch, H., Uniform Refinement of Curves, Linear Algebra and Applications, 114/115(1989), 841-870.

[8] Micchelli, C.A. and Prautzsch, H., Refinement and Subdivision for Spaces of Integer Translates of a Compactly Supported Function in Numerical Analysis, edited by Griffiths, D.F., and Watson, G.A., (1987), 192-222.

[9] Yosida, K., Functional Analysis, Springer-Verlag, Berlin, 1966.

Charles A. Micchelli IBM T.J. Watson Research Center Mathematical Sciences Department Yorktown Heights, NY 10598 U.S.A.

Approximation of Measures by Fractal Generation Techniques

s. Demko·

ABSTRACT In this paper we discuss the constructive approximation aspects of a measure generating technique that was developed in the last decade by fractal geometers. Measures are generated as stationary distributions of Markov chains that are related to function iteration. The general framework provides a robust machine for constructing parametric families of measures and is, thus, very well suited for measure approximation problems. This formalism - called iterated junction system theory -has found application in computer image synthesis and image compression. In fact some of the early work was motivated by ~ desire to create more realistic images for flight simulators. Other areas of potential application include dynamical systems, signal processing - especially chaotic signals, and quadrature theory.

1 A Machine for Making Measures

The paper is organized as follows. In this section, we give basic definitions and examples. In Sections 2 and 3 we discuss the important inverse problem: given a probability measure how can we construct a Markov process of a given type whose stationary distribution is close to the given measure? Geometric aspects are treated in Section 2 and analytic aspects in Section 3. This is a highly non-linear problem and a completely satisfactory answer is not known. In Section 4 we discuss the polynomial eigenfunctions of a fundamental linear operator associated with the measure generation scheme. In some cases, these eigenfunctions form an Appell sequence and share some important properties with the classical Bernoulli polynomials. In particular they arise naturally in expansions of integrals with respect to the invariant measure of the associated Markov chain. In Section 5 we propose some problems for future research.

Let K be a compact metric space and W1, ... , WN Borel measurable maps from K to itself and P1, ... ,PN positive numbers that sum to 1 (probabilities). We consider the random walk defined on K by the transition rule:

"move from x to Wi (x) with probability Pi"

·Work supported by a NATO Grant.

PROGRESS IN APPROXIMATION THEORY (A.A. Gonchar and B.B. Saff, eds.), ©Springer-Verlag (1992) 241-260. 241

242 S. Demko

and the problem of describing its long term behavior in terms of an invariant probability measure.

To study this and related matters we introduce the linear operator T defined on Borel measurable functions on K by

N

(TI)(x) = Lpd(Wi(X». (1.1) i=1

If T takes continuous functions into continuous functions then its adjoint T" takes M(K) into itself where M(K) is the space of countably additive regular Borel measures on K. In this case for II E M(K) and B a Borel set

N

(T*II)(B) = LPill(W;1(B». (1.2) i=1

Definition. The triple (K, {wil, {pill where Wi'S and Pi'S are as above is called an iterated function system (IFS) if the operator T in (1.1) takes C(K) into C(K). Here C(K) is the space of complex-valued continuous functions on K. Sometimes the underlying set K will not be mentioned.

Since the Pi'S are probabilities, T" takes P(K) into P(K) where P(K) is the set of probability measures on K: P(K) := {II E M(K) : II(B) 2: o for all Borel sets and II(K) = I}. The Markov-Kakutani (or Schauder) fixed point theorem then guarantees the existence of a I' E P(K) for which I' = T" 1'. It is possible for T" to have more than one (and thus infinitely many) fixed points in P(K). However; if the Wi'S are all contractions, there is a unique I' and it is computable.

Theorelll 1.1 Let (K { Wi}, {pil) be an IFS. Assume that each Wi is a strict contraction: there is 0 ~ Si < 1 such that d(Wi(X),Wi(Y» ~ Sid(X,y) for all x, Y E K. Then, there is a unique probability measure I' such that T" I' = 1'. I' is characterized by

N J f(x)dl' = ~Pi J f(wi(x»dl' 1=1

for all f E C(K). (1.3)

Furthermore, I' is attractive in the sense that for all Xo E K and for all continuous f

J Idl' = lim (TI: I)(xo), (where TI: 1:= T(T"-1 I». (1.4) 1:_00

Equivalently, (r)1:1I converges to I' in the weak" topology for any initial II E P(K).

Approximation of Measures by Fractal Generation Techniques 243

Proof: See [BD].

The support of JJ is of interest by itself.

Definition. The support of the measure JJ appearing in Theorem 1.1 is called the attractor of (K, {Wi}, {Pi}).

The attractor can be defined independently of the probabilities. To discuss it we will use the notion of Hausdorff metric for compact subsets of K.

Definition. Let (K, d) be a compact metric space and define 1l(K) to be the set of non-empty compact subsets of K with distance function

d1f.(B, C) = max {sup d(z, B), sup d(z, C)} (1.5) :l:EC :l:EB

where d(z,A) = inf{d(z,y) : YEA}. It is known that 1l(K) is a complete metric space with the metric (1.5),

[Dug, p. 253]. The existence and uniqueness of an attractor is guaranteed by the following.

Theorem 1.2 (Hutchinson). Let (K, {Wi}, {Pi}) be an IFS in which all Wi'S are contractions. Then, the map from 1l(K) to 1l(K) defined by

N

W(B) = U wi(B) (1.6) i=l

is a contraction mapping with contraction factor equal to the maximum of those of the Wi'S. Consequently, there is a unique non-empty compact subset of K satisfying

N

A = UWi(A). (1.7) i=l

.A = limn_oo Wn(B) for any initial B E 1l(K) . .A is the support of the measure whose existence is guaranteed by Theorem 1.1. .A can be further characterized as the closure of the union of the fixed points of the maps {Wi} and all possible finite compositions of them.

Proof. See [H].

Examples

1. Brolin [Bro]. Take K to be the closed unit disk, {z : Izl ~ I}, Wl(Z) = JZ, W2(Z) = -JZ, and P1 = P2 = 1/2. Notice that WI and W2 are not continuous on the branch cut {z: Re z ~ O} but that ! f( JZ) +

244 S. Demko

!J( -.jZ) is continuous if J(z) is continuous. There are two extremal invariant measures: Lebesgue measure on the unit circle and the point

f21r mass at o. If Zo :j:. 0, then T" J(zo) converges to 2~ Jo J(ei9 )dO. If

Zo = 0, then (T" f)(zo) = J(O) for all k. In this example, the Wi'S are the inverse branches of the polynomial p( z) = Z2. More generally, if Wb ... ,WN are the inverse branches of an Nth degree polynomial q(z), then the IFS with Pi = -k has the equilibrium measure (in the sense of potential theory) on the Julia set of q as an invariant measure. This measure is attractive in the sense that for any initial Zo - with two possible exceptions - (T" f)(zo) converges to the integral of J with respect to the equilibrium measure.

2. Let K = [0,1], W1(Z) = rz, W2(Z) = (1 - r)z + r, P1 = r and P2 = 1 - r for some 0 < r < 1. Theorems 1.1 and 1.2 apply. Since wdO, 1] U W2[0, 1] = [0, r] U [r,I], we see that the attractor is [0,1]. Since

11 J(z)dz = r 11 J(rz)dz + (1 - r) 11 J(r + (1 - r)z)dz,

we see that the invariant measure must be Lebesgue measure on [0,1]' This example shows that there is not necessarily a unique IFS generating a particular measure. This can also, more easily, be seen by noting that the maps and probabilities that correspond to the operator T2 generate the same measure as those corresponding to T.

3. Let K = [0,1], W1Z = lz and W2Z = lz + ~. We consider the set iteration:

W(K) [o,~] U [~, 1] ,

W 2(K) = [o,~] U [~,~] U [~,~] U [~,1], in general Wi(K) is the set obtained in the ph step of the standard construction of the classical Cantor set. Thus, the attractor is the classical Cantor set. If we had started our iteration with So = {OJ, we would end up constructing the Cantor set "from the inside": W(So) = {O,H, W2(SO) = {O,~,~,~}, W3(SO) = to, i7'~' 287,i, ~~,~, ~~} , .... The assignment of probabilities P1 and P2 to the maps can be viewed as a method of "texturing" the Cantor set.

4. [B, p. 105] In the complex plane consider the affine maps Wi(Z) = 8iZ+ (1 - 8i)ai where 81 = 82 = 0.6; 83 = 84 = 0.4 - 0.3i; a1 = 0.45 + 0.9i; a2 = 0.45 + 0.3i; a3 = 0.6 + 0.3i; a4 = 0.3 + 0.3i. Since these are all


contractions, ~here is a compact subset which is mapped into itself by each Wi. If we simulate the associated random walk with Pi = 1/4 for each i, we obtain an attractor that resembles a leaf. In the next section we will indicate how these Wi'S were obtained.

The next two examples show how familiar objects in Approximation Theory can be cast in IFS terms.

5. Let Mle be the L1-normalized kfh order B-spline on the knot set {0,1, ... ,k} and let 0 = Zo < Z1 < ... < ZN = k be a uniJorm

Z'+i: - z· 1 refinement of this knot set. Let Wj(z) = J k J z+Zj = NZ+Zj

for 0 :5 j :5 N - k. Then, the function MIe(wj1(z» is a kfh order B-spline on the knot set .{Zj, . '.', Zj+i:}' Since Mle is in the span of the MIe(wj1(z»'s, 0:5 j :5 N - k, there exist constants Cj so that

N-Ie

MIe(z) = L: CjMIe(wj1(Z». j=O

In IFS terms, Mle is the density of an absolutely continuous IFS meaN Ie c·

sure: T* Mle = Mle where (TJ)(z) = Ej =-;' !vJ(Wj(z». From (1.4) we see that integrals against Mle can be simulated by

J J(z)MIe(z)dz = lim (Tie I)(zo). Ie-+oo

6. [Bj p. 213] Let

w, ( : ) = (::: 0.:5)(:) ~ ( : ) = (~:5 o.~)(:) + ( : )

and P1 = P2 = 1/2. The attractor is the graph of the quadratic q(z) = 2z - z2 on [0,2]. To see this observe that

for 0 :5 Z :5 2. Apropos to this example, Berger [Ber] is studying IFS aspects of subdivision algorithms. The paper of Micchelli in this volume contains related material and references.

246 S. Demko

The conditions on K, Pi, Wi can be relaxed a great deal and still yield attractive (i.e., constructable) measures. K can be locally compact, the Wi'S

need only satisfy an "average contractivity" condition and the Pi'S can be continuous functions whose moduli of continuity satisfy a Dini condition. See [BDEG] for details. The paper [EY] contains general results on the continuous dependence of the invariant measure on the parameters of the IFSj it also shows that every probability measure on a compact metric space is an IFS measure if we make our definition of IFS general enough. Finally, we note that every regular countably additive probability measure can be approximated in the weak· -topology by an IFS measure. Simply partition the support of the target measure into a finite number of sets of small diameter and take Wi(Z) to be a constant function taking a value in the if" set and Pi to be the measure of the t"f" set. As the diameter of the partition goes to zero, the IFS measure approaches the target measure.

2 The Inverse Problem I: A Geometrical Solution

The fixed point argument that gave Hutchinson's characterization of the attractor can be perturbed to yield a very useful approximation result.

Theorem 2.1 (Collage Theorem) Let K be a compact metric space and T E 1l(K). Suppose that Wl, ••• , WN are contraction maps from K to itself such that (cf. (1.6))

d1(W(T), T) ~ £, (2.1)

then if A is the attractor for the IFS (K, {Wi}, {Pi}) (Pi> 0),

£ d1(T,A) ~ 1- s (2.2)

where s is the largest of the contraction factors of the Wi'S.

Remark. Note that A is a constructable approximation to the target set T.

Proof. See [B, p. 96].

The IFS in Example 4 of Section 1 was constructed by Michael Barnsley by covering a real ivy leaf with 4 smaller ivy leaves and manually estimating the shrinking and rotation factors of the affine maps that take the larger leaf onto the 4 smaller leafs. In a similar manner he calculated from a botanical drawing the parameters that would make the attractor resemble a fern. For more elaborate pictures produced by collage theorem methods and algorithmic details, see [B] or [BJMRS].

If the Wi'S are constant maps whose values form an £-net for the target set 'I, then (2.1) is automatically satisfied and (2.2) holds with s = O. This


typically requires that the number of maps on the order of ( ~)d 'where d is the fractal dimension ofT [1, Chapter 10]. For example, for the unit interval

d = 1 and for the classical Cantor set d = !::. However, each of these sets can be constructed with only 2 maps as Examples 2 and 3 of Section 1 show. In the general case, one would hope to be able to approximate a given T with an IFS attractor generated by a fairly small number of affine maps. No geometrical approximation results for IFS beyond the Collage Theorem seem to be known. A characterization of those sets which can be approximated at a given rate (consider £ as a function of the number of affine maps) would be a welcome addition to the theory. A verifiable sufficient condition for approximation at a given rate might even be useful in practice.

Implementation of the Collage Theorem appears to require either a "human-in-the-loop" (to select the maps Wi) or human cognitive powers. For example, several of the more complicated images in [BJMRS] were created by people experienced with the Collage theorem interacting with a computer program over a period of about 2 weeks. Several groups of researchers have taken steps to automate the fractal approximation process by focusing on the invariant measure and the characterizing functional equation (1.3) rather than on the Collage approach. We discuss this in the next section. Finally, Jacquin [J], has developed an automated geometric method with roots in IFS theory. This method, which we will not discuss in detail here, relies on acanol\ical cl~ification of image segments.

3 The Inverse Problem II: Analytic Solutions

We recall the functional equation for I' (1.3):

N J Jdl' = J ~pd(Wi(Z»dl' for all J E C(K) (3.1) 1=1

and view it as an analytic form of the Collage Theorem. In this section we assume K ~ Rand Wi(Z) = aiz+hi with lail < 1 although some extensions to Rn are possible. With J(z) = eitl: we obtain a functional equation for the Fourier transform of I'

N

jJ(t) = J eitl:dl'(z) = LPjei6;tjJ(ajt). j=l

(3.2)

One might hope to use this identity to obtain IFS parameters Pj, hj , aj from the Fourier transform of a target measure 1'. In the first comprehensive treatment of measure approximation by IFS methods Elton and Van [EY] simplified (3.2) by requiring all aj 's to be identical. They called the resulting

248 S. Demko

IFS a homogeneous IFS and characterized the closure of the set invariant measures for homogeneous IFS with fixed scale a. Before stating their result we rewrite (3.2) in the homogeneous case with aj = a for all j

N

pet) = peat) :Epjeib;f = p(at)u(t) (3.3) j=l

where 0" is the discrete measure with mass points bj and corresponding weights Pj. From (3.3) if the scale factor a is known then the measure 0"- and thus the IFS parameters bj, Pj - must be the inverse Fourier

transform of ~«t». Now, not every measure is the invariant measure of a I-' at

homogeneous IFS with scale parameter a. Elton and Van prove

Theorem 3.1 Let I-' be a probability measure on R. Then, I-' is the limit of invariant measures for finite homogeneous affine IFS's with linear part

a if and only if ~«.» is positive definite. I-' a·

There seem to be no published accounts of the computational aspects of determining 0" from the Fourier transform of 1-'. One potential drawback of this approach is the fact that it requires an estimate of the scale factor a. This is important in order for the method to be efficient in recovering just the class of homogeneous IFS measures. For example, the classical Cantor set can be exactly reconstructed with a = (l)j for any j ~ 1. The value a = 1/3 is optimal in some natural sense since it gives the IFS with the fewest number of maps, 2. A value of a different from (l)j would either

force ~«. » to be not positive definite or give rise to an IFS with infinitely I-' a·

many maps. The situation for moment matching or more generally polynomial sam

pling is much better. The idea of exploiting (3.1) with the functions fez) = zn, n ~ 0 was suggested in the early· papers of Barnsley and Demko and Diaconis and Shahshahani [BD], [DiS]. In fact, in [BD] an attempt is made to reproduce a I'dragon fractal" by using empirical moments to determine the IFS parameters. The case of general IFS seems to be intractable (but see [HaM] for some potentially useful ideas). The homogeneous case has been studied by these groups of researchers: Handy and Mantica [HaM], Abenda and Turchetti [AT], Bessis and Demko [BeD]. The first two groups study the problem of determining the scale factor a and the IFS parameters

bj,pj,1 ~ j ~ N, from the moments Tn = f ZndT, n = O,1, ... ,2N, ofa

target measure T. In this case (3.1) becomes

(3.4)


where (1'j = L~1 Pi~ is the ph moment of the discrete measure (1' having mass points bi and weights Pi. Given TO, ... , T2N and a number a we compute (1'0, ••• , (1'2N-1 from the lower triangular system that (3.4) determines for n = 0, 1, ... , 2N - 1. In fact

n-1 ( ) (1'n = (1- an)Tn -?: 7 ai Tj(1'n_j.

.=1 (3.5)

Next, from the moments (1'0,"" (1'2N-1 we attempt to recover the the measure (1' by the classical Pade-Stieltjes method [BaG]. It could happen that there is no probability measure '(1' satisfying (3.5); this will happen if the as

sociated Hadamard matrix «(1'i+j )O$i,j $N -1 is not positive definite. If there is such a (1', then upon computation of (1'2N we check the difference:

2N-1 ( ) -2N 2N -1 -i (1'2N - (1 - a )T2N + L . a Tj(1'2N-j·

i=1 l

(3.6)

If this is 0, we take a to be the linear part of our maps. If not, we revise our guess at a. Typically, we start with two values a1 < a2 so that (3.6) has opposite signs for a1 and a2 and use a root bracketing method to determine a. There are other equivalent ways of implementing this idea; for example, Abenda and Turchetti work with the Hadamard determinant directly.

The above approach is conceptually simple but is generally ill-conditioned because the problem of determining the mass points and weights of (1' from its moments is typically ill-conditioned. For example, if the initial moments of (1' happened to agree with those of Lebesgue measure on [0,1]' we would be faced with a problem equivalent to that of inverting a section of the Hilbert matrix, see [Bag]. This can be avoided by using either the Sack Donovan method [SD] or Gautschi's method [G] for recovering mass points and weights of measures from orthogonal polynomial samples of the measure. This approach was taken by Bessis and Demko and while it does not completely stabilize the problem (other difficulties arise when we confront the analogue of (3.4» it does allow for the reliable solution of problems with 2 to 3 times as many maps as the moment method does.

In the Bessis-Demko approach (3.4) is replaced by

J Pn(X)dT(X) = t cn,j J Pn(b)d(1'(b) J=O

(3.7)

where Po, Pl, ... are polynomials orthogonal with respect to some measure v; (1', as above, is the discrete measure giving the distribution of the bi'S;

and cn,j = J qn,j(X)dT(X) where qn,j is the polynomial of degree n - j in

the addition formula n

Pn(x + z) = L qn,j(x)Pj(z). (3.8) j=O

250 S. Demko

The qnJ's can be generated from recurrence formulas derived from the 3-term recurrence formula for the Pn's. For the Chebyshev polynomials of the first kind these formulas are given explicitly in [BeD]. The implementation of this method follows the lines of the implementation of the moment method as sketched above with the Sack-Donovan-Gautschi method being

used to recover 0' from the samples J Pn(b)dO'(b) , n = 0, ... , 2N - 1. If

the Pn's are the Chebyshev polynomials of the first kind, then this recovery process is in general well-conditioned. Unfortunately, the triangular system given by (3.7) becomes ill-conditioned as the size grows. In spite of this, we found that we could obtain meaningful results for problems that were at least twice the size of those considered in [AT] and [HaM]. Perhaps a mod-

ification of this method in which the continuous samples J Pndr generate

discrete samples J QndO' with respect to a different class of orthogonal

polynomials would give rise to a well-conditioned linear system of the form

(3.9)

In the above approaches we are trying to determine a homogeneous affine IFS {Wi}, {Pi} so that for a given target measure r the functional equation (3.1) holds for all polynomials of degree 2N. It is not hard to see that the invariant measure I' for this IFS must satisfy

2N J Idl' = J Idr for I(z) = ~ajzi. )=0

(3.10)

So standard approximation theory argument gives

Theorem 3.2 [BeD] Let r be a given probability measure and let I' be a probability measure so that (3.9) holds, then for all continuous g

(3.11)

The bound (3.11) suggests viewing" as providing an approximate inte

gration formula for r. (The question of how to efficiently evaluate J 1 d" is

discussed in the next section.) If the parameter a is set equal to 0, then I' is a discrete measure. In this case we have I'n = Tn for all ° ~ n ~ 2N -1 and I' gives the classical Gaussian quadrature for T. SO, homogeneous IFS measures obtained by matching a maximal number of moments can be viewed as generalization of Gaussian quadratures.


The error bound in (3.11) seems to indicate that a homogeneous IFS with N maps gives rise to a measure that approximates a target measure a little better than an N-point Gaussian quadrature and a little worse than an (N + I)-point Gaussian quadrature. It does not take into account the self-similar aspects of the IFS measure or the fact that the support of the IFS measure is an uncountable set (unless a = 0). It is not evident how to incorporate these facts into a rigorous error theory. However, a numerical experiment in [BeD] suggests that there are important measures - or measure-function pairs - for which the IFS measure is much superior to the Gaussian quadrature.

The integral 1 r

Uo = 2 Jo vzG(z)dz (3.12)

represents the zere>point vibrational energy of a face-centered cubic crystal with nearest neighbor control force constants. Here G(z)dz = dT is the fraction of normal modes in [z, z + dz]. Wheeler and Gordon [WG] computed the first 30 moments of T using combinatorial methods and obtained the rigorous bounds

0.3408807 < Uo < 0.3408883. (3.13)

Using 10 moments, and Gaussian quadrature they obtained a relative error of about 10-3 • Bessis and Demko used the moment data of Wheeler and Gordon and obtained a relative error < 6.10-5 with a 4-map homogeneous IFS. Possibly the homogeneous nature of the crystal was exploited by the IFS to give this improved estimate.

4 Eigenfunctions of the Operator T and Their Applications

We now consider the eigenvalue problem

(TI)(z) = >'I(z)

where N

(4.1)

(TI)(z) = LP;/(ajz + bj). (4.2) j=l

It is clear that the spaces of polynomials of fixed degree are invariant under T and thus that T has polynomial eigenfunctions. With In(z) = Ej=o cjzi, we compute

N n n

(Tln)(z) = LPiLcj(aiz+biY = L'Yjzi (4.3) i=l i=O i=O

252 S. Demko

where

Setting (Tfn)(x) = Anfn(x)

we see from (4.4) with j = n that

N

An = :LPiai' i=l

(4.4)

(4.5)

(4.6)

and choose Cn = 1. Next, from (4.4) we see that the conditions 'Yj = ACj j = n - 1, n - 2, ... ,0 lead to the recurrence

N n

Cj :LPiai + :L CJ:WJ: = AnCj. (4.7) i=l J:=j+l

where

(4.8)

Therefore, if the numbers Aj = E~l Pia1, j = 0,1,2, ... , n are distinct, then there is a unique monic polynomial of degree n with Tfn = Anfn. The Pi'S do not have to be probabilities in this analysis, they don't even have to be positive. However, for the remainder of this paper we will take the Pi'S to be probabilities and the ai's to be equal and of absolute value less than 1. In summary, we have

Proposition 4.1 Let T be the operator (4.2) associated with a homogeneous IFS whose maps all have contraction factor a. Then, for each k = 0,1, ... there is a unique monic polynomial PJ: such that

(4.9)

Example. Take a = ~'Pi = ~, bi = (i~l) for 1 ~ i ~ n. Then, (4.9) reads as

(4.10)

Letting x = ny we see

(4.11)


which is Raabe's multiplication theorem for the Bernoulli polynomials, see Lehmer's paper [L] for a development of the Bernoulli polynomials based on this identity. The fact that the Bernoulli polynomials were eigenfunctions for this special T was also noticed by Daubechies and Lagarius [Da La].

We collect some important properties of the Pn's here.

Theorem 4.2 Let Po, Pl , ... be the polynomials of Proposition ./.1 and let I' be the invariant measure for the associated IFS. Then,

(1) J Pndl' = 0 forn = 1,2, ... .

(2) nPn-l = P~ for n = 1,2, ... .

(3) Pn(z + 1) - Pn(z) = nzn-l for n = 0,1,2, ....

(4) with jj(t) = J etll:dl'(z) , we have the generating /unction

(4.12)

Proof.

(1) By (1.4) and (4.9) we have for any z, JPndl' = lim T'" Pn(z) = "'-00 lim an'" Pn(Z) = O. "'_00

(2) Differentiating E~l PiPn(az + b) = an Pn(z) we see that Pn is a polynomial of degree n - 1 that satisfies TP~ = an-lp~. So, p,. is a multiple of Pn - l • The multiplier must be n since Pn is monic.

(3) As in [L] this is proved by induction starting from Po and using

(4) The Laplace transform jj(z) = J eZlI:dl'(z) is an entire function with

jc(O) = 1. So there is a convergent expansion in a z-neighborhood of 0:

(4.13)

254 S. Demko

where Qn is a polynomial of degree n. Since jj(z) = E:=o Pn :~ , we have

ConsequentlYI

zn = t (~)pjQn_j(z) j=O J

which forces Qn(z) to be monic.

Now, applying T to (4.13) in the z-variable we obtain

( ZS) 1 N T ~ = -_-LPiellzsed; p(z) p(z) i=1

ellZS = jj(z) U(z)

ellzs = jj(az)

00 anzn = EQn(Z)-,

n=O n.

with the next to the last equality following from the identity jj(z) = U(z)jj(az) which is the Laplace transform analogue of (3.3). From this string of equalities we obtain TQn = anQn. Since Qn is monic of degree n and Pn is unique, we obtain (4). •

Let {Wi}, {Pi} and {Ui}, {qi} be two homogeneous IFS with strictly contractive maps, say Wi(Z) = az+hi , 1::; i::; N, Uj(z) = ez+dj , 1::; j::; M with lal < 1, lei < 1. It is not hard to find an interval K which is invariant under all of the maps Wi, Uj and which contains the associated attractors. We can then view the linear operators

N

(Td)(z) = LPi/(az + hi) i=1

(4.14) M

(T2I)(z) = Lq;/(ez + dj )

j=1


as acting on C(K). It is easy to see that ifT1T2 = T2T1, then the two IFS's have the same invariant probability measure:

if TiP = p, then T2Tip = Tip but T2Tip = Ti(T;p) so T2 p is a fixed probability measure of Ti so T; p = p.

The converse of this statement is also true! It follows immediately from (4.12) since the eigenfunctions of T are uniquely determined by p. These facts are summarized in

Corollary 4.3 Let

N M

(Td)(z) = LP;f(az + bj) and (T2f)(Z) = Lqi/(ez+dj) j=l i=l

where qi ~ 0, Pj ~ 0, Eqi = 1 = Ep;. and lal < 1, lei < 1. Then, T1T2 = T2T1 if and only if there is a probability measure p so that Ti p = p = T2 p.

The expansion (4.12) shows that the sequence {Pn} is an "Appell sequence" - the generating function is of the form g(z)e·u :. As such, identities (2) and (3) of Theorem 4.2 as well as many other interesting properties are consequences of (4.12), see [1]. It would seem natural to call the Pn's "Bernoulli polynomials with respect to p" since the general form of g(z) so strongly resembles the form for the classical Bernoulli polynomials in which case

g(z) = --z=:- = [11 eZfdt]-l e -1 0

Another similarity between the Bernoulli polynomials and the Pn's is their appearance in error expansions for quadrature rules.

We mentioned in Section 3 the possibility of developing a theory of approximate integration by homogeneous IFS measures and left open the

question of how to efficiently evaluate J fdp. There are two methods which

immediately come to mind. The first is based on an ergodic theorem. If we start with any Zo and compute Zn = Wi .. (Zn-t} with probability Pi .. , then (cf. [ED

f 1 n

fdl' = lim -1'" f(zj). n-+oo n + L..J

j=O

(4.15)

This algorithm works in quite a general framework and has been used in computer graphical applications of IFS, see [BJMRS]. Unfortunately the smoothness of / has little effect on the convergence rate.

The second integration method is based on (1.4): if z is chosen arbitrarily, then

(4.16)

256 S. Demko

The approximation J Idl' ~ (Tic I)(x) is very much like a Riemann sum

with NIc subintervals for an N-map IFS. The choice x = 1'1 gives a quadra

ture rule very much like the mid-point rule; J fdl' = 1(l'd if I is a linear

polynomial and (Tic f)(l'd behaves like a composite mid-point rule. If the attractor for the IFS is A and if wi(A) n wj(A) consists of at most one point, then we have a true composite mid-point rule. There are cases of interest when wi(A) n wj(A) is uncountable; for example, the 4-map IFS approximation to (3.12) is like this. In these cases we don't have a true composite formula. While the approximation based on (4.16) is demonstrably better than that of (4.15) for C1 and C2 functions, it is a low-order of accuracy formula even with x = 1'1. This can be improved in the case of homogeneous IFS where one can develop a Richardson extrapolation procedure for smooth I. The possibility of doing this was suggested in [D)

based on formal expansions of J fdj.t. We give a self-contained alternate

exposition here.

Proposition 4.4 Let T be the operator (4.2) associated with a homogeneous affine IFS with common contraction factor a and invariant measure 1'. Let {Pn}~=o be the associated monic polynomial eigenfunctions. Then, if f is a polynomial of degree n

(4.17)

fork ~ o.

Proof. Consider the case k = 0 and write f(x) = ,,£7=0 CjPj(x) and com

pute the integral of the kfh derivative on each side. For J P?)dl' we easily

obtain 0 if k > j and j! if k = j since Pj is monic. For k < j we use parts (2) and (1) of Theorem 4.2 to see that

J P?)dl' = j!O;k.

Therefore J f(lc)dl' = kIck

which gives (4.17) for the case k = O. The case k ~ 1 follows from Tk Pj = aikPj. •


From (4.17) we obtain the expansion formula for J IdJ.' if I is a poly

nomial n J (j) J IdJ.' = (Tie I)(z) - [; Ii! dJ.' aile P;(z). (4.18)

If I is analytic we can write down the formal expansion

(4.19)

If M J IdJ.' r:d ~ 'Y;/(e;)

;=1 (4.20)

is a quadrature formula exact for degree d, then (4.18) becomes

which implies O(ale(M+1» accuracy for polynomials of degree n. This formula can be extrapolated in the Romberg fashion to give higher order formulas which integrate higher degree polynomials exactly. Further details and computational aspects can be found in [D].

5 Some Open Problems

The theory of measure approximation by IFS is just beginning to be developed and while a few technical questions were mentioned in this article, the most important unsolved problem is "the inverse problem": construct efficient IFS approximations to' give measures. We collect here some problems directly related to this problem and some other problems of general interest. See the remarks that follow for additional information.

Pl Develop sampling methods for non-homogeneous IFS on the line.

P2 Develop sampling methods for measures in R2 or Rn.

In particular

P3Find IFS approximations or representations of Lebesgue measure on nice subsets of Rn (e.g., convex polyhedra).

P4 Investigate the use of non-affine maps. For example, numerical conformal mapping techniques might be useful at times.

258 S. Demko

P 5 Operators like T and T* appear in other contexts: wavelets and stationary subdivision. Explore these connections.

P6 With a = 0 the homogeneous affine IFS measures of Section 3 can be viewed as Gaussian quadratures. Explore this connection.

Remarks on

Pl Locally homogeneous methods based on piecewise polynomial sampling are possible. We are developing this idea with G. Turchetti.

P2 It might be possible to reduce this problem to a one-dimensional problem by mapping the n-dimensional measure to the line and applying one-dimensional techniques.

P3 A representation of Lebesgue measure on an n-dimensional set by an IFS with contractive maps gives a Monte-Carlo integration method for this set. It might be necessary to use non-constant probabilities however.

P4 Bessis and Mantica have some unpublished results along this line.

P5 See Micchelli's contribution to this volume and [DaLa].

References [AT] S. Abenda and G. Turchetti, Inverse problem for fractal sets on

the real line via the moment method, n Nuovo Cimento, Vol. 104 B, No.2, pp. 213-227, 1989.

[BaG] G. A. Baker, Jr. and P. Graves-Morris, Pade Approximants, Addison-Wesley, 1981.

[B] M. F. Barnsley, Fractals Everywhere, Academic Press, 1988.

[BD] M. F. Barnsley and S. Demko, Iterated function systems and the global construction of fractals, The Proc. of the Royal Soc. of London A, Vol. 399, pp. 243-279, 1985.

[BDEG] M. F. Barnsley, S. Demko, J. Elton and J. Geronimo, Markov processes arising from functional iteration with place dependent probabilities, Annales de l'Institut Henri Poincare, Probabilities et Statistiques, Vol. 24, pp. 367-394, 1988.

[BJMRS] M. F. Barnsley, A. Jacquin, F. Malassenet, L. Reuter, and A. D. Sloan, Harnessing chaos for image synthesis, Computer Graphics, Vol. 22, pp. 131-140, 1988.

[Ber] M. A. Berger, Random affine iterated function systems: smooth curve generation, preprint.

[BeD]

[Bro]

[DaLa]

[D]

[DHN]

[DiS]

[Dug]

[E]

[EY]

[G]

[HaM]

[H]

[J]

[L]

[Lor]

[SO]


D. Bessis and S. Demko, Stable recovery of fractal measures by polynomial sampling, CEN-Saclay preprint PhT 89-150, 1989.

H. Brolin, Invariant sets under iteration of rational functions, Arkiv for Matematik, Vol. 6, pp. 103-144, 1965.

I. Daubechies and J. C. Lagarius, Two-scale difference equations, I. Existence and global regularity of solutions, to appear in SIAM J. on Math. Anal.

S. Demko, Euler Maclauren type expansions for some fractal measures, Fractal '90 Proceedings, to appear.

S. Demko, L. Hodges, and B. Naylor, Construction of fractal objects with iterated function systems, Computer Graphics, Vol. 19, pp. 271-278, 1985.

P. Diaconis and M. Shahshahani, Products of random matrices and computer image generation, Contemporary Mathematics, Vol. 50, pp. 173-182, 1986.

J. Dugundji, Topology, Allyn and Bacon, 1966.

J. H. Elton, An ergodic theorem for iterated maps, Ergod. Th. and Dynam. Sys, Vol. 7, pp. 481-488, 1987.

J. H. Elton and Z. Yan, Approximation of measures by Markov processes and homogeneous affine iterated function systems, Constructive Approximation, Vol. 5, pp. 69-87, 1989.

W. Gautschi, On the construction of Gaussian quadrature rules from modified moments, Math. of Comp., Vol. 24, pp. 245-260, 1970.

C. Handy and G. Mantica, Inverse problems in fractal construction: moment method solution, to appear in Physica D.

J. Hutchinson, Fractals and self-similarity, Indiana J. Math., Vol. 30, pp. 713-747, 1981.

A. E. Jacquin, A novel fractal block-coding technique for digital images, Proceedings of ICASSP '90, to appear.

D. M. Lehmer, A new approach to Bernoulli polynomials, Amer. Math. Monthly, December, 1988, pp. 905-911.

G. G. Lorentz, Approximation of Functions, Holt, Rinehart, and Winston; 1966.

R. A. Sack and A. F. Donovan, An algorithm for Gaussian quadrature given modified moments, Numer. Math., Vol. 18, pp. 465-478, 1972.

260 S. Demko

[WG] J. C. Wheeler and R. G. Gordon, Rigorous bounds from moment constraints, In: The Pade Approximant in Theoretical Physics, editors G. A. Baker, Jr. and J. 1. Gammell, Academic Press, 1970.

Stephen Demko School of Mathematics Georgia Institute of Technology Atlanta, Georgia 30332

Nonlinear "Wavelet Approximation in the Space C(Rd) R.A. De Vore* P. Petrushev X.M. Yu

ABSTRACT We discuss the nonlinear approximation of functions from the space C(R d) by a linear combination of n translated dilates of a fixed function cpo

Dedication. We dedicate this paper to the memory of our friend and colleague Vasil Popov. It was Vasil who brought us together. His work on nonlinear approximation and the use of wavelets was our inspiration.

1 Introduction

There have recently been developed, [6], [7], new methods for the nonlinear approximation of a function f E Lp , ° < p < 00, based on the wavelet decomposition of f. This approach. recovers most classical results for free knot spline and rational approximation and also allows their natural extension to several variables. The purpose of the present paper is to use wavelet decompositions for nonlinear approximation-in the space C(Rd ).

Approximation in the uniform norm is not covered by previous work and requires some significant new ideas.

The terminology 'wavelet' originated with Yves Meyer; he used it to denote a univariate function <.p such that the normalized translated dilates 2k/2<.p(2k x - j), j, k E Z, are an orthonormal basis for L2(R). We shall call such functions orthogonal wavelets. Their simplest example is the function <.p(x) := -1 on [0,1/2), <.p(x) := 1 on [1/2,1]' and <.p(x):= 0, otherwise. The translated dilates of <.p are the Haar basis. Examples of orthogonal wavelets with higher smoothness have been given by Meyer [11] and Daubechies [3]. The wavelets, as constructed by Daubechies, are closely connected to subdivision algorithms in Computer Aided Geometric Design as can be

'The first author was supported by NSF Grant DMS 8620108 and AFOSR Contract No. 90-0323


262 R.A. DeVore, P. Petrushev, X.M. Yu

seen in the work of Cavaretta, Dahmen, and Micchelli [2]. In several space dimensions, one uses more than one function tp to generate a complete orthonormal system.

The orthogonal wavelets are important not so much that they are a basis for L2(R) but because they are also a basis for many function spaces such as the Sobolev, Besov, H" and L, spaces. Moreover, the norm of a function in these spaces can be. described in terms of the coefficients in its orthogonal expansion. This allows many questions concerning these spaces to be handled on the sequence level. This is also one of the compelling reasons for using wavelet decompositions in approximation.

There is a broader definition of wavelet used in [6] in which orthogonality is not required. Let tp be a function defined on Rd. It is convenient to index the translated dilates of tp by dyadic cubes. For k E Z and j E Zd, we let tpI(Z) := tp(2~z - j), where the dyadic cube 1 = j2-~ + 2-~O with 0 := [0,1]d. We shall say that j2-~ corresponds to 1. This indexing avoids double subscripting and indicates roughly the main support of tpl.

We shall also use the notation 1)~ to denote the set of dyadic cubes 1 whose sidelength £(1) is 2-~ and 1) to denote the union of the 1)~, k E Z.

By a wavelet for a space X of functions on R d, we mean any function tp E X such that each / E X has a representation

/ = E aI(f)tpI (1.1) Ie'"

where the al are linear functionals on X not depending on / and where (1.1) holds in the sense of convergence in the topology of X. Thus, for us, a wavelet need not generate an orthonormal basis. Indeed, the translated dilates tpI, 1 E 1), need not even be linearly independent.

There are two primary ways (different from the orthogonal wavelets) for constructing wavelets. The first of these, given by Frazier and Jawerth [9],[10], shows that (1.1) holds for a very general class of functions tp with

dual functionals aI(f) = (1/111) J /(z)'If;r(z) dz described by the trans

lated dilates of a function t/J on Rd. The representation (1.1) holds among other things for all / E L" 1 S p S 00.

We give an idea of how their method works with the following simple example. We refer the reader to [10] for details. Suppose the function tp is in Ll(Rd) and its Fourier transform <,3(z) := fa. tp(u)e-ic.u du satisfies the following conditions for some 6 > 0:

(i) <,3(z) = 0, Izl S 6 and Izl ~ '11",

(ii) 1<,3(z)1 ~ c > 0, 1/2 S Izl S 2. (1.2)

The function ~(z) := E~ez 1<,3(2~z)12 > 0 is defined for all z #: 0 because for each such z the sum defining ~ has only a finite number of nonzero

Nonlinear Wavelet Approximation in the Space G(Rd ) 263

terms. We let '1(z) be the function with Fourier transform i! = ((J/Cb. Then, we have the following identity

1 = 2: ((J(2I:z )i!(2I: z ), I:EZ

It follows that for each distribution J with mean value 0

J = 2: CPI: * ,pI: * J (1.3) I:EZ

with CPI:(z) := 2I:dcp(2I:z ) and ,pl:(z) := 2I:d,p(2I:z ) and ,p(z) := 'ij(-z). Each convolution appearing in the sum (1.3) can be evaluated by the Poisson summation formula (this is where condition (1.2)(i) is used):

and formula (1.1) follows with aI(1) := 1/111 J J(z),pI(-z)dz .

The conditions (1.2) can be weakened significantly (see §4 of [10]). For example if u is a continuous function satisfying (1.2)(ii) and

(1.4)

then for some sufficiently small ~ > 0, (1.1) is valid for cp(z) := u(~z) and X = Lp , 1 < p < 00. Thus some dilate of u is a wavelet. However, the dual functionals can not . generally be described by the translated dilates of one function,p. Among other things, the Frazier - Jawerth construction admits rational functions. For example (1 + IzI2)-m, m > d clearly satisfies (1.4) and therefore one of its dilates is a wavelet.

Again, one is not only interested in the decomposition (1.1) but also in describing smoothness norms of J in terms of the coefficients of the CPl. This is possible provided cP is sufficiently smooth and decays sufficiently rapidly as Izl -+ 00. We do not describe these results precisely here (the reader may consult [10]) since we will not use them. However, their analogues are given in §2 for the next wavelets which we now describe.

The second method for constructing wavelets is more directly connected with approximation and will be the subject of this paper. Let cP be a continuous function with compact .support such that

cp(z) = 2: cjcp(2z - j) j

(1.5)

for some finite set of numbers Cj. Property (1.5) says that CPI for any dyadic cube 1 E VI: can be rewritten as a finite sum of CPJ, J E VI: +1 , i.e. in terms


of <PJ at the next finer dyadic level. This property is the starting point for both subdivision algorithms and the Daubechies construction of orthogonal wavelets.

In the study of subdivision, one is interested in the relationship between the coefficients Cj and the function <po For example, for which Cj do we know that there exists a solution <p satisfying (1.5) and when does <p have certain smoothness or other properties. The monograph of Cavaretta, Dahmen, and Micchelli [2] gives a detailed study of these and related questions. In this paper, we are assuming that we already have a function satisfying (1.5).

Let S := So := span {<p(. ~ j) : j E Zd} be the span of the integer translates of <po Then, we can form the dilated spaces Sk := {/(2k z): / E S}, k E Z. It follows that Sk is spanned by the functions <p(2kz - j). The condition (1.5) implies that

kE Z. (1.6)

As k gets larger, the spaces Sk get thicker. Instructive examples are the B-splines in one or several variables or the box splines. They are well known to satisfy (1.5). For example, if 1), then SA: is the space of piecewise linear functions with knots at the dyadic integers j2-A:, j E Z.

To go further and obtain a wavelet decomposition (1.1), we assume that USk is dense in X. Then each / E X is the limit of a sequence SA: E Sk: 11/ - Skllx -+ O. It follows that / = So + (S1 - So) + .... Each of the terms Sk-SA:-1 appearing in this sum is, by virtue of (1.6), in Sk and therefore can be written as Sk - Sk-1 = E/E"D. bl<PI. This gives the decomposition (1.1) except that we do not know anything about the coefficients bl. However, if we take a specific method of approximation given by some linear operators, then the coefficients become clearer.

Let P be a bounded projector from X onto So. For example, when X = L2 we can take the orthogonal projection. By dilation, we obtain for each k E Z a projector I\ from X outo SA:, 'and IIPk II :$ IIPII. Hence, under these assumptions, Pk(f) -+ / and we have

Now, it is easy to see that, in all cases of interest, one can let ko -+ -00 and thereby obtain from (1.7), the wavelet decomposition (1.1) with coefficient functionals CI given by translation and dilation of 2d functionals (these functionals depend on the relative position of I in its parent; there are 2d such relative positions). In §3, we mention additional conditions on <p which guaranteed the existence of (quasi-interpolant) projectors Pk.

Nonlinear Wavelet Approximation in the Space C(Rd) 265

The choices of projectors Pic lead to many interesting possibilities. For example, using box splines and the cardinal interpolant is the starting point for the surface compression algorithms introduced in [4], while using quasi-interpolants is useful in both surface and image compression [5]. In Daubechies construction of orthogonal wavelets, one chooses the projector P as the orthogonal projection of L2 onto So. Then the spaces Vic := Sic e PIc-l(SIc), k E Z, are mutually orthogonal. To construct an orthogonal wavelet, one finds a function cp E Va whose integer translates are an orthogonal basis for Va.

2 Nonlinear Approximation

Our main interest in this paper is not wavelet decompositions per se but rather problems in nonlinear approximation. Suppose that cp is a given function. We let :En be the nonlinear space consisting of all

s= ~CICPI' IAI ~n. (2.1) lEA

If cp is a univariate B-spline of order r, the elements in :En are splines with at most (r + l)n knots. If cp is a rational function in d variables then the elements of :En are rational functions of degree ~ Cn. Thus, we see that for these cp, the space :En is interesting for approximation.

We shall be interested in approximating I E Lp(Rd), 0 < p < 00, (f E C(Rd), P = 00), by the elements of :En. Accordingly, we introduce the approximation error

(2.2)

There are two questions of interest to us. How can we construct good or near best approximants from :En; and secondly for what class of functions do we have a given error of approximation (like O(n-a ». These questions were studied extensively in the case 0 < p < 00, by DeVore, Jawerth, and Popov [6], for wavelets of the three types mentioned above. A generic description of their results is the following.

To find a good Lp approximation to a function I, one should find a suitable wavelet decomposition. (1.1) for I in terms of cp and then choose the n-terms for which 1c1(f)IIII1/p is the largest. We underline suitable because in the general wavelet case, such as splines, there is not a unique wavelet decomposition. For example, different projections P as described in §1 give different decompositions. One needs that this decomposition by compatible with Lp, for example that the projections P be bounded on Lp. We should also mention that this general idea has limitations. It needs additional conditions on cp and I which will be described in §3. Moreover, what is important for this paper, this technique does not work for p = 00


(and in fact deteriorates as p gets close to infinity) as the following simple example shows.

Let I{) be the Haar function of §1 and let for 0 < £ < 1

n n

/(z) := 2: 1{)(2kz) + (1 + £) 2: I{)(z - j). k=1 j=1

If we want to approximate / by the elements of :En, then the strategy of choosing largest coefficients would choose the second sum in the definition of / producing an error of n (at z = 0) in Loo while the better choice of the first sum produces an error of only 1 + £. This shows that the case p = 00 (or p close to 00) requires a more sophisticated construction of a good approximation. This will be the subject of this paper.

The results which characterize the approximation order involve in an essential way the Besov spaces. A Besov space B; (Lp), a > 0, 0 < q, p ~ 00

is a smoothness space in Lp. The parameter a gives the smoothness order, much like the order of differentiation, while the second parameter q is a fine tuning parameter. It makes subtle distinctions between the different smoothness spaces with the same a and p. The parameter q is vital in discussions about nonlinear approximation.

To define the Besov spaces, let a > 0, 0 < p,q ~ 00. Then, B;(Lp) consists of all functions / defined on Rd such that

(2.3)

where r > a and wr(f, t)p := sup lI~h(f, ')lIp

Ihl9

is the modulus of smoothness of order r of / in Lp(Rd) which is defined using the r-th differences, ~h(f), of /. In the definition of the Besov space, r can be taken as any integer larger than a. While the semi-norms (2.3) are different, the norms 1I/IIB:<Lp) := II/lip + 1/IB:<Lp) are equivalent for different values of r.

Here is the generic characterization theorem proved in [6]. If a > 0, 0< p < 00 and r := r(a) := (aid + 1/p)-1 then

00 1 2: [na/dun (f)pr - < +00 +-+ / E B~(LT)' (2.4) n=1 n

One would prefer to obtain the characterization for arbitrary r on the left side of (2.4) since the particular choice of r = 00 would characterize the functions with approximation O(n-a / d ). However, this is not valid. Instead we obtain a characterization theorem for each a, but for only this one value of r = r( a) once a is selected. We shall use the abbreviated

Nonlinear Wa.velet Approxima.tion in the Space C(Rd) 267

notation BQ := B~-(L .. ), T:= r(a) := (a/d+ l/p)-l where the space Lp where approximation takes place is understood to be fixed.

The characterization (2.4) needs some restrictions on the function 11'. We refer the reader to [6] for the precise statement of the theorems but make the following remarks. The characterization is known in the case of orthogonal wavelets provided II' is smooth enough, essentially a little smoother than the condition II' E BQ. In the case of interest to us, where II' satisfies the refinement equation (1.5), the characterization is valid if II' is smooth enough (II' E C'" with r> a) provided two additional conditions hold. The one is a Strang - Fix condition while the other is a local linear independence condition. We describe these in the next section. However, none of the results of [6] have been shown for p = 00.

To prove the characterization (2.4), one follows the following general recipe. We prove for some (3 > a that the following two inequalities are valid:

D'n(f), ~ Gn-P/dlfIB~·

ISIB~ ~ GnP/dIlSII,·

(2.5)

(2.6)

Inequality (2.5) is a direct theorem of approximation, sometimes called a Jackson inequality and (2.6) is an inverse theorem (Bernstein inequality). Once, (2.5) and (2.6) have been established then the characterization (2.4) follows from general results which relate interpolation spaces to approximation (see DeVore and Popov [8] or Petrushev [12]).

In this paper, we establish (2.4-6) for the case p = 00, i.e. approximation in the space G. Results of this type for spline approximation were announced by Petrushev and serve as the motivation for this work. In the process of deriving our results, we develop new techniques for proving direct and inverse theorems of nonlinear wavelet approximation.

In extending the results of [6] to p = 00 there are actually two possible choices for the approximation norm, namely G or BMO. Our techniques will apply to either. Since the results for G are somewhat easier to prove, we give them here. We shall report on the approximation in BMO in a later work.

3 Approximation in C

We shall now describe the main results of this paper. We let D'n(f) .D'n(f)oo denote the approximation error (2.2) for approximation in the space G(Rd). In addition to the condition (1.5) on 11', we shall also require that II' satisfies two additional properties. The first of these, known as the Strang - Fix property, is that for some positive integer r, we have

(i) cp(o) = 1, cp(2,..j) = 0, j E Zd, j =/: 0,

(ii) Dl/cp(2,..j)=O, jEZd, j=/:O, Ivl<r. (3.1)


Conditions of this type were first introduced by Schoenberg [14] and later studied by Strang and Fix [15] and de Boor and Jia [1] among others. They imply that the spaces SA: contain the space P r of polynomials oftotal degree < r. In particular from the Poisson summation formula and (3.1) (i), we see that the 1('1 are a partition of unity:

I: 1('[ == 1, k E Z. IE'D"

Our last main assumption about I(' concerns linear independence. Let Q be a cube in R tl and let AQ denote the set of all j E Ztl for which 1('(. - j) is not identically zero on Q. We shall assume that

for all Q E 1), the functions 1('(. - j),j E AQ ,

are linearly independent over Q. (3.2)

It follows from this that the functions 1('(. - j), j E ztl, are globally linearly independent and hence are a basis for So. The primary examples of functions I(' are box splines and multivariate B-splines (see §7 of [6]). With these assumptions on 1(', we shall prove the following for the spaces BfJ := B~(LT)' T:= dlf3. Theorem 3.1. Let I(' E C 6 (Rtl) have compact support and satisfy the refinement condition (1.5), the Strong-Fix condition (3.1) for some r > 0, and the linear independence condition. Then, for any d ~ f3 ~ min(r,s), and f E BfJ, we have

(3.3)

Theorem 3.2. Let I(' satisfy the conditions of Theorem 3.1. If SEEn and 0< f3 < min(r,s), then

(3.4)

Theorem 3.3. If I(' satisfies the conditions of Theorem 3.1, then for any d < a < min(r,s) and T:= dla, we have

00 1 ~)naltl(Tn(f)oor ;; < +00 ..... f E B~(LT). (3.5) n=1

We will explain the reason for the restriction f3 ~ d later in this section. The remainder of the paper will be devoted to a proof of these theorems.


If cP satisfies the conditions of Theorem 3.1, then the semi-norm I/IBI' is equivalent to a sequence norm applied to the coefficients of the wavelet decomposition of I. To be more precise, in what follows the coefficients aI(f) denote those that are obtained by using the quasi-interpolant projectors to obtain the wavelet decomposition (1.1) by writing I as a telescoping sum as in (1.7). The following lemma was proved in [6].

Lemma 3.4. Let cP satislY the conditions 01 Theorem 3.1. II I E BfJ has the representation I = L:/E'D bjCP/, lor some constants b/ , and ilO < 13 < min(r,s), then with T:= dlf3,

IfIB' S c (~16'1') 'I' (3.6)

with C depending only on cpo In addition, il al(f) are the coefficients 01 I given in the representation (1.1) by using quasi-interpolants, then

I/IBI' ~ (I: lal(f)l'r) liT

IE'D

(3.7)

where the constants 01 equivalency depend only on cP and T il T is small.

It follows easily from (3.7) that if I E BfJ, 13 ~ d, then I E C(Rd). Indeed, if I = L: aI(f)CPI is the wavelet decomposition (1.1) of I and T = dlf3, M := IIcplioo then

(3.8)

where we used the fact that the iT norms increase with decreasing T. The embedding (3.8) does not hold for 13 < d. This explains the restriction 13 ~ d (a ~ d) in Theorem 3.1 (Theorem 3.3). However, in the case 13 < d the corresponding embedding into BMO is valid. Moreover, Theorem 3.3 holds for the full range of a < min(r,s) with C(Rd) replaced by BMO as will be reported on elsewhere.

We conclude this section with some remarks about the space SA: := span{CP/: I E VA:} which were proved in [6]. The CP/, I E VA:, are a basis for SA:. Hence, each S E SA: has the unique representation

S = L: C/(S)cpI (3.9) IE'Dk

The coefficient functionals CI can be chosen to have support on any cube J E V on which cP I does not vanish. In particular, we have


(3.10)

where the constant C depends only on tp and III/IJI. The Lp norm of S can also be compared with the coefficients. We shall

only need this for p = 00, in which case, we have

(3.11)

with C ~ 1 a constant which only depends on tp.

4 Decomposition of Trees

The main difficulty to be overcome in proving direct estimates for the uniform approximation of f by the elements of En is the overlapping of the supports of the tpI, I E 'D. To handle this, we shall need the following combinatorial lemmas which deal with dyadic cubes.

Let r be a collection of dyadic cubes. If IE r, we let BI(r) denote the collection of cubes J E r with J e I and J maximal, i.e. J is not contained in a larger cube with these properties. The following lemma, which will be used in the proof of the inverse Theorem 3.2 was given in DeVore and Popov[7].

Lemma 4.1. If r = {I} is an arbitrary finite collection of dyadic cubes, then there is a second collection I' of dyadic cubes with the following properties:

(i) reI', (ii) IBI(r)l:5 2d, for all IEI',

(iii) Irl:5 241rl·

Moreover, for any cube IEI', each child of I contains at most one cube from BI(r).

We are assuming that the function tp has compact support. By working with an integer translate of tp in place of tp (which has no effect on the spaces Sk or En), we can assume that tp vanishes outside of the cube [O,i]d, for some odd integer i. HIE 1)k, 1:= j2-k + 2-kO with 0 = [0, l]d, j E Zd,

tpI will vanish outside of I' := j2-k +2-kO' with 0 ' = [O,ijd. We shall call lithe "support cube" of I (however it does not follow that tpI is nonzero on all of I').

The following lemmas will show that it is possible to separate the cubes IE 1) into disjoint classes such that if I, J are in the same class then either I' n JI = 0 or one of the I', JI is contained in the other.


LeIllllla 4.2. Let f be an odd integer. Then eachz = j2-1I:, j E Z and k = 0,1, ... has a unique representation

(4.1)

where io E Z, iv = 0 or 1, II = 1, ... , m and 0 :$ m < 00. In this unique representation, we have m :$ k.

Proof. We prove the existence of such a representation (with m:$ k) by induction on k = 0, 1, .... For k = 0 the statement is obvious. Suppose that we have established the existence ofthe representation (4.1) whenever k < K and consider any z = j2-K . If j is even, j = 2j', then z = j2-K = j'2-K +1 has by our induction hypothesis a representation of the desired form with m :$ K - 1 :$ K. On the other hand, if j is odd, then j = 2j' + f and z = j'2-K +1 + f2- K . Therefore, from the representation of j'2-K +1, we obtain a representation of the desired form (4.1) for z with m :$ K.

To prove the uniqueness of the representation (4.1), we suppose that

m m ·2-11: - . + ~. M\-V _ ., + ~ ., D2-v J - lO L.J 'v~":; - '0 L.J 'v~ .

v=1 v=1

Let r be the largest integer for which ir =f. i~. Clearly, r ~ 1 and

r-1 (ir - i~)f = ~(i~ - iv)f2r-v + 2r(i~ - io).

v=1

Since the right side is divisible by 2 and the left side is not, we obtain a contradiction. •

We can use Lemma 4.2 to "color" each point z = j2-1I:, j E Zd, k E Z as follows. If k = 0, then ~ = j and we assign :c the color 7'(z) = (7'1 (z), ... , 7'd(Z» where 0 :$ 7'1'(z) < f is congruent to jl' modulo f. This has also the effect of coloring all points z = j2-11: , k :$ O. If k > 0, we express each component jI'2-11: as in (4.1): jI'2-11: = io(p) + .... We give z the color 7' := 7'(z) := (7'1(:C)' ... ' 7'd(Z» where 0 :$ 7'1'(:c) < f is congruent to io(p) modulo f. From the uniqueness of the representation (4.1), it follows that the coloring is well defined.

We shall frequently make use of the following important property of our coloring.

j2-11: =f. j'2-11: have the same color if and only if j == j'(modf). (4.2)

Indeed, if k :$ 0, then j2-11: and j'2-11: have the same color if and only if (j - j')2-11: == O(mod f) and this is equivalent to j == j'(mod f) because 2 is a unit mod f. If k ~ 0, and if 7' is the color of j2-11: and 7" is the color of j'2-11: , then j2-11: = 7' + D, + L~=1 2-v ftv and j'2-11: = 7" + fA' + E~=1 2-v ft~.


Where the components of the vectors ~ and ~, are integers and those of the vectors fv and f~, v = 1, ... , Ie are 0 or 1. Hence j - j' = 2·('Y - 'Y') + IN with N E Ztl. Since 2· is a unit mod l, 'Y = 'Y' if and only if j == j'(mod l).

The following lemma gives our partition of1). For a cube I, int (I) denotes the interior of I.

Lemma 4.3. Let, lor each I E V, I' be defined as above where l is an odd integer. Then, there exists disjoint sets T('Y), 'Y E Ztl n [O,l)tl, such that

(i) 1) = UT('Y),

(ii) T('Y) nT('Y') = 0, (4.3)

(iii) iII,JET('Y), eitherI'~J' orJ'~I' orint(I')nint(J') =0.

Proof. We first claim that if I E V, then all the vertices of I' have the same color. Indeed, if I E 1). and j2-· is the smallest vertex of I', then the other vertices v of I' are of the form v = j'2-· = (j +le)2-· where e E Ztl has all components either 0 or 1. Hence j' == j(mod l) and therefore, by (4.2), all these vertices have the same color.

Now, given a 'Y E Ztl n [O,l)tl, we let T('Y) be the collection of all cubes I E 1) whose vertices have color 'Y. This is our partition of V and it clearly satisfies (i) and (ii). To verify property (iii), we suppose that I, J E T('Y) with 2-· = leI) ~ l(J). Let j2-·, j E Ztl be the smallest vertex of J. We consider the collection of points (j + IN)2-· with N E Ztl. These are exactly the points of the form j'2-· which share the color of j2-· by (4.2). Since the vertices of I' share this color they are in this set. Now, either all the vertices of I' are contained in J' in which case I' C J' or else some vertex v of I' is not in J in which case int (I') n int (J') = 0.

5 Proof of the Direct Theorem 3.1

We fix I E BfJ with P ~ d. We wish to construct an approximant S E :Ecn such that

(5.1)

with C a constant independent of I and n and with II . II := II . 1100 here and throughout the remainder of the paper. We begin· with the wavelet decomposition (1.1) of I based on the quasi-interpolant projectors. We use the abbreviated notation al := aI(f) for the coefficients in this expansion.

According to Lemma 3.4, I/IBII ~ (LIE,!) laIIT)1/T with r := diP. It is therefore enough to prove (5.1) under the additional assumption that

(5.2)

Nonlinear Wavelet Approximation in the Space C(R4) 273

We use Lemma 4.3 to decompose 1) into the union of the disjoint sets 7(-y) and obtain the decomposition for I:

1= E I.." ..,ezdn[O,l)d

1..,:= E al'PI· leTC..,)

(5.3)

It is therefore enough to prove (5.1) for each ofthe functions I..,. Let I.., be one of these functions and let 7(-y) be the corresponding tree. We recall property (4.3)(iii)

if I, J E 7(-y), either I' ~ J' or J' ~ I' or int (I') n int (J') = 0. (5.4)

For a fixed no, let 7* ( -y) denote the collection of all cubes in 7 ( -y) such that either i(I) $ 2-no or i(I) ~ 2n o. In view of (5.2), for no sufficiently large, we have

E lall'" < lin. lePC..,)

It follows that for 1-1 := ElePC..,) al'PI, we have with M := II'PII,

Here, we used the fact that the i1 norm does not exceed the iT norm because T $ l.

At most n cubes I E 7(-y) have coefficients satisfying lallT ~ lin. We let Ao denote the set of these cubes and So := EleAo al'PI. Then So E En and therefore it will be enough to approximate the function 10 := I.., - 1-1 - SO, i.e. to show that 10 satisfies (5.1). We further let To:= 7(-y) \ (T*(-y)UAo). Then 10 = EleTo al'PI and all coefficients in this sum satisfy lall T $ lin.

Given a cube I E To, if there is a cube J E To such that I' C J', then by (5.4), there is a unique smallest cube J E To with this property. We call J the predecessor of I and I is a successor of J. By a chain, we mean a collection of cubes C = {II:} C To such that for each k, 11:-1 is the predecessor of II:. The largest cube I E C it is called the top of C, similarly, the smallest cube in C is the bottom of C. All other cubes in C are called intennediate.

We shall further decompose the sum representing 10 into sums over chains of a certain type. Given a collection of cubes 7 C To, we say that C is a primitive chain for 7 if C c T and C is a chain (for To) and for the top I of C, we do not have I' C J' for any other cube JET.

Now let C1 C To be a smallest primitive chain for To such that

E lallT ~ lin. (5.5) lee l


Here smallest means that this chain does not contain another primitive chain with this property. We note that there may not be any such chains. We let 'Ii := To \ C1 and repeat this process. That is, we let C2 be a smallest primitive chain for 'Ii which satisfies EIEC~ laIIT ~ l/n and we let T2 := 'Ii \ C2 • Continuing in this way, we obtain chains CII , k = 1, ... , N satisfying

L laIr ~ l/n. (5.6) IEC~

Each chain CII is primitive in 1i,-1. Since these chains are disjoint, from (5.2) we have that N ~ n.

We also claim that L laIr ~ 2/n (5.7) IEC~

Indeed, if J is the bottom of CII then removing J from CII produces a chain which violates (5.6). Since the coefficient aJ of the cube removed satisfies laJ IT ~ l/n, We have (5.7).

In what follows, we shall make use of the following property:

a chain C is primitive in 1i, if and only if

the predecessor of its top is not in 1i, (5.8)

This is proved by induction on k. For k = 0, it is clear by the definition of primitive chain. Suppose that this has been shown for 1k-l. If CII is a primitive chain in 1i, and J is its top cube then by the very definition of primitive, the predecessor of J. cannot be in 1k. Conversely, suppose that the predecessor of J is not in CII and consider the cubes Jo := J,Jl,'" where for each j, Jj is the predecessor of Jj_l. We know that It is not in 1k. Suppose that for some v, JII is in CII and let v be the smallest index with this property. Then JII-l is in some chain Gi, with i < k (since it has been removed). Then JII-l must be the top of 4 because its predecessor is not in Ci . But this contradicts that Gi is primitive with respect to 7;,.

We need one more further processing of the chains CII. Suppose Cj is one of our chains and J is its top. If the predecessor I of J is either the top or an intermediate cube of a chain CII, we then break CII at I, thereby producing two disjoint chains: one of them has I as its bottom cube. The other has as its top cube one of the successors of I. There will be at most n such breaks (at most one for each of the original chains). After completing this process, We shall have at most 2n chains. We denote the new collection of chains by C:, k = 1, ... ,N with N ~ 2n. We also let rt := I' where I is the top of C: and Ii; := J' where J is the bottom of CZ.

This collection of chains has the following properties. For all j, k =


1, ... , N one of the following is true

(i) rt c Ij- or It C Ik (ii) int (I') n int (J') = 0 for all IE C;, J E CZ.

(5.9)

To prove (5.9), we first observe that if CZ is one of our chains and I is its top and if the predecessor J of I is in To, then J is the bottom of one of our chains. Indeed, if I is the top of one of the original chains Ci then J must be in a chain C;/, i' < i,for otherwise Ci would not be primitive. But then Cil was broken at J and J is the bottom of some chain C; . If I itself was created by a break then its predecessor is the bottom of some C; .

Now suppose that (ii) does not hold, that is there are I E C;, J E CZ satisfy int (J') nint (I') f 0. We shall show that (i) must be satisfied. Now, by property (5.4), either J' C I' or vice versa. We can assume that the first is true; the other case is identical. Let Jo, J}, ... be the sequence of cubes with Jo := J and Ji the predecessor of Ji_l for each i = 1,2, .... One of the cu bes in the sequence J~, Jf, ... is I' and another is It. Since the chains Ct are disjoint, as we transverse the cubes J~ in order of increasing index, we first meet rt and then IT and then I' which verifies (i).

Let Ik := LlEC* allPl, k = 1, ... , /f, and let IN+l := 10-(11 + ... + IN)' k

It will be convenient to let CN+1 := To \ U19:5NCZ even though this is not (necessarily) a chain. Then,

10 = II + 12 + ... + IN+l.

We next observe that with M := IIIPII,

IIlklloo ~ 2Mn-P1d ,

Indeed, from (5.7), we have

k= 1, ... ,N +1 (5.10)

k=l, ... ,N.

(5.11) because of (5.7). To verify (5.10) for k = N + 1, we let z be a point in the support of IN +1 and let Tm be the last set in our construction of the original chains C1 , C2 , • •• • If J 0 is the smallest cube in Tm which contains z, we construct the chain Co: : Jo, J1o •• • , J/I where each Ji is the predecessor of Ji-l and v is the smallest integer for which the predecessor of J/I is not in Tm. From (5.8), we see that Co: is primitive in Tm. Since our construction stopped at Tm , we must have LIEC., lallT ~ lin. Then as in (5.11), we obtain IIN+l(z)1 ~ Mn-fJ1d . Since z is arbitrary, we have (5.10) for j = N + 1 as well.


Our last step is to disjointify the supports of the fie, k = 1, ... , N. If the bottom cube J of C; has sidelength 2- 11 , then using (1.5), we can write

fie = L bII()I. (5.12) l(I)=2-"

It is easy to see (and was shown in [6, (4.9)]) for any I appearing in (5.12) with bI =1= 0, we have

(5.13)

Now, let Ale be the set of all I appearing in (5.12) such that int (I') n int (Ik) =1= 0. There are at most C cubes in Ale with C depending only on l (i.e. the support of I()) and d. We let Sle := L:IEA" bII()I. Then f; := fie -Sle vanishes on I k . Because of (5.13), supp U;) c rt. Hence, from (5.9), the f;, k = 1, ... , N have disjoint supports. Since, IIflell ~ 2Mn-P1d , it follows from (3.10) that the coefficients IbII of fie are all ~ Cn-P1d and hence so are those of f;' Using (3.12), we see that IIf:1I ~ Cn-P1d , k = 1, ... , N.

Now, the function S:= L:f"=1 Sle is in ECn and

fo - S = J; + ... + fN + fN+l.

Therefore,

lifo - SII ~ l~SN IIf;1I + IIfN+111 ~ Cn-P1d + 2Mn-P1d .

This completes the proof of Theorem 3.2.

6 Proof of the Inverse Theorem 3.2

We begin with the following lemma concerning the space S Ie := span 1 E'D" I() 1·

LelllIlla 6.1. There exist constants e > 0 and C > 0 such that for each S E Sle and I E'DIe, we have for any set E with lEI ~ elII :

IISII(1) ~ CIISII (I \ E). (6.1)

Proof. We have S = L:JEV" CJI()J with CJ := cJ(S). If I E 'Die, we let A(I) denote the set of all J E 'Die such that I()J does not vanish identically on I. Since I() has compact support, we have IA(I) I ~ Co with Co depending only on I(). From (3.10), we can estimate the coefficient CJ , J E A(I),

ICJ(S)I ~ 1~lllSI ~ I~I {l'E lSI + lnE lSI}

< 1~ll\E lSI + C II~i IISII(I) ~ CIISII(I \ E) + CeIlSII(I). (6.2)

Nonlinear Wavelet Approximation in the Space G(Rd ) 277

Now, IISII(1) $ MEJeA(I) ICJI, with M := 1I1P1i. Using this together with (6.2), we obtain

IISII(I) $ CIISII(1 \ E) + CeIlSII(I) (6.3)

with C depending only on IP. If we take e sufficiently small the last term on the right of (6.3) can be moved to the left side and we obtain the lemma.

Now, let SEEn, S = EleA bIIPI with IAI $ n. We want to estimate ISIB~ in terms of IISII. If I E A, we are interested in dyadic cubes J such that l( J) = l( I) and IP I does not vanish identically on J. We let r denote the collection of all such cubes J for all the lEA. Clearly, If! $ Cn. For this r, we let r be the collection of cubes given by Lemma 4.1. It is important to distinguish the cubes J E f from the cubes lEA. The latter correspond to the IPI which appear in representing S, the former relate to the support ofthe IP I.

Now, let T = d/P and let p := miner,s). Since, we are assuming that P < p, we have

where

Wp(S,t)~:= sup f 1~'S(zWdz. O<lhl9 i Rd

We are going to estimate wp(S, t)~. Let t > 0 be fixed for the moment and let Ihl $ t also be fixed. We divide

the cubes in r according to their size. Let ro := {I E f: lei) $ pt}, rl := {I E f: lei) > pt} and AD = UleroI. We say a cube I E r 1 is "gootf' if IInAol $ 2-de21II with e > 0 the constant of the Lemma 6.1 (this constant e is now fixed and later constants may depend on e). Otherwise, we say I is "bad". Now let r 2 := {I E r 1 : I is "bad"} and Bo := Uler:J. We further define B := {z: dist(z, Bo) $ pt}. We claim that IBI $ CIAol. Indeed, let r; be the collection of all maximal cubes in r2, i.e. I E r; means IE r 2 and I is not contained in any other J E r 2. Then the cubes I E r; are pairwise disjoint and Eler; III = IBol. Now if z E B, then there is a cube I E r; such that dist(z, I) $ pt. Moreover lei) > pt and therefore z E I where I is the cube with the same center as I but side length 31( I). Hence

IBI ~ L: III ~ C L: III· Ier; Ier;

Since IE r 2, we have lAo n II > (1/2d)e2III and then

C2d C2d

IBI $ -2- L: lAo nIl $ -2 IAol· e Ier; e

Here we have used the fact that the I E r; are disjoint.


Define A := U:=o(Ao-kh). We have IAI :$ CIAol. For D := Rd-(AUB), we can write

(6.4)

We shall estimate the two integrals on the right side of (6.4). The first is quite easy since

1 :$ CIISWIAUBI:$ CIISII"'IAol:$ CIISII'" L: III· (6.5) ~B U~

We shall next estimate L which is a bit tricky. The idea of the proof is

that at the points z,z + h, ... ,z + ph which contribute to a:(S,z), only !.pI, with 1 large, contribute. So we should be able to estimate la~(S,z)1 by terms IhIPIID:SII(I) with 1 large. Here D: is the p-th directional derivative in the direction TJ := h/lhl. In turn, we can estimate the last type of a term by IhIPIISlIl(1)-p. The problem is that even though S(z), ... , S(z + ph) only involve !.pI with 1 large, the segment [z, z + ph] may still meet s~l cubes. The way around this is to introduce some auxiliary sums S and S. Here are the details.

We first note that if zED, then [z, z + ph] n Bo = t/J. We define

S = L: bI!.pI.

IEA,t(I»pf

If ZED, then S = S at z, ... ,z + ph because none of these points is in Ao. Hence, for zED, we have

la~(s,z)1 = la~(S,z)1 :$lhIPIID:SII[z,z+ph]. (6.6)

We shall next estimate the derivative appearing on the right side of (6.6). Given zED and y E [z,z+ph], we let 1, be the smallest cube in r1 which contains y. If no such 1, exists, then S vanishes at y and the estimates that follow are trivial. Let r be the child of 1, which contains y. Then from the definition of 1" 1* is not in r 1. We let

E,:= 1*- U J.

Actually, E, has a simple structure: from Lemma 4.1, either E, = r or E, = 1* \ 10 for some 10 E r 1. By the definition of 1" we have y E E,.

We shall next estimate D:S on E, which in turn gives an estimate for ID:S(y)l. We note that from the very definition of 1" we have

S(u) = S(u) := bJ!.pJ(u),

Nonlinear Wavelet Approximation in the Space O(R") 279

Using our assumption (1.5) about cP, we can rewrite S at levell(II/):

S(u) = ~ CJ (S)cpJ (u), u EEl/ (6.7) JEA"

where

A.rJ := {J E'D: l(J) = l(II/) and CPJ is not identi~ally zero on E,,}.

We next estimate the coefficients appearing ~ (6.7). Let J E AI/. We

shall find a cube K C J' n E" on which S and S agree and where we can

estimate the coefficient CJ(S). We consider the following cases. If E" = [*,

we can take K := r. If E" 'I r, then by Lemma 4.1, E" = r \ 10 with 10 E r 1 . If 1101 ::5 el1*l with e as above, then we can take K := r again. Finally, if 1101> el1*l, then we can write EI/ as the union of dyadic cubes of measure 10. We take K as one of these cubes on which CPJ does not vanish identically.

In all these cases CPJ is not identically zero on K and IKI/III ~ e. Therefore, from (3.10), we have

(6.8)

where CE is a constant depending on e (and on cpl. Since K ~ EI/, we

have -S = =s on K. Also =s E S" where 2-" = l(K). Therefore, setting k = K - (K n Ao) and noticing that II/ is "gootl', by Lemma 6.1, we have

because IKnAol ::511" nAol::5 2-de2III/I::5 elKI·

But S = S on k, therefore

"=Sll(K) ::5 CII-SII(k) = CIISII(k) ::5 CIISII(II/). (6.9)

From (6.8) and (6.9), we obtain

ICJ(S)I ::5 CIISII(II/). (6.10)

Returning to estimating ID:S(y)l, we have

ID:S(yW ::5ID:S(yW ::5 C~~ ICJ(S)nD:CPJ(y) I" ::5 Cl(II/)-P"IISII"(II/).

Hence,

IID:S",.[z, z + ph]::5 CLl{I,,)-P"IISII"(I,,). (6.11) I"


Now the cubes I" appearing in (6.11) are from rl. Therefore from (6.11) and (6.6), we obtain

where if [x, x + rh] n I # 0

otherwise.

Since f Q(x, I)dx ~ CIJI, IE r l , we obtain

lld~(S,xWdx ~ Ctpr L l(I)d-prllSlnI)· D lerl

From this, (6.4) and (6.5), we have

wp(S,t)~ ~ CIISW (L l(I)d +tpr L l(I)d-pr) . lero lerl

Recall that ro and r 1 depend on t. At last, we obtain

ISIBII = {100 rd-1wr(S, t)~dt }

< CIISW {Ll(I)d 100 rd-Idt let l(I)/p

+ ~l(I)d-pr l l(I)'P rd-l+pr dt}

ler 0

~ CIISlir L 1 ~ CnllSll r ,

let

where we have used the fact that -d - 1 + pT > -1, i.e. p > {3. This completes the proof of the inverse theorem.

7 Proof of Theorem 3.3

For a > 0 and 0 < q ~ 00, let A~ := A~(C) denote the approximation space which consists of all functions f E C(R d) such that


with the usual modification for q = 00. The expression in (7.1) is a quasiseminorm for A~, we obtain the quasi-norm for this space by adding lillie to this expression. The Jackson inequality in Theorem 3.1 and the Bernstein inequality in Theorem 3.2 characterize the spaces A~ (see [8]) as interpolation spaces. Namely, for each 0 < q ~ 00 and 0 < a < {J < min{r, s), we have

(7.2)

with equivalent norms. In some cases, we can identify the interpolation spaces which appear on

the right side of (7.2) as Besov spaces. For this, we shall use two theorems on interpolation. The first of these is the following theorem of Peller [13] for interpolation between BMO and Bf3:

(BMO,Bf3)alf3.f = Ba, provided q = r{a):= d/a. (7.3)

The second is a theorem of DeVore and Popov [8] which says that the family A~ is invariant under interpolation: if 0 < ao, a1 < 00 and 0 < qo, q1 ~ 00,

we have

where a := {1- 9)ao + 9a1. (7.4)

It follows from Peller's theorem and the reiteration theorem of interpolation that

provided a:= (1-9)ao+9a1 and q = T{a) := d/a. (7.5)

We shall need the following.

Lemma 7.1. For any d:$ {J < min{r,s) we have the continuous embeddings

(7.6)

Proof. The right embedding in (7.6) is an immediate consequence of the Jackson inequality of Theorem 3.1. For the left embedding, we suppose that IE Ae with r := r{{J). We let S" E E2" satisfy III - S"II ~ 2U2"{/)00, k = 0,1, .... Then, with T" := S" - S"_1, k = 1,2, ... and To := So we have / = E~=o T". Since T :$ 1, the seminorm I·IB~ is subadditive and we obtain from the Bernstein inequality of Theorem 3.2:

00

I/IB~:$ :EIT"IB~:$ C:E2"f37"IIT"II~:$ CII/II:A~. (7.7) "=0 n=O

Here, for the last inequality, we use the triangle inequality to find for k = 1,2, ... ,


while IIToll $ Cllflloo. The inequality (7.7) gives the left embedding of ~~. .

If we use the embeddings (7.6) with P = d and d < P < miner, s), we obtain from (7.4) and (7.5) the continuous embeddings for d < ex < P,

A~(a) = (A~(d),A~(p»aIPI .. (a) c B a C (A:!o,A~)aIPI .. (a) = A~(a)' That is, we have A~(a) = B a which is Theorem 3.3.

References

[1] C. de Boor and R.Q. Jia, Controlled approximation and a characterization of the local approximation order, Proc. Amer. Math. Soc. 95(1985), 547-553.

[2] A. Caveretta, W. Dahmen, and C. Micchelli, Stationary Subdivision, preprint.

[3] I. Daubechies, Orthonormal basis of compactly supported wavelets, Communications on Pure & Applied Math. 41(1988), 909-996.

[4] R. DeVore, B. Jawerth, and Brad Lucier, Surface compression, preprint.

[5] R. DeVore, B. Jawerth, and Brad Lucier, Image compression through transform coding, preprint.

[6] R. DeVore, B. Jawerth, and V. Popov Compression of wavelet decompositions, preprint.

[7] R. DeVore and V. Popov, Free multivariate splines, Constr. Approx. 3(1987), 239-248.

[8] R. DeVore and V. Popov Interpolation spaces and nonlinear approximation, In: Functions Spaces and Approximation, (Eds.: M. Cwikel, J.Peetre, Y. Sagher, H. Wallin), Vol. 1302, 1986, Springer Lecture Notes in Math (1988), 191-207.

[9] M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Math. J., 34(1985), 777-799

[10] M. Frazier and B. J awerth, A discrete transform and decompositions of distribution spaces, to appear in J. of Functional Analysis; also in MSRI reports 00321-89, 00421-89 (1988).

[11] Y. Meyer, Ondelletes et Operateurs, Hermann Pub!., France, 1990.


[12] P. Petrushev, Direct and converse theorems for spline and rational approximation and Besov spaces, In: Functions Spaces and Approximation, (Eds.: M. Cwikel, J.Peetre, Y. Sagher, H. Wallin), Springer Lecture Notes in Math, Vol. 1302, 1986, pp. 363-377.

[13] V. Peiler, Hankel operators of the class 9r and their applications (Rational approximation, Gaussian processes, majorant problem for operators), Math. USSR-Sb., 122(1980), 538-581.

[14] I.J Schoenberg, Cardinal Spline Interpolation, SIAM CBMS 12 (1973).

[15] G. Strang and G.F. Fix, A Fourier analysis of the finite element method, In: Constructive Aspects of Functional Analysis, G. Geymonant, ed., C.I.M.E. II Cilo, 1971, pp. 793-840.

Ronald A. DeVore Deptartment of Mathematics Univ. of South Carolina Columbia, SC 29208 U.S.A

Xiang Ming Yu

Pencho Petrushev Mathematics Institute Bulgarian Academy of Sciences 1090 Sofia, P.O. Box 373 BULGARIA

Department of Mathematics Southwest Missouri State University Springfield, MO 65804 U.S.A.

Completeness of Systems of Translates and Uniqueness Theorems for Asymptotically Holomorphic Functions

A.A. Borichev

1

We consider some completeness problems for systems of right and arbitrary translates in certain weighted spaces of functions on the real line. A generalization of the Titchmarsh convolution theorem and a tauberian theorem for quasianalytic Beurling-type algebras are obtained.

The solution of these problems involves the usage of the so-called generalized Fourier transform. After that, completeness problems turn into uniqueness problems of the theory of functions, which are interesting in themselves.

This report is a short version of the work, one part of which is published in [3], and the other will appear in [1].

2

We deal with problems concerning completeness of systems of translates {rtf},

(rtf)(:c) = f(:c - t),

in spaces A of functions on the real line. Via the Hahn-Banach theorem, such problems are transformed into problems on solving convolution equations: on the whole line,

a * b = 0,

if we take all translates, and on the half-line,

if we take only right translates, where a E A, b EA· ,

Suppose that the Fourier transforms:Fa and:Fb ofthe convolutors a and b (as usual,


286 A.A. Borichev

are well-defined on certain sets. If the intersection of these sets is non-empty, then the equations a * b = 0, (a * b)+ = ° can be rewritten accordingly as :Fa·:Fb = 0, :Fa·:Fb E :F(A * A*)+; that is, harmonic analysis problems turn into multiplicative problems from the theory of functions.

In studying the possibility of extending this method to the case when these sets are disjoint (in particular, when the Fourier transform doesn't exist for one of the spaces A, A*) it turns out that one can construct an analogue of the Fourier transform, that maps convolutions into products. It is applicable to very rapidly growing functions for which the usual Fourier transform cannot be used. Though the Fourier images of these functions are not necessarily analytic, they are asymptotically holomorphic, i.e. they satisfy

/ E C1(0), 111/(z)1 < w(dist(z, a~», w(o) = 0,

where a = ! (-Ix + i/y). We further remark that this "generalized" Fourier transform is consistent; that is, it coincides with the usual Fourier transform on the domain of definition of the latter.

Unfortunately, this technique doesn't allow us to work with Banach spaces. Thus we are lead to study problems of completeness in projective (inductive) limits (of weighted spaces). In addition, for the weighted spaces under consideration, we sometimes need very strong conditions on the regularity of the weight (which shall not be written down precisely in this paper).

3

The usage of generalized Fourier transform is particularly efficient in problems of spectral analysis; that is, for cases of empty spectrum. See [13] about the spectral analysis-synthesis problem.

This paper is devoted to two problems, stated in [7], [12, Problem 7.18], and in [12, Problem 7.19].

(a) The classical Titchmarsh convolution theorem claims that the convex hull of the support of the convolution of two functions with compact supports is equal to the sum of their supports. The condition of compactness cannot be omitted in general (it is sufficient to consider u == 1, and a function v with supp v C (0,1), f01 v(z)dz = 0).

In the papers of Domar [7] and Ostrovskii [15] the Titchmarsh theorem was extended to the case of functions, decaying rapidly on the negative halfline (approximately as exp(-z2), exp(-lzl1oglzl» and possibly, having

Completeness of Systems of Translates 287

some growth on the positive half-line:

inf supp u + inf supp v = inf supp u * v. (3.1)

It should be mentioned that these papers were motivated by problems from such different domains as radical Banach algebras and probability theory.

When u and v are bounded or their growth is at most exponential, one can apply the Fourier transform and standard analytic techniques such as those in [15] to prove (1).

We introduce a self-adjoint topological algebra U,

U = {f E LI.,(R) , Vc > 0, ia.1f(Z)I'(P(Z»-'dZ < 00,

3c> 0, J~ If(-zW(p(z)Ydz < oo}, where p E C(~), log p(z) is a convex function, lim3:_oo z-llogp{z) = 00

and for some c < 00 the function z-Clogp(z) decreases (for large z). A weakened version of (I) for U,

u E U+, v E U_ ~ inf supp u + inf supp v = inf supp u * v (3.2)

is equivalent to the fact that elements f of the algebra U+ are cyclic (that is clos£{rd, t ~ O} = U+) if aJ}d only if 0 E esssupp f.

The statement on the non-existence of zero divisors in U,

u, v E U, u * v = 0 ~ u = 0 or v = 0 (3.3)

is equivalent to the fact that every non-zero element f of U is cyclic; that is clos£{rd, t E R} = U.

Finally, the equality (1) is equivalent to the fact that

clos £{ rd, t ~ O} = U {::} inf supp f = -00.

(b) The general tauberian theorem of Wiener claims that every closed ideal Ie Ll(R) such that the Fourier transforms :Ff(t), f E I do not have common zeros on R is equal to Ll(R) itself.

A. Beurling, in the late thirties, introduced a class of function algebras, so called Beuding algebras,

L!(R) = {f: fp E Ll(R)},

where p(z)p(y) ~ p(z + y), p(z) ~ p(O) = 1, p(tz) ~ p(z), t ~ 1. These conditions on the weight p imply the existence of two limits

1. logp(z) (l'± = 1m . 3:_±OO z

288 A.A. Borichev

Beurling divided these weights into three groups; the analytic case, Q+ > Q_; the quasi analytic case,

100 logp(x) - a+1x1d 1 2 X = 00;

-00 +x

and the non-quasianalytic case,

100 logp(x) - Q+1x1d 1 2 X < 00.

-00 +x

This classification is natural enough because the Fourier transform maps elements of L;(R) into functions which are continuous in the strip region S = {z : Q_ ~ 1m z ~ Q+}, and are analytic in its interior.

Further, the divergence of the integral

100 logp(x) - Q+1x1d 1 2 X,

-00 + x

is equivalent to the quasi analyticity of the Fourier transforms of elements of L;(R) on the boundary of S (in the sense of the Denjoy-Carleman theorem).

Beurling proved that an analog of Wiener's theorem is valid in the nonquasianalytic case.

A closed ideal I in L;(R) is said to be primary at 00 if

n {z E S: :FJ(z) = O} = 0. lEI

In 1950 Nyman [14] proved that in some particular (analytic and quasianalytic) cases there exist primary ideals at 00 in L;(R). Later, Korenblum [11], Vretblad [16], and Domar [5] demonstrated chains of primary ideals in the general case, Hedenmalm [8] described all primary ideals at 00 in the analytic-non-quasianalytic case (for p(x) = exp clxl it was made by Korenblum [10)).

Here the analytic-non-quasianalytic case is the following one:

Joo logp(x) - Q+xd 1 2 X < 00,

+x 1 logp(x) - Q-Xd

---::;..=....;~--=2-- X < 00. -00 1 + x

The ideals are parameterized by two numbers as follows:

It = {f E L;(R): c±(f) ~ c},

c±(f) =}~ 2; log+.log+ I:FJt±x) 1- x,

(3.4)

(3.5)

(3.6)


where, for the sake of simplicity a = a+ = -a_. The corresponding tauberian theorem is formulated as follows:

Let la E L}(R), a E A. Then closC{rda, a E A, t E R} = L}(R) ¢}

¢} inf 6+(fa) = inf L(fa) = -00 & n {z E S: Fla(z) = O} = 0. aEA aEA aEA

The fundamental steps in these works concern the possibility of analytic continuation of Carleman's transform of a functional annihilating an ideal I primary at 00 in L}(R), and the log -log theorem of Levinson. Specifically, if IfJ E (L}(R»·, then the elements 1fJ+ and 1fJ- have Fourier transforms which are analytic, respectively, in C+ and C_.

One should prove that if IfJ is.orthogonal to I, then FIfJ+ can be continuated analytically across the strip S to -FIfJ-. This entire function is called the Carleman's transform of IfJ and is denoted by FIfJ. Then, one should evaluate the growth of this function by using Levinson's log -log theorem. In the quasianalytic case the first step could be made by a method from the theory of commutative Banach algebras, offered by Domar [6].

We show a way of extending both of these steps and, accordingly, give a description of primary ideals at 00 in the quasianalytic case for the space

L;,:r:(R) = {f: 'tin, l(z)(1 + IzIR) E L;(R)}.

One more problem, which can be treated in a similar way, was formulated by Gurarii [12, Problem 7.19] and concerns ideals in L}(R+).

Question: Is it true that if 1 E L}(R+), 0 E esssupp/, FI(z) 1: 0, 1m Z ~ 0, then closC{rd, t ~ O} = L}(R+)?

We assert that the answer is positive if L}(~) is replaced by the space L;,:r:(R+).

4

Let us introduce the generalized Fourier transform. It can be defined by different methods. We use a construction proposed by A. Volberg. It should be noted that when solving convolution equations, one can replace the spaces U from (a) and L},:r:(R)"from (b) by spaces of smooth functions.

In the case (a) put

p·(r) = max(rz -logp(z», p·(r) = rv(r) -logp(v(r»,

j ll(Im z) j:1(z) = -00 I(z)ei:r:zdz, imz ~ O.

It can be proved that j: is an isomorphism between the convolution algebraU1 ,

U1 = {f E COO(R) : 3c> 0 'tIk 3Cl, 111:)(-z)1 ~ Cl(p(Z»-C, z ~ 0,

290 A.A. Borichev

and Q / J, a function algebra relative to pointwise multiplication, where

Q = {f E C1(C+) : Vc < 00 Vk 3Cl, 181(z) I ~ cl(l+ IRe zl}-I:

x exp(-p*(cImz)),

3c < 00 Vk 3Cl, I/(z)1 ~ cl(l+ IRezl}-k

x exp(p*(cImz»)},

J = {f E Q: Vc < 00 Vk 3cl. I/(z)1 ~ cl(I+IRe zl}-k exp( -p*(cImz»)}.

So the problems (3.1) - (3.3) turn into questions on multiplicative structure ofthe algebra Q/ J.

Further by theoretical-functional methods in [3] the following results can be proved.

Theorem A. The implications (3.2) and (3.3) are valid lorU. The implication (3.1) holds il

lim logp(z) = 00.

"'_00 zlogz

In the case (b), for the sake of simplicity, let p be even, a = a+ = -a_. We define p*, v, f:1 :

p*(r) = max{logp(z) - rz), p*(r) = logp(v(r)) - rv(r),

f:o/(z) = 100 l(z)ei"'Zdz, Imz > a,

-tl(Imz)

l t1 (-Imz)

f:o/(z) = -00 l(z)ei"'Zdz, Imz < -a.

It can be proved that for I E L~''''(R) the generalized Fourier transform f:1(z) (which, by the definition, is equal to FI(z) for z E S and to f:o/(z) for other z) belongs to C1(C) and

Vn, 18(f:f)(z)lexp p*(IImzl) = o«IImzl- ct(1 + IRezl)-2).

Further, for <p E (L~''''(R))* * Cgo

IFep±(z)lexp(-p*(IImzl) = 0«1 + IRezl)-2).

Finally, it can be proved that a functional ep is orthogonal to a primary ideal I at 00 if and only if Fep = Fep+ = -Fep_ is an entire function and f:1· Fep E LOO(C).

So by function theoretic methods in [1] the following results are proved (under some regularity conditions).


Theorem B. In the quasianalytic case all primary ideals at 00 are described as in (3.4), (3.5), where instead 01 (3.6) we have

c5±(f) = 1~ [R (log+ IFlt±Z) D -z] ,

and R(z) = (2/tr) J~ (logp(y»/(1 + y2) dy.

Theorem C. II I E L!J~(~), 0 E esssuppl, FI(z) i: O,Imz ~ 0, then

closC{rt/, t ~ O} = L~J~(~).

Similar results can be stated in the analytic-quasianalytic case, at least for even weights (P(z) = p(-z»,

Joo logp(z) - a+z d 1 + z2 Z = 00.

5

We now present some uniqueness theorems for analytic and asymptotically holomorphic functions that arise in proofs of Theorems A-C.

(a) The implications (3.1) - (3.3) are equivalent to the following ones for the algebra Q:

I, 9 E Q, Ig E J :::} IE J or 9 E J,

IE Q,g,lg E QnLoo(C+)\J:::} for somecl/(z)I+lg(z)1 < c.exp(cImz),

I, 9 E Q, Ig E QnLoo(C+) \J :::} for somec I/(z)l+ Ig(z)1 < c·exp(cImz).

In the proofs of these statements the usual asymptotically holomorphic technique (estimates on the harmonic measure and balayage, see [2],[4]) is used to reduce them to uniqueness theorems for analytic functions (in particular, to some theorem of Ostrovskii [15]).

(b) When proving Theorem B, we state three uniqueness theorems: on asymptotics of quasianalytic functions on R, on asymptotics of entire Carleman's transforms, and on that of their products:

Theorem D. Let I E L~(R), I i: 0 and p quasianalytic. For some set E C~, such that m(En(z,z+ 1)) < 1/3 lor all z, there exist the limits

292 A.A. Borichev

which are either finite or equal to -00. There is a function f such that R±(f) = o. Theorem E. Let tp E (L;(R))*, tp ::j:. 0, and let an entire function :Ftp be the Carleman's transform of tp. If p is quasianalytic, there exist the limits

Rf (tp) = lim [R (log max l:Ftp(z)l) - xl ' x-±oo Rez:::x

which are either finite or equal to ±oo. Rf (tp) is equal to -00 if and only if:Ftp is bounded in the half-plane {z: ±Re z > OJ. There is a functional tp such that Rf(tp) = O.

Theorem F. If f and tp satisfy the conditions of Theorems D and E, then

Here, besides the asymptotically holomorphic technique, some sharp form of the Warshawskii theorem on the asymptotics of conformal mappings of infinite strips is employed.

It should be noted that weaker estimates on asymptotics of quasianalytically smooth functions were earlier stated in [9], [11], [16]. Theorem E can be considered as an extension of the log-log theorem of Levinson and the Phragmen-Lindelof theorem for the strip.

Acknowledgement. The author thanks N.K. Nikolskii for his attention to this work.

References

[1] A.A. Borichev, Beurling algebras and generalized Fourier transformation, LOMI Preprints E-4-90, Leningrad, 1990.

[2] A.A. Borichev, Boundary uniqueness theorems for asymptotically holomorphic functions and asymmetric algebras of sequences (Russian), Mat. Sbornik 136(1988), no. 3, 324-340. English transl. in Math. USSR Sbornik 64(1989), no. 2, 323-338.

[3] A.A. Borichev, Generalized Fourier transformation, Titchmarsh theorem, and asymptotically holomorphic functions (Russian), Algebra and Analysis, 1(1989), no. 4, 17-53; English transl. in Leningrad Math. J., 1(1990), no. 4, 825-857.

[4] A.A. Borichev, A.L. Volberg, Uniqueness theorems for asymptotically holomorphic functions (Russian), Algebra and Analysis, 1(1989), no. 1, 146-177; English transl. in Leningrad Math. J., 1(1990), no. 1, 157-191.


[5] Y. Domar, Bilaterally translation-invariant subspaces of weighted LP(R), Radical Banach algebras and automatic continuity, In: Lect. Notes in Math., 975(1983), Springer-Verlag, 210-213.

[6] Y. Domar, On the analytic transform of bounded linear functionals of certain Banach algebras, Studia Math., 53(1975), no. 3, 203-224.

[7] Y. Domar, A solution of the translation-invariant subspace problem for weighted LP on R, R+ or Z, Radical Banach algebras and automatic continuity, In: Lect. Notes in Math., 975(1983), Springer-Verlag, 214-226.

[8] H. Hedenmalm, On the primary ideal structure at infinity for analytic Beurling algebras, Arkiv for Mat., 23(1985), no. 1, 129-158.

[9] I.I. Hirshman, Jr., On the behavior of Fourier transforms at infinity and on quasi-analytic classes of functions, Amer. J. Math., 72(1950), no. 1,200-213.

[10] B.I. Korenblum, A generalization of Wiener's tauberian theorem and harmonic analysis of rapidly increasing fucntions (Russian), Trudy Moskov. Mat. Obsc., 7(1958), 121-148.

[11] B.1. Korenblum, Phragmen-Lindelof type theorems for quasianalytic classes of functions (Russian), Investigations on Contemporary Problems of Theory of Functions, 1961, Moscow, 510-514 ..

[12] Linear and Complex Analysis Problem Book, 199 Research problems, In: Lect. Notes in Math., 1043(1984), Springer-Verlag.

[13] N.K. Nikolskii, Invariant subspaces in operator theory and function theory, Itogi Nauki i Techniki: Mat. Anal., 12, VINITI, Moscow, 1974, pp. 199-412; English transl. in J. Soviet Math. 5(1976), no. 2.

[14] B. Nyman, On the one-dimensional translations group and semi-group in certain function spaces, Thesis, Uppsala, 1950.

[15] LV. Ostrovskii, Generalizations of the Titchmarsh convolution theorem and the complex-valued measures, uniquely determined by their restrictions to a half-line, 8th Int. Semin. Uzhgorod, In: Lect. Notes in Math., 1155( 1985), Springer-Verlag, 256-283.

[16] A. Vretblad, Spectral analysis in weighted L1 spaces on R, Arkiv for Math., 11(1973), no. 1, 109-138.

A.A. Borichev Steklov Mathematical Institute St. Petersburg Branch Fontanka 27 St. Petersburg, 191011 RUSSIA

Approximation by Entire Functions and Analytic Continuation * N . U. Arakelyan

ABSTRACT This article deals with the application of results from the theory of approximation by entire functions to, classical problems about the analytic continuation of analytic functions given by their Taylor series. Generalizations and completions of well known results due to E. Lindelof, F. Carlson, and others are obtained.

1 Introduction

From the mid-sixties on, the methods and results ofthe theory of tangential approximation by entire functions and its generalizations found important applications in a number of branches of complex analysis such as the theory of boundary behavior, R. Nevanlinna's theory of value distribution and multi-dimensional problems all of which required the construction of examples with pathological behavior.

In the present paper, we shall discuss applications of the theory of approximation by entire functions to classical problems about the analytic continuation of power series.

Section 1 gives the formulation of our problem, discusses an approach to its solution, recalls some classical results, and states a lemma.

Before we can state our theorems in convenient form, we need some notations and some geometrical notations which are defined in Section 2. Section 3 gives the statement of our theorems and Section 4 gives their proof.

The letter A will denote a positive "world constant" (i.e. independent of the variables of the problem under consideration), not necessarily the same at all occurrences.

1.1 THE PROBLEM

In Weierstrass' theory of analytic functions, such a function is defined by

*The editors are grateful to Wolfgang Fuchs for his extensive revisions of the original manuscript.


296 N. U. Arakelyan

an "analytic element"; the complete analytic function consists of the totality of analytic continuation of the analytic element. Typically an analytic element is a convergent power series

00

fez) = L fnzn. (1.1) n=O

It is an important problem to give methods by which the properties of the complete analytic function can be read off from (1.1). In particular, one would like to be able to describe the location of the singularities of f( z) in terms of the fn.

1.2 THE COEFFICIENT FUNCTION METHOD

Under the assumption that

fn = <,hen) (n> no), (1.2)

where <,h is a function belonging to some class of holomorphic functions, explicit results have been found by Hadamard and many others (see [7] and [3]).

1.3 CLASSICAL RESULTS

In this section we state three classical results which were obtained by the coefficient method. For explanations of the notations and the terms used in the statement of these theorems, see Section 2.

Theorem A. (Leau [18], LeRoy [19]) The function element (1.1) can be analytically continued to C\[l,oo], if the function <,h in (1.2) is holomorphic in the neighborhood of 00.

Theorem B. (Carlson [9], [10]) (a) Let <,h be an entire function of exponential type with indicator diagram I. Then (1.2) implies that (1.1) has a single-valued continuation to the component of the open set C\ exp( -I*) which contains the origin.

(b) If the width of the set I in the direction of the imaginary axis is less than 211", then f admits analytic continuation to the point at infinity along a radius and the expansion

00

f(z) = fo-<,h(O)- L<,h(-n)zn n=l

is valid near 00.

(c) If the width of the compact set J C C in the direction of the imaginary axis is less than 211" and if (1.1) has a single-valued analytic continuation to the domain C\ exp( -J*) which contains 0 and 00, then there exists an

Approximation by Entire FunCtions and Analytic Continuation 297

entire function of ezponential type whose indicator diagram is contained in J and which satisfies (1.2) with n = 1.

Theorem C. (Leau [18], Wigert [25]) The complete analytic function fez) is an entire function of 1/(z - 1), if and only if

00

fez) = L: ¢(n)zn, n=O

where ¢ is an entire function of order zero.

Noting that the theorem applies to f(l/z) and putting z = 8/80 we deduce

Corollary. f(8) is an entire function of 1/(s - so), if and only if

00

f(8) = L:exp(won),p(n)s-n (lsi> 1801), n=O

where ,p is an entire function of order zero and Wo = log so.

Theorem C is the first example of a theorem on analytic continuation which gives necessary and sufficient conditions. Our theorems will be generalizations of Theorems A and B giving necessary and sufficient conditions. To state them conveniently we shall need the notions introduced in Section 2.

1.4 RESULTS FROM ApPROXIMATION THEORY

The approximation by the entire function g(z) to the function fez) on the unbounded set X C C is called tangential or asymptotic with speed a( z), if

If(z) - g(z)1 < ea(z) (z EX),

where e > 0, fez) E C(X) is holomorphic in the interior of X, and a(z) -+ 0 asz-+ooinX.

We shall need the following result from approximation theory:

Theorem A. (M.V. Keldish [17], [21]) If X is a closed, unbounded region in the complex plane bounded by a Jordan curve in the extended complex plane, then tangential approximation on X is not possible for arbitrary a(z), but it is possible when e > 0 and

a(z) = exp( -lzI6) (0 < 6 < 1/2).

298 N. U. Arakelyan

1.5 A LEMMA

The coefficient function t/J is by no means unique. The following lemma is useful in the choice of a suitable t/J.

Lemma 1. Given {gn} (n = 0,1,2, ... ) with Ignl1/n -+ 0 as n -+ 00 and a real number m, one can find an entire function of order < 1 and of interior exponential type ~ 11' in the hal/-plane

11' m = {z = x + iy : x ~ my}

such that

t/J(n) = g(n) (n = 0,1,2, ... ), t/J(n) = 0 (n = -1, -2, ... )

h(8,t/J) = -00 (exp(i8) E 11'!).

Here 11'! is the interior of 11' m. The proof of Lemma 1 is essentially the same as that of Lemma 1.2 in [3]. Therefore, it is omitted.

2 Some Geometrical Notations

2.1 NOTATIONS

For any set A of complex numbers we write

A*={z=W:WEA}

and, generally, for any function G: C -+ C, G(A) = {z = G(w) : wE A}. We define

A = A(a,[J) = {z : a ~ argz ~ [J} (a ~ [J ~ a + 211'), 1I'm = {z = x + iy: x ~ my}.

2.2 SOME GEOMETRIC NOTATIONS.

The support function of the set A of complex numbers is defined by

K(8) = K(8,A)'= Sup"Re(ze-i8 ). (8 E R) zEA

The function K : R -+ (-00,00] has period 211' and is lower semicontinuous. It also has the property of trigonometric convexity:

If a < f3 < a + 211' and K(a) and K(f3) are finite, then

K(8) ~ AK(a) + I'K([J) (a < 8 < [J)

where

A = sin(f3 - 8)/ sin(f3 - a), I' = sin(8 - a)/ sin([J - a).

Approximation by Entire Functions and Analytic Continuation 299

Therefore the set K-l(oo) is either empty or it consists of intervals of length 1r whose endpoints mayor may not be included.

Vice versa, if the function K has all the enumerated properties, then

A = n {w E C : Re(we-i9 ) ~ K(O)} 0$9$211"

(2.1)

is a closed convex set uniquely determined by the finite values of K(O). For a proof of these facts, in the case that K is finite-valued, see [20, Ch. 1, 19]. The generalization to all K presents no difficulty.

2.3 THE INDICATOR

A function holomorphic in the sector

is of exponential type in A, if

rT = lim sup Izl-1log I</>(z) I < 00

as z -+ 00 in A. A function holomorphic in A 0 , the interior of A, is locally of exponential type in A, if it is of exponential type in every sector with vertex at the origin which is contained in A.

If </> is holomorphic in A, the indicator function of </> in A is

h(O, </» = lim sup r-1log 1</>(rei9)I. r-oo

The function

K(O) = { h(~</» where h is finite, otherwise,

has all the properties of a support function. The associated set

is called the indicator diagram of </> (see [20, Section 15]).

2.4 LOG-CONVEX SETS

A set E C C is logarithmically convex (log-convex), if there is a closed, convex set L = L(E) such that

a) E\{O,oo} = exp(L) = {z: z = expw, wE L}. b) The mapping z = exp w is univalent in the interior of L.

Any such L(E) will be called a logarithmic diagram of E.

300 N.U. Arakelyan

It follows immediately from the definition that a line Re w = U either has an empty intersection with L(E) or meets it in a vertical interval

The convexity of L also implies that the functions Vj are defined in a closed, finite or infinite interval S of the u-axis. The interval is unbounded on the left, if and only if 0 E Ej S is unbounded on the right, if and only if 00 E E.

We discuss the geometry of L(E) in more detail in three cases. (a) If E is bounded, 0 ¢ E, and its complement is connected, then, by

the compactness of S,

V2(U) - Vl(U) ~ C < 2'1f (u E S).

A simple geometrical argument shows that in this case L( E) lies in the parallelogram bounded by lines Re w = const through the endpoints of S and by two parallel support lines through the points (uo, Vj(uo», where Uo is chosen so that the maximum of V2 - Vl is attained for u = Uo.

(b) If 0 E BE, choose P E L(E). By considering the straight line segments PQ, where Q -+ -00 along BE, we see that there is a ray

v = mu + const (u < u(P» (2.3)

passing through P and lying entirely in L(E). By (2.2), the real number m is unique and independent of the choice of P. We shall call it the parameter of L(E). The set L(E) is the union of the lines (2.3).

This and (2.2) imply that

Vj(u) = mu + Cj + 0(1) (u -+ 00) (j = 1,2) (2.4)

and that L( E) may be enclosed in a strip of vertical height ~ 2'1f. (c) 0 is an interior point of E. Let Zo = exp Wo, where Zo is one of the

boundary points of E closest to O. Then we must have

By continuity (2.5)

but, by the choice of wo,

(2.6)

unless E is the disk Izl ~ exp Uo. If we exclude this case, the convexity of L(E) shows tha.t Wo is unique.

Any two support lines of L(E) through (UO,Vl(UO» and (UO,V2(UO» respectively can not intersect in Rew < Uo. Therefore

D+V2(UO) ~ D+Vl(UO).

(By co~vexity these differential coefficients exist.)


Vice versa, if M is a closed convex set contained in the region

'1.&0 ~ Re w ~ bj '1.&0 + iVl('I.&O), '1.&0 + iV2(UO) EM,

which can be enclosed in a strip of vertical height 211" bounded by straight lines of slope m, then M together with the part '1.& < '1.&0 of the strip is the logarithmic diagram of a log-convex set E, provided that

(2.7)

Similar results hold for sets containing 00. They are quickly deduced from the above results by the transformation z ~ 1/ z which transforms E into a log-convex set with diagram -L(E).

3 Statement of Results

3.1

Theorem 3.1. Let E be a log-convex compact set with connected complement in C, which contains 0 as a boundary point and whose pammeter is m. Let L(E) be a logarithmic diagram of E. The formal power series

(3.1)

represents a holomorphic function in a neighborhood of 00 which admits a single-valued analytic continuation to C\E, if and only if there is an entire function ¢ of order at most 1 which is locally of exponential type in the half-plane 1I"m and which satisfies

¢(n)=fn (n=O,I,2, ... ) (3.2)

and h(O,¢) ~ K( -0, L(E)) (exp(iO) E 1I"m) (3.3)

Theorem 3.2. Let E be a log-convex set containing the origin as an interior point and let L(E) be a logarithmic diagram of E. The power series (3.1) represents a function holomorphic in a neighborhood of 00 and admitting a single-valued analytic continuation to C\E if and only if there is an entire function ¢(z) of exponential type in the complex plane satisfying (3.2) and

h(O, ¢) ~ K( -0, L(E)) (0 E R). (3.3')

302 N. U. Arakelya.n

As a corollary of Theorem 3.2, we can prove

Theorem 3.3. The power series (1.1) with radius of convergence one can be analytically continued across the arc

'Y = {z: Izl = 1, 7r ~ largzl > u},

if and only if there is an entire function T/J( z) of exponential type satisfying (3.2) and the conditions

h(O,¢) = 0, D±h(O,¢) $ u. (3.4)

Remarks. 1. The novelty and significance of these theorems is that they give necessary and sufficient conditions.

2. Condition (3.3) is the equivalent to I C L(E)* where I is the indicator diagram of ¢ (= indicator diagram of ¢ restricted to 7r!!.).

3. In two of the theorems the "analytic element" is a power series with center 00, in the third theorem it is a power series with center z = 0. This distinction is of no significance. The transformation z -+ 1/ z changes one type of series into the other and it changes L(E) into -L(E), so that Theorems 3.1 and 3.2 are easily changed into statements about ordinary power series. For example, the choice

yields

so that

E = {z: z E .6.(-17,17), Izi ~ I}

L(E) = {w : u ~ 0, Ivl $ u},

K(O,-L(E» = { "1:01 101 < 7r/2,

otherwise.

This gives as a corollary of Theorem 3.1.

Theorem 3.4. If the power s-eries (1.1) has radius of convergence one, then it admits analytic continuation to C\.6.(-u,u) if and only if there is an entire function ¢ satisfying (3.2) and

h(O,¢) $ ulsinOI (101 < 7r/2).

4 Proofs

4.1 PROOF THAT THE CONDITIONS OF THEOREM 3.1 ARE

NECESSARY

Suppose that f(z) is holomorphic near 00 and that it has a single-valued analytic continuation to C\E.


Since 0 E 8E, L(E) is as described in 2.4(b). In particular L(E) has boundary curves

v = Vj(u) (u::5 hj j = 1,2).

Since the complement of E is connected,

It is obviously possible to find a Jordan curve r outside L(E) consisting of two curves v = V; ( u) (u ::5 hj j = 1, 2) and a rectifiable connecting piece joining (h, VI (h» to (h, V2(h» and satisfying the conditions

V2(U) < V2(U) < Vl(U) + 211'j V2(U) - 211' < Vl(U) < Vt(u)

and Vj(U) - V;(u) -+ 0 (u -+ -00).

Also we can choose r close to L(E), i.e. for each w E r

inf Iw-tl < f. tEL(E)

(4.1)

(4.2)

In view of (2.4) we can assume that the element of length of r satisfies

Idwl < Aidul (w = U + iv, U < h). (4.3)

Also, by the construction of r

Iwl<A (wEr,u~O), (4.4)

and in view of (2.4)

Iwl < -Au, + A (w E r, U < 0). (4.5)

The mapping s = exp w maps r on a Jordan arc 7. Also 7 U {O} is a closed Jordan curve C. Let D be the closed region in the extended complex plane bounded by C and containing 00. The function f(s) is holomorphic in D. By Theorem B applied to f(l/s) in 1/ D we can find an entire function

00

g(s) = L:gnsn (4.6) n=o

such that If(l/s) - g(s)1 < exp(_lsI1/ 3 ) (s E l/D).

Therefore F(s) = f(s) - g(l/s)

satisfies IF(s)1 < exp(_lsl-1/ 3 ) (s ED). (4.7)

304 N.U. Arakelyan

Consider the function

X(z) = (1/211"i) 1r F(eW)eWZdw

= (1/211"i) i F(s)sZ-lds (s = eW).

We prove first that X is an entire function. By (4.7) (with s = expw) and (4.4),

(4.8)

(4.8')

IF(expw)expwzl<explwzl<expAlzl (wEr, O~u). (4.9)

By (4.7) and (4.5) for wE r, u < 0,

IF(expw)expwzl < Aexp(-exp{-u/3}) exp(-Aulzl). (4.10)

We split the integral in (4.8) into the integrals over

r 1 = r n {w : u ~ O} and r 2 = r n {w : u < O}.

We note that r is rectifiable and that (4.3) holds on r. The estimates (4.9) and (4.10) now quickly show that

1~(z)1 < Aexp(Alzl) + A 100 exp( _t)tA1z1dt,

where we have put exp( -u/3) = t. Evaluating the last integral we have finally

1~(z)1 < Aexp(Alzl) + Ar(Alzl) < Aexp(Alzllog Izl). (4.11)

For any z in the complex plane the estimates (4.9) and (4.10) can easily be replaced by estimates which hold· uniformly in some neighborhood of z, it suffices to replace Izl by a slightly larger number. This remark allows us to recognize that ~ is a holomorphic function of z, by applying the test that

H(z) = [G(w, z)dw (G(w, z) E C(r x D»

is a holomorphic function of z in the domain D, if G( w, z) is a holomorphic function of z in D for every w E r and if IG(w,z)1 < K(w) for all z in D and IrK(w)ldwl < 00.

We have shown that ~ is an entire function of z satisfying (4.11). We show next that the indicator diagram of X satisfies

h(~,X) < K(-8,L(E».

By (4.2) on r

lexpwzl = expRewz < Aexp [ sup Retz+ fIZI]. fEL(E)

. (4.12)


Or, if z = rexp(iO),

I exp wzl < exp[K( -0, L(E»r + fr] (w E r).

Therefore, using (4.8) with 8 = expw,

Ix(z)1 < exp[K( -0, L(E) + f)r] [ exp( - exp[_lwl-1/3])ldwl·

This proves (4.12), since f is arbitrarily small and the integral is finite. The shape of L( E) implies that

K(-O,L(E» < 00 (expiO E 7r!),

where 7r! is the interior of the half-plane 7r m, and otherwise

N ext we show that

x(') = {

K( -0, L(E» = 00.

In -gn

o (n = 0,1,2, ... )

(n = -1, -2, ... ).

(4.13)

(4.14)

(4.15)

We use the formula (4.8'). Let "I' be a closed contour in D with winding number 1 with respect to 8 = 0 which is obtained from "I by replacing the part of "I inside lsi < f by a suitable curve in this disk. By (4.7), the integral in (4.8') changes by arbitrarily little, if "I is replaced by "I', provided f is small enough. By Cauchy's Theorem the integration along "I' can be replaced by an integration along the circle lsi = R, on which I(s) and g(l/s) are given by their power series. This proves that for z = n the integral along "I' has the value given by the right hand side of (4.15) and (4.15) follows.

By Lemma 1 we can find an entire function 'I/J(z) of exponential type at most 7r such that

satisfies (3.2) and

Because

¢(z) = X(z) + 'I/J(z)

h(O,X) ='-00 (expiO E 7r!).

log la + bl < max (log lal,log Ibl) + log 2,

h(O,4J) ~ max(h(O, X),h(O, 'I/J» and (3.3) follows from (4.13) and (4.16). This completes the proof.

(4.16)

306 N.U. Arakelyan

4.2 PROOF OF THE NECESSITY OF THE CONDITIONS OF

THEOREM 3.2

The proof is along the same lines as the proof given in the previous paragraph, but the details are simpler and we shall only sketch them.

We assume that fez) is holomorphic near 00 and that it has a singlevalued analytic continuation to C\E.

Assume for the moment that E is not a disk lsi ~ Isol. The set L(E) is described in 2.4( c) and it is easy to see that there is a curve r joining the points (uo, Vl(UO» and (uo, V2(UO» lying in the half-plane u ~ Uo and outside L(E) which satisfies (4.2) for every w E r. In addition we may assume that s = exp w maps r onto a closed, rectifiable Jordan curve C with winding number 1 around the origin. In the case that E is the disk {lsi ~ Ison, we may take for r any curve which lies in lsi> Isol except for the point s E r and which has winding number 1 with respect to the origin. Let So = exp(uo + iVl(UO» and write 'Y for C\{so}. The exterior D of C and the curve'Y are outside E. The map

t = 1/(s - so)

maps DU'Y on an unbounded closed region T bounded by the Jordan curve t('Y) U {oo} in C where C = C U {oo}. The function /(so + (l/t» [= /(s)J is a holomorphic function of t in T\ { oo}. By Theorem A we can find an entire function get) such that

I/(s + (l/t» - g(t)1 < exp( -ltll/3) (t E T).

Or F(s) = /(s) - g(I/(s - so»

satisfies IF(s)1 < exp[-Is - sol-1/3] (s E D\{oo}).

For lsi> Isol, 00

g(I/(s - so» = L gns-n. n=O

We put

X(z) = (1/211") 1 F(s)sZ-lds,

where SZ is defined as exp wz, we r. Since I log sl is bounded on 'Y,

IF(s)sZ-ll <'Aexp(Alzl)

on 'Y and X is an entire function of exponential type.

(4.17)

By repeating the reasoning used in 4.1 with obvious, minor modifications we see that (3.2) and (4.12) imply (4.15).


As in 4.1, (4.2) implies

h(fI, x) $ K(-fl,L(E». (4.18)

By the corollary of Theorem C we can find an entire function "iIi of order zero such that the coefficients in (4.12) are given by

gn = t/I(n),

where t/I(z) = exp(woz)'\l'(z).

Then h(fI,t/I) = Re(woexpifl).

Since Wo E L(E), we have by the definition of K

Re(woexpifl) $ sup Re(wexpifl) = K(fI,L(E». wEL(E)

We conclude, as in Section 4.1, that

q,=x+t/I

satisfies h(fI,q,) $ max[h(fI, X), h(fI, t/I)] = K(-fl,L(E».

This proves (3.3) and (3.2).

4.3 SUFFICIENCY OF THE CONDITIONS OF THEOREMS 3.2 AND 3.1

We shall actually prove a slightly stronger statement: The conclusions of Theorem 3.2 follow from the following hypotheses:

(a) The function q, is holomorphic and of locally finite exponential type in the interior 1I"!!a of 11" m.

(b) (3.2) is satisfied for n ~ 1. (c) (3.3) is satisfied. Let k( w) be the function

00

k(w) = Lq,(n)exp(-nw). n=l

By (3.3') with fI ::: 0,

lim sup log Iq,(n) lin $ K(O, L(E». n-oo

(4.19)

308 N.U. Arakelyan

By the discussion in 2.4(c), there is a point Wo E L(E) at which sup Rew wEL(E)

is attained and K(O, L(E» = Re Wo = Uo < 00.

[We note in passing that the case L(E) = 0, Wo = -00 remains meaningful. In this case Theorem 3.1 becomes Lemma 1.]

Therefore k( w) is a holomorphic function in

Rew> Uo.

It is not hard to see that the substitutions

W = w - Woj tPl(Z) = tP(z)exp(-Woz)

reduce the case Wo ::f. 0 to the case

Wo=O

with L(E) replaced by L(E) - Woo From now on we shall assume that (4.20) holds. Since

k(w + 2'lri) = k(w),

(4.20)

the analytic continuation of k(w) from the right half-plane to the point Wl + 2n'lri (n an integer) is possible, if it is possible to the point Wl' Let

M = U {L(E) + 2n'lri}. nEZ

We assert that k(w) can be analytically continued from u > 0 to any point Wl in u < 0 which lies in the complement of M. Out proof will be based on an application of the residue theorem to the integral of

g(s,w) = g(s) = tP(s)exp(-ws)j[exp(2'lris) -1] (w E 'lr!) (4.21)

around a suitable contour. Before we can give the details, we need a geometrical consideration.

By the discussion in Section 2 there is a uniquely determined a E [0,2'lr] such that we may assume that

L(E) C {w : -a ~ v - mu ~ 2'lr - a}.

Without loss of generality we may assume that Wl lies in

{w : -a ~ v - mu ~ 2'lr - aj u ~ O}

and since domains of holomorphy are open, we may further restrict ourselves to the case

Wl E {w: -a < v - mu < 2'lr - aj u < OJ.


Let I(a, t) = {w : w = t + rexp ia, 0 < r < oo}

and let m = tan.8; 1.81 < '11"/2.

Then ei (/ E 'II"!!" if and only if 19 -.81 < '11"/2. Let W1 = U1 + iV1 and suppose that

The vertical line u = U1 intersects L(E) in an interval with endpoints (U1, V1 ) and (U1, V2 ). Since (U1,mu1) E L(E),

The line 1(.8+'11"1, u1+iVd contains inner points of L(E) (because -a < VJ.). It follows that any support line of L(E) through U1 +iV1 must have a slope tan(.8 + T),

o < T, 1.8 + TI < '11"/2.

Let A be such a support line. The point W1 is below A and we can find fJ > 0 so that one of the parallels, A', to A and at distance fJ from A separates the point U1 + iVI from a neighborhood of W1. Let H be the closed halfplane bounded by A' which contains WI. Notice that H is at a distance ~ fJ from L(E). Let W2 = W1 + 2'11"i. Notice that W lies above L(E), i.e. V2 - mU2 ~ O. By repeating our reasoning, we see that there is a half-plane H' containing W2 and at a distance ~ 6 from L(E) which is bounded by a line of slope tan(.8 - T'),

T' ~ 0, 1.8 + T'l < ('11"/2).

Therefore the boundary lines of H and H' are not parallel, they intersect in a point t. The point t is the vertex of the sector H n H' and of its opposite sector which contains L(E). Any line II through t which does not meet these two sectors lies in H U H' and therefore has a distance ~ fJ from L(E). If exp(/d) gives the direction of the perpendicular from the origin to II, this implies that II is at a distance ~ K (IC, L(E»+6 from the origin and a fortiori for any point of the half-plane H(IC) bounded by II and containing HnH'

Rew exp( -lCi) ~ K(IC,L(E» + 6 (w E H n H').

Taking account of the position of the origin we find that IC can take all values in

{3 + T - ('11"/2) ~ IC ~ {3 + T' + ('11"/2).

Now we are ready to estimate the function (4.21).

310 N.U. Arakelyan

If s = rexp(8i) E 1I'!!a, then, by (3.3) and a well known uniformity theorem about the indicator function [20, Ch. I, Theorem 28]

Iq,(s)l:5 exp{[K(-8,L(E» + £(r)]r} , (4.22)

where £(r) -+ 00 as r -+ 00 and

{J - (11'/2):5 a:5 8 < (J < a+ (11'/2). (4.23)

If wE H(-8), s = rexp(8i), then

I exp(sw) I ~ expRe[rwexp(8i)] ~ exp[(K(-8,L(E) +6)r]. (4.24)

The condition wE H(-8) will be satisfied for all 8 satisfying (4.23), if Iwl >A and

18+ argwl < 11'/2. (4.25)

Let S( N) be the sector

1/2:5 lsi :5 N + (1/2), I argsl :5 7],

C(N) its boundary described in the .positive sense. If (4.25) is satisfied, then, for sufficiently small 7], (4.22) and (4.24) are satisfied. Also on C(N),

11/[exp(211'Bi) - 1]1 < A(7]).

Therefore, on C(N)

Ig(s)1 < Aexp{-Rer[6 - £(r)]).

On the other hand (1/211'i) times the integral of g(s) around C(N) is equal to the sum of the residues of g(s) at the poles in SeN). Letting N -+ 00, we have

k(w) = (1/211'i) L g(s, w)ds

In

Iwl > A, largwl < (11'/2) - 27].

The contour C consists of the lines

args = ±7]

and the circular arc lsi = 1/2, largsl < 7].

(4.26)

We may suppose that 7] is so small that (4.25) is satisfied for arg w = (J+T and 191 :5 7]. By our construction a neighborhood of 1({J + T, WI) is in H({J + T - (11'/2». For all points of this neighborhood the integral (4.26) converges and the absolute value of the integrand is uniformly bounded by an integrable function. This shows that k(w) can be analytically continued to WI along 1({J + T, WI).

The proof in the case that VI - mtAl > 0 follows the same lines.


4.4 PROOF OF THEOREM 3.3

(a) The conditions of the theorem are necessary. By the conditions of the theorem we can find a simply connected open set containing the unit disk and the arc {z : z = expi9, u < 9 < 2'11" - u} in which fez) is holomorphic. We may assume that this set is symmetrical with respect to the real axis and that its complement is a log-convex set E with diagram L(E). The set -L(E) lies in u ~ 0, it intersects the imaginary axis in the interval with endpoints -iu and iu and it is confined to the strip Ivl ~ '11". By Theorem 3.2 there is an entire function of exponential type satisfying

h(9, t/J) ~ K( -9, -L(E» (9 real). (4.27)

Putting 9 = 0 in (4.27) gives h(O,t/J) ~ O. Since (1.1) has radius of convergence equal to 1, we must have

h(O,t/J) = O. (4.28)

For e > 0 there are support lines of -L(E) through ±i(u + e) making the angle ±6, 6 ~ 0 with the imaginary axis. - L( E) is contained in the sector S bounded by these support lines and the imaginary axis. Therefore

K(-9,-L(E» ~ K(-9,S) = (u+ e) sin 191 (191 ~ 6).

Conditions (3.4) follow from this and (4.28) by letting e -- O.

(b) Proof of the sufficiency of the conditions. Now we assume the existence of an entire function t/J of exponential type with the properties described in the theorem. We must prove that the function E:=l t/J(n) exp( -nw) can be continued analytically to points u+iv with u < 0 along v = constant, provided that this constant is congruent to a number in (u,2'11"-u) modulo 2'11". By the Remarks 2 and 3 after Theorem 3.3 and the sufficiency part of Theorem 3.2, this will be the case, if the indicator diagram 1 of t/J has the following property:

(a) The convex set -1* contains the origin, but it does not contain points Vi ofthe imaginary axis with IVI > u.

Since h(O, t/J) = 0 and since the set E contains points s of arbitrarily large absolute value, -1* contains the origin and it lies in the half-plane u ~ O. Suppose that Vi with V> u belongs to -1. Then

K(9,-r) ~ K(9, J) = Vsin9 (0 < 9 < '11"/2).

But this contradicts (3.4). For negative V the proof is the same. This proves (a) and completes the proof of the theorem.

312 N.U. Arakelyan

References [1] Agmon, S., On the singularities of Taylor series with reciprocal coef

ficients. Pac. J. Math. 2(1952), 431-453.

[2] Arakelyan, N.U., Approximation complexe et proprietfs des fonctions analytiques, In: Proc. Internat. Congr. Math. (Nice, 1970), Vol. 2, Gauthier-Villars, Paris, 1971, 595-600.

[3] Arakelyan, N.U., On efficient analytic continuation of power series, Math. USSR Sb., 52(1985), No.1, 21-39.

[4] Arakelyan, N.U., Gauthier, P.M., On tangential approximation by holomorphic function, Izv. Acad. Nauk Arm. SSR, Mat. 17, No.6, (1982), 421-441.

[5] Arakelyan, N.U., Martirosyan, V.A., The localization of the singularities of power series on the boundary of the circle of convergence. Izv. Acad. Nauk Arm. SSR, Mat. 22, No.1, (1987), 3-21.

[6] Arakelyan, N. U., Martirosyan, V .A., The localization of the singularities of power series on the boundary of the circle of convergence. II. Izv. Acad. Nauk Arm. SSR, Mat. 23, No.2, (1988), 123-137.

[7] Bieberbach, L., Analytische Fortsetzung, Springer-Verlag, Heidelberg, 1955.

[8] Carleman, T., Sur un theoreme de Weierstrass, Ark. Mat. Astr. Fis. 20 B, (1927), No.4, 1-5.

[9] Carlson, F., Sur une Classe de Series de Taylor, Diss., Upsala, 1914.

[10] Carlson, F., fiber ganzwertige ganze Functionen, Math. Z. 11(1921), 1-23.

[11] Cowling, V.F., A generalization of a theorem of LeRoy and Lindelof, Bull. Amer. Math. Soc. 52(1946), 1065-1082.

[12] Dienes, P., The Taylor Series, Clarendon Press, Oxford, 1931.

[13] Dufresnoy, J., Pisot, Ch., Prolongement analytique de la serie de Taylor, Ann. Sci. Ecole Norm. Super., Ser. 3, (1951), 68, 105-124.

[14] Gawronski, W., Trauthner, R., Analytische Forsetzung von Potenzreihen, Serdica, Bulgar. Math. Publ., 2, No.4, (1976), 369-374.

[15] Hadamard, J., Essai sur I'etude des fonctions donnees par leur developpement de Taylor, J. Math. Pures Appl. (4),8(1982), 101-186.

[16] Hadamard, J., La Serie de Taylor et son Prolongement Analytique, ColI. Scientia, Carre et Naud, Paris, 1901, No. 12, 102s.


[17] Keldish, M.V., Sur I'approximation des fonctions holomorphes fonctions entires, C.R. (Dokl.) Acad. Sci. USSR 47(1945), 239-241.

[18] Leau, L., Recherches sur les singularities d 'une fonction definie par un deve/oppement de Taylor, J. Math. Pures Appl. (5), 5(1899), 365-425.

[19] LeRoy, E., Sur les series divergentes et les fonctions definies par un deve/oppement de Taylor, Ann de la faculte des sci. de Toulouse (2), 2(1900),317-430.

[20] Levin, B. Ya., Distribution of Zeros of Entire Functions, GITTL, Moscow, 1956; English transl. in Amer. Math. Soc., Providence, R.I., 1964.

[21] Lindelof, E., Le Calcul des Residus et ses Applications a la Theorie des Fonctions, Paris, Gauthier-Villars, 1905.

[22] Mergelyan, S.N., Uniform approximations of functions of a complex variable, Uspekhi Mat. Nauk 7(1952), No.2 (48), 31-122; English transl. in Amer. Math. Soc. Transl. (1) 3(1962), 294-391.

[23] P6lya, G., Untersuchungen tiber Ltinken und Singularitiiten von Potenzreihen, Math. Z., (1929), 29, 549-640.

[24] P6lya, G., Szego, G., Aufgaben und Lehrsat aus der Analysis, B. 1, 2. Berlin, Springer-Verlag, 1925.

[25] Wigert, S., Sur les fonctions entieres, Oefversigt af svenska Vetenskaps. Forhandl., 57(1900), 1007-1011.

N.U. Arakelyan Institute of Mathematics Armenian Academy of Sciences Marshal Bagramian Ave. 24-B 375019 - Yerevan ARMENIA

Quasi-Orthogonal Hilbert Space Decompositions and Estimates of Univalent Functions. II

N .K. Nikolskii V.I. Vasyunin

1 Introduction

This paper is the second part of a report on an investigation of vecto

rial Cauchy-Bunyakowskii-Schwarz (CBS) inequality and its applications

to estimates of Taylor coefficients of univalent functions. The first part is

published in [13] and contains a description of the main general ideas of

our approach: CBS inequality for operator measures, quasi-orthogonal (co

isometric) decompositions with respect to complementary metrics, multiplicative averaging of solutions of general evolution equations. The detailed

exposition of the theory is contained in [17].

Here we describe in brief the contents of paper [18] devoted to coefficient

estimates of univalent functions on the unit disc D = {z E C : Izl < I}. The main goal of [18] is to propose an explanation of de Branges' proof of

the Bieberbach conjecture [3] from the operator theory point of view and

show a joint source of this and some other estimates. Several of the next

sections play the role of an introduction to the subject.

2 A Glimpse of the Coefficient Problem

We consider the well-known class S of univalent function (i.e. one-to-one

and analytic),

f is univalent in D, f(z) = z + I: f(n)zn} , n~2


316 N.K. Nikolskii, V.1. Vasyunin

where i(n) stands for Taylor coefficient of I. The general coefficient prob

lem consists· of describing the n-dimensional body filled in by Taylor coef

ficients of S-functions:

((i(2), ... , i(n + 1» : 1 E S}.

The problem has remained open since the beginning of the century and, as

a substitute many, (in fact, an enormous amount of) believers in geometric

function theory deal with some modulus estimates of coefficients or combi

nations of coefficients of S-functions. Several profound theories were con

structed to support such estimates, and many brilliant monographs were

written in the field. Apologizing to the experts, we mention only one of

them, namely [15].

One of the main touchstones of the field was the famous Bieberbach

conjecture which later was turned into L. de Branges' theorem. The con

jecture was raised by L. Bieberbach in 1916 and consisted of the following

inequality:

li(n)l~n , IES, n=2,3, ... ; (2.1)

it was also conjectured that an equality holds iff 1 coincides with one of

the so-called Koebe functions z/(I- (z)2, 1(1 = 1. During the conjecture's

first fifty-six years, the first five coefficients (for n = 2,3,4,5,6) were suc

cessfully attacked by Bieberbach, Loewner, Garabedian, Schiffer, Pederson

and Ozawa, taking on the average more than 10 years for each. In 1984 de

Branges completely beat the conjecture [3], [4]. The reader can find various

information about the intriguing story of the proof and its verification in

[2], [6], [8]; in a series of popular expository papers [1], [9], [10], [12], [19];

and even in the mass media [11].

de Branges' proof is based on three important standings of geometric

function theory:

-Loewner's brilliant idea to include a given univalent function 1 into

a flow {/th>o of univalent functions such that 10 = f,/t(z) = etz + ... , oftiOt = Pt· zO/t/oz with analytic Pt ,RePt(z) > 0 (Loewner equation);

-Robertson's hilbertizaton of the problem: the following inequality

(Robertson conjecture) implies (2.1),

n

L Ig(2k + 1)12 ~ n for odd functions 9 E S; (2.2) 1:=1

Quasi-Orthogonal Hilbert Space Decompositions 317

-Lebedev's and Milin's exponentiation (inequality) which shows that the following estimate for logarithmic coefficients (Milin conjecture) implies

(2.2),

t Ih(k)12. k· (n - k + 1) $ 4 t n -! + I, (2.3) k=l k=l

where h(z) = log(J(z)jz), f E S.

3 de Branges' Breakthrough

Formally, the main idea of the proof [4] is to consider inequality (2.3) as an

initial point of a one-parameter family of inequalities in weighted Dirichlet

spaces Q(u(t)) , (3.1)

k~l

and using a Loewner family {lth~o follow the corresponding estimates along with the How. The main technical trick is to subject Uk to a very special system of differential equations,

, , Uk uk+l T+ k+1 +uk- uHl=O, l$k$n (uk=O,k>n) (3.2)

and use it together with the Loewner equation for {It h~o. The proof finishes after a series of somewhat mysterious computations and making use of an inequality of Askey and Gasper for Jacobi polynomials.

4 An Explanation Through CBS Inequality

In fact, throughout several of de Branges' papers some hints are scattered that point to the operator theoretical nature of the proof, see [3], [6], [8]. Following these tracks we discovered a theory published in [13], [17]. As

an application of the theory we now propose a new approach to coefficient

estimates which gives, in particular, the results of [4], [5], [16]. Let us start

with a list of questions arising naturally from the proof outlined in Section

3. The questions are supplied with answers resulting from our approach

and which are satisfactory from the operator theory point of view. For the

reader's convenience we reproduce a fragment of our theory in the next

section.

318 N.K Nikolskii, V.I. Vasyunin

Q A 4.1 What is the nature of de Branges' Inequality (2.3) is a (very) special inequality (2.3) and how can new ones case of the CBS inequality for an be produced? operator measure related to the

Littlewood subordination principle. Changing the measure we produce new inequalities.

4.2 Asking for an estimate of. a To calculate and handle vector-given univalent function, why valued integrals involved in should we consider Loewner Bows? CBS inequalities. The fact is

that general evolution equations are an inevitable tool to find the Radon-Nikodym derivative of an operator measure; for composition operators these equations become the Loewner differential equation.

4.3 What is the meaning of de Branges' The system is a coordinate form system of equations (3.2)? of a vectorial differential equation

for so-called isometric trajectories in the weighted Dirichlet space.

4.4 Why does the proof so essentially Because the Radon-Nikodym depend on the properties of some derivative mentioned in Answer 4.2 special functions? turns out to be a Jacobi matrix

(depending on a parameter), and so its positiveness can be treated by using orthogona,J. polynomials, etc.

5 Short Course of Quasi-Orthogonal Decompositions

Referring to [13], [17] for all details let us restrict ourselves to a very brief recalling of the main notions related to the CBS inequality and quasiorthogonal decompositions.

5.1 GENERAL CBS INEQUALITY

The general CBS inequality is concerned with a Hilbert space operator

valued measure E defined on a measurable space X and such that 0$

E(6) $ I for every 6, 6 c X. Denoting by H the main Hilbert space,

let :F be the set of all H-valued (measurable) step functions on X. For


IE:F, I = E~=l X6iZi, Zi E H, let us define

Then

IIE(X)-1/2 Ix dE/1I2 $ Ix (dEI, I) . (5.1.1)

for every I, I E :F, and moreover both integrals can be defined for I from the completion :F of:F with respect to the scalar product given by the right

hand side of (5.1.1). The operator W,

WI= L dEl

can be continued to :F and acts as a co-isometry from:F onto Range E(X)1/2

(Range T stands for T H endowed with the range-norm IlziiT = min{ Ilyll : Ty = z}j co-isometry means that the adjoint W· is an isometry).

In order to apply CBS inequality (5.1.1) it is important to describe the

completion :F, and the only way we know to do this is to assume the existence of the Radon-Nikodym derivative of E with respect to a scalar

measure. Let us realize this plan for a measure on the real line R or, equivalently, for the restriction of a given measure to a u-subalgebra generated

by a totally ordered collection of sets.

5.2 STIELTJES MEASURES AND EVOLUTION EQUATIONS

The computation and handling of the Radon-Nikodym derivative of a mea

sure on R depends on a multiplicative structure of its indefinite integral. Omitting details let us consider a measure on an interval (a, b] C R defined

by a right continuous family of operators T"

E«r,sD = TrT,:' - T,T,·, a $ r$ s $ bj To = I.

Then there exist contractions Tr• such that T, = TrTr, and (with a slight

additional assumption) Til == I. These contractions form an evolution fam

ily because they satisfy the evolution identity Trt = Tr,T6f , r $ s $ tor,

equivalently (under a differentiable condition), the evolution equation

()Tr, I"\() 1"\( ) del {)Tr, I -a = u r Tr" u r = -a . r r .=r

(5.2.1)

320 N.K. Nikolskji, V.1. Vasyunin

In this notation, T. == T<I. and for every r E [a, b],

Before using these computations for specifying the CBS inequality, we gen

eralize it by introducing a weight.

5.3 WEIGHTED QUASI-DECOMPOSITIONS

One can consider the same evolution family Tr , but in the scale of Hilbert spaces:

Tr• : H(s) -+ H(r), a:$ r:$ s :$ b.

Let us endow H(s) with a scalar product ofthe form (z,y), = (u(s)z,y).

The scaled Tr• are contractions iff A( s) ~O where

A(s) = u'(s) + 2Re(u(s)0(s», a:$ s :$ b.

Finally, after a Hermitian factorization of the weighted Radon-Nikodym

derivative u-1Au-1 = rr" (we assume the invertibility of u(s) for the

sake of simplicity) one can transform the general CBS inequality into the

following proposition: the operator

9 1--+ 16 T<I.tg(s)ds (5.3.1)

is a co-isometry from the space defined by the norm u: IIg(s)II~(,)ds)1/2 onto the complementary space 1£<111 of Range T<l6 in the space H(a) (by

definition, the complementary space of Range T is Range ~*, DT* = (I _ TT*)1/2).

In other words, we always have

(5.3.2)

and an arbitrary vector z from 1£<16 can be represented in the form z = f: T<I,g(s)ds , and there exists a unique such representation for which the

inequality becomes equality.


5.4 REFORMULATION FOR EVOLUTION EQUATIONS

The last proposition can be restated in the following way. Let a vector

function z obey the equation

z' =Oz-g

with an operator function 0 generating an evolution family Tr6 j if a weight

function u satisfies the aforementioned condition A(s) ~O, a::::; s ::::; b , we

have

The equalities hold iff

(uz)' + O*uz == 0 on [a,b].

5.5 MULTIPLICATIVE AVERAGING

In fact, for applications we need some modification of the proposition stated

in Section 5.4. We get this modification using a process called multiplicative (or chronological) averaging of solutions of evolution equations. It is

important to note in advance that this trick is aimed at estimating evolu

tion operators themselves (i.e. the fundamental solution of the equation)

rather than vectorial solutions corresponding to specific inhomogeneous

parts. Now, we describe the method in a special setting needed for what follows.

Let us consider an operator-valued function 0 0 , 2ReOo(s) ~ 0,

a ::::; s ::::; b, a weight function u, a vector-valued function Zo and, finally,

a collection {u(') : , E Z} of unitary operators indexed by a measurable space Z. They are subject to the following relations:

u(s)u«) = u«)u(s) for a ~ s ~ b, (E Zj

(uzo)' + Oouzo == 0,

Ao ~f U' + 2Re(uOo) ~ 0 on [a,b].

Let 0 be a generator of an evolution Tr• such that

O(s) E conv(u«)Oo(s)u«)*: (E Z),

(5.5.1)

(5.5.2)

(5.5.3)

322 N.K. Nikolskii, V.I. Vasyunin

(closed convex hull) and hence is represented in the form

O(s) = L u«()Oo(s)u«()*dl'.«()

with a family of probability measures 1'., a:5 s :5 b. Now if we put

g(s) = l u«()go(s)dl'.«(),

where go = Oozo-z~, we can claim that for every solution z of the equation z' = Oz - 9 the following inequalities hold

2

IIz(a)lI~ -llz(b)II~:5 1b To.g(s)ds :5 IIzo(a) II; -lIzo(b)II~. (5.5.4) ?tob

If 1'. = 6(, a:5 s :5 b (i.e. if z == u«()zo) the inequalities become equalities. A few remarks are needed here. In applications (see Section 9 below) we

have Zo and 0 0 fixed in advance and so equations (5.5.2) and (5.5.1) and an initial value O'(a) will be considered as sources to find 0'. Then condition (5.5.3) has to be checked and after that we may be sure of the validity of inequalities (5.5.4).

6 Back to Univalent Functions

The Littlewood subordination principle consists (in particular) of the inequality

for every univalent mapping B : D - D with fixed point zero (B(O) = 0)

and for every norm 1I/1I~~(w .. ) = Ln~11/(n)12Wn satisfying wn/n !. In particular, it is true for the Dirichlet space g,

g:= {f: I(z) = E/(n)zn, 11/113 = E 1/(n)12n < oo}. n~l n~l

Moreover, the same is true for the indefinite Dirichlet space Ig,

(6.1)

where Ig consists of all Laurent series 1 = Lnez /(n)zn with

Ln>o nl/(n)12 < 00 and with a finite set of non-zero negatively indexed


coefficients j(n); the indefinite norm is defined by the equality 11/111" = EnEZ nli<n)12 •

In fact, inequality (6.1) characterizes univalent functions: if a function B is holomorphic in D, B(O) = 0 and if (6.1) holds, then B is univalent. This proposition is proved by L. de Branges [6] but ascended to the well-known

descriptions of univalent functions in terms of quadratic forms (Grunsky,

Golusin, Nehari; see [15]).

7 Operator Measures Generated by Subordination Principle

Let us try to combine the subordination principle with the techniques of

quasi-decompositions outlined in Section 5. To this aim we consider a to

tally ordered chain {C.} of composition operators on the Dirichlet space

g. Let C.I = loA., s ~ 0 where A. stands for a univalent function,

A,,(D) C D normalized by the conditions A,,(O) = 0, (8A"j8z)lz=o = e-·. Let us assume C:C" ::5 C;Cr for r ::5 s which are equivalent to the inclu

sions Ar(D) :::> A,,(D). Then an evolution family c;. is well-defined by the

equalities C: = C;C;" , r ::5 s. In order to deal again with composition

operators let us consider the dual evolution family .: fixing a positive real

b we put

(7.1)

Now, following Section 5, let us consider quasi-decompositions generated

by the evolution families {Tr • }o~r~.~b on the Dirichlet space g. Elementary

computations show that the corresponding generator 0 has the form

d O(s) =q,(s,.) zdz' s~O, (7.2)

where q,( s, .) are analytic in D and

Re q,(s, z) ~ 0 for Izl < 1; q,(s,O) == 1. (7.3)

·It is easy to see that the oper~tors Cr. do not form an evolution family.


Evolution equation (5.2.1) reduces to

oBr6 _..I.( .) oBr6 or - "I' r, z oz' or

and is said to be the Loewner equation.

8 Loewner Quasi-Decompositions

(7.4)

In order to get an explicit example of general decomposition (5.3.1) -

(5.3.2), let us write down its unweighted version generated by the evolution

Tr3 of Section 7. To this aim, fix an interval (a, b), 0 ~ a ~ b and denote

by g(B) the complementary space of Range Tab in the space g, B ~f Bab.

Claim: the space g(B) consists of all functions 1 of the form

1 = 1b l(s,Ba.)Ba.ds, (8.1)

where I(s, .) is analytic in the unit disc and such that the right-hand side

of the following inequality

II/II~(B) ~ ~ 1b IITie1j(:,./(s, ·)II:~ ds (8.2)

is finite. Here H2 stands for the Hardy space H2 := {f : 1 = 2:n>O J(n)zn, 11/112 = 2:n>o lj(n)j2} and T.p for the Toeplitz operator with symbol ,p : T.pl = PH2~/, 1 E H2. Every 1 admits a unique representation which

turns the inequality into an equality.

Let us remark that the above decomposition is also presented in [7] but

in a different form; the proof in [7] differs from ours. It is also interesting

to note that inequality (8.2) can be rewritten as a series of inequalities in

the usual g-norm if we take into account a formula due to Aronszajn and

de Branges for the complementary norm:

IIxll~T. = sup{llx + Tyl12 -llyl12 : y E H},

see [7], [14], [17] for details.

9 Applying Multiplicative Averaging

The aim of the paper is to obtain estimates for univalent functions, hence

for Loewner chains (7.4) or equivalently for evolution families (7.1). On


the other hand, we obtained a large collection of inequalities for functions

ofthe Dirichlet space (see (8.2» representing them through evolutions Tr•

by means of formulas (8.1), (5.3.1). In fact, we are not able to extract all information on Br• from inequalities (8.2) as well as all information on general evolutions Tr• from (5.3.1). Instead, we pick out a part of inequalities (8.2) to make transparent some estimates for special expressions in Br ,

like 10g(Br./z), (Br./z)" and (some day) for some others. We do this by

using the process of multiplicative averaging outlined in Section 5.5, and so

the matter is about a special kind of control theory: some information on

evolutions Tn can be discovered by estimating the solution of the equation

z' = Oz - 9 for a special control function g.

It is easy to see how to adapt the constructions of Section 5.5 for Loewner

evolutions (7.1) - (7.4). Indeed, the estimates we need (i.e. inequalities

(2.3» are rotation invariant as is the case for the class of generators (7.2)

- (7.3) and evolutions (7.1), (7.4). So, it is very natural to try the circle

group as {u( () : ( E Z} from Section 5.5:

u«()f = f«(z), (E T ~r {e E C : lei = I}. (9.1)

Further, by the well-known Herglotz theorem one can represent an arbitrary

Loewner generator (7.2) - (7.3) by an integral

O(s) = l u«()Oo(s)u«()*dlt.«(), s ~ 0,

where the probability measure It. is taken from the representation

and 0 0 is independent of s :

l+z d Oo(s) == -1- z -d • -z z

(9.2)

Now according to Section 5.5 we have to fix a basic "isometric trajectory"

zoo Let it also be independent of s : zo(s) == zoo

Finally, assuming conditions (5.5.1) - (5.5.3) to be fulfilled (which will

be the subject of forthcoming sections) we can claim that

lIy(a, b) II! ~ L nlzo(n)12(un (a) - un(b», (9.3) n~l


where

yea, b) = 16l ~ ~ ~!:: Baa' (x~«Baa)djja«)ds (9.4)

and Br• stands for the solution of (7.4) and 11·lla for er( a)-weighted Dirichlet

norm: er(a) = diag{ern(a)}n~1' Ilxll~ = En>1 Iz(n)12 . n· ern (a). Inequality (9.3) turns into an equality for y(a,b) = x~«z) - xO«B~6(z» where B;a is defined by the equation

B;, er-, z

(1 + (B$.)2 = (1 + (z)2'

10 Isometric Trajectories Cause de Branges' Equations

(9.5)

Commutation relations (5.5.1) in circle group case (9.1) mean that operators er(s) are diagonal with respect to the power basis {Zn}n~1 : er(s) = diag{ern{s)}n>1' Since the function Xo is assumed to be constant, equation

(5.5.2) defines the functions ern(s)zo(n), n ~ 1, whenever the initial data

ern(a) are chosen. Now, let us assume zo{n) ::f; 0, n ~ 1. Then the diagonal functions ern are completely defined by the values ern{a), and hence

there exists a unique (diagonal) weight function er which turns the given

(constant) vector function Xo into an "isometric trajectory":

where Brt is equal to B~t from (9.5) and other notations are taken from

Section 5.3-5.4. Similarly, all rotations xo{(z) are "isometric trajectories".

It is of interest to write down equation (5.5.2) in the coordinate form:

er~zo{n)-er~+1zo(n+1)+nernzo(n)+(n+1)ern+1Zo(n+1) = 0, n ~ 1. (10.1)

These equations can be called generalized de Branges' equations because

they turn into (3.2) for a special choice of Xo, namely for Xo = -210g(1+z).


11 de Branges' Inequality

de Branges' inequality follows from (9.3) with the last choice of Zo. Namely,

(9.4) gives

(11.1)

and then (9.3) coincides with the de Branges' inequality for logarithmic

coefficients [4]. Taking a = 0, b -+ +00,0'A:(0) = n - k + 1 for 1 ~ k ~

n,O'A: = 0 for k > n and taking into account the homogeneity of (11.1) with

respect to B = Bab we can conclude (just repeating reasonings from [4])

that inequality (2.3) holds for every f from S. The inequality becomes an

equality for limb_oo(xo(z)-zo(Bob)) = log(z/(l + z)2) and for its rotations

log(z/(l + (z)2) (i.e. for f being equal to Koebe functions).

12 Positiveness of a Jacobi Matrix

Inequality (9.3) holds provided Ao ~ 0, Ao = 0" +2 Re O'no. The quadratic

form of Ao is defined by a Jacobi matrix J,

J = 0 J1.3 A3 J1.4 (12.1)

o

with J1.n = -O'~/n, An = O'~/n + O'~+1/(n + 1) + 2(O'n - O'n+l), n ~ 1. Several general rules are known to find the spectrum of a Jacobi matrix

(e.g. in term of zeros of the corresponding orthogonal polynomials or poles

of the Pade approximants). Instead of this for the de Branges' case (Section

11) we can use an elementary induction argument to reduce the problem

to a well-known Askey-Gasper inequality, see [4], [2]. Namely, without loss

of generality one can assume that O'A: = 0, k > n, and then it can be

easily checked that det JA:/ det JA:+1 = -O'~/k where JA: stands for the


k-th cut-off vf matrix (12.1)

o

o

So, the positiveness .:J ~ 0 is equivalent to the inequalities O"~ < 0, 1 ~

k ~ n which are just partial cases of the mentioned Askey-Gasper theorem

provided 0"1:(0) form a convex sequence:

13 Other Inequalities and Concluding Remarks

13.1 Two DIMENSIONAL CASE

For the two dimensional case (0"" = 0, k > 2) we are able to compute

(9.4) and (9.3) for every (constant) :1:0 and to find all admissible weights 0"

subjected to the positiveness condition .:J ~ O. It results in the following

one-parameter inequality

AER, A f 3/4

13.2 ANOTHER POINT OF VIEW

In fact, one can see from Section 5.5 that the method of multiplicative aver

aging works without the sharpness condition (5.5.2). The only consequence

of this change is omitting the assertion about the cases of equality.

For example, let 0" = diag{O"n}n>l and O"~ == 0 (Le. O"n = const for every

n ). Then one can check that condition (5.5.3) for no given by (9.2) is

equivalent to O"n ~ O"n+1(n ~ 1), and we have from (5.5.4) that

IIIOg z~:(O) II: ~ 20"1(8 - r),


where IHlr stands for u(r)-weighted Dirichlet norm and Br • for an arbitrary Loewner chain. This inequality is obtained in a different way in [16].

13.3 OTHER EVOLUTION FAMILIES

Our approach can be used for estimates of so-called power coefficients of

univalent functions, i.e. estimates for (f(z)/z)//, f e S where v stands for a

real number. To this aim, we have to consider the dual evolution family Tr •

(defined in Section 7) related to operators Cr., Cr./ = (Ar,/zA~,(O))// .

(f 0 A r.) where Br• = Ab-.,b-r is a Loewner chain. This evolution has the

function all, where o//(s) = o(s) + v(q,(s,·) - 1) as the generator (with 0 taken from (7.2) - (7.3)). Applying the same method and choosing the basic

(constant) isometric trajectory :Co in the form :Co = v-1«1 + z)-2// - 1) it

is possible to reduce equation (10.1) to

n (' ) 2v + n ( I ( )) -- un + nUn + 1 un+! - n + 1 + 2v Un+l = 0, n+v n+v+ n ~ 1,

and the limit case of inequality (5.5.4) for a = 0, b -+ 00 transfers into

11 -1((f)// )112 (r(k+2V))2 v -; - 1 ~ 4 ~ k .001:(0) k!r(2v + 1) , feS,

where II . II stands for u(O)-weighted g-norm. The inequality becomes an

equality for functions u«():co, ( e T.

These results are obtained in a different way in [5], [16].

References

[1] Berenstein, C.A. and Hamilton, D.H. Et la conjecture de Bieberbach

devint Ie theor~me de Louis de Branges ... - La Recherche, mensuel

No. 166, mai (1985).

[2] The Bieberbach conjecture. Proceedings of the Symposium on the Oc

casion of Proof. Providence: AMS, (1986).

[3] de Branges, L. A proof of the Bieberbach conjecture. Preprint LOMI,

E-5-84, Leningrad: LOMI, (1984).

[4] de Branges, L. A proof of the Bieberbach conjecture. - Acta Math.

(1985), 154 : 137-152.


[5] de Branges, L. Powers of Riemann Mapping Functions. - The Bieber

bach Conjecture. Proceedings of the Symposium on the Occasion of

the Proof. Providence: AMS, (1986) : 51-67.

[6] de Branges, L. Underlying Concepts in the Proof of the Bieberbach

Conjecture. - Proceedings of the International Congress of Mathemati

cians. Berkeley, California, USA, 1986, Berkeley, (1987) : 25-42.

[7] de Branges, 1. Square summable power series. Heidelberg, Springer

(to appear).

[8] de Branges, L. Das mathematische Erbe von Ludwig Bieberbach (1886-

1982). - Address to Mathematisches Institut der Universitat Basel,

May 4, 1990 (in the occasion of receiving of Alexander Ostrowski

Prize).

[9] FitzGerald, C.H. The Bieberbach Conjecture: Retrospective. - Notices

Amer. Math. Soc. (1985) 32 : 2-6.

[10] Fomenko, O.M. and Kuz'mina, G.V. The Last 100 Days of the Bieber

bach Conjecture. - Mathematical Intelligencer. (1986) 8, No.1: 40-47.

[11] Kolata, G. Surprise Proof of an Old Conjecture. - Science. (1984) 225:

1006-1007.

[12] Korevaar, J. Ludwig Bieberbach's conjecture and its proof by Louis de

Branges. - Amer. Math. Monthly. August-September, (1986) 93, No.

7: 505-514.

[13] Nikolskii, N.K. and Vasyunin, V.1. Quasi-orthogonal Hilbert space de

compositions and estimates of univalent functions. I. Proc. of Sympo

sium on Functional Analysis and Applications, Sapporo, 1990, (Math.

Reports of Sapporo University; to appear).

[14] Nikolskii, N .K. and Vasyunin, V.1. Notes on two function models. - The

Bieberbach Conjecture. Proceedings of the Symposium on the Occasion

of the Proof. Providence: Amer. Math. Soc., (1986) : 113-141.

[15] Pommerenke, Ch. Univalent functions. Gottingen: Vandenhoeck and

Ruprecht, (1975), p.376.

[16] Rovnyak, J. Coefficient estimates for Riemann mapping functions. -

J. d'Analyse Math. (1989) 52 : 53-93.


[17] Vasyunin, V. and Nikolskii, N. Quasi-orthogonal decompositions with

respect to complementary metrics and estimations 0/ univalent func

tions. Algebra and Analysis, (1990) 2, No.4: 1-81 (Russian).

[18] Vasyunin, V. and Nikolskii, N. Operator measures and coefficients 0/ univalent/unctions. Algebra and Analysis, 1991, (to appear, Russian).

[19] Zemanek J. Hipoteza Bieberbacha, 1916-1984. - Roczniki Polskiego Towarzystwa Matern., Seria II: Wiadomosci Matern. XXVII, 1986 :

1-14.

N.K. Nikolskii and V.1. Vasyunin Steklov Mathematical Institute St. Petersburg Branch

Fontanka 27

St. Petersburg, 191011

RUSSIA

On the Differential Properties of the Rearrangements of Functions

V.I. Kolyada

1 Introduction

Let I be a measurable function on a set E C Rn. In the case lEI = 00, we suppose that I{z E E : I/(z)1 > y}1 =: A,(y) < 00 for all y > O.

The nonincreasing rearrangement of I is defined to be the function ret) that is nonincreasing on (0, lEI) and equimeasurable with I/(z)l. The rearrangement ret) can be given by the equality

r(t):= sup inf I/(z)l, 0 < t < lEI. eCE,lel=t :z:Ee

For any measurable set E C R n with lEI < 00, we denote by E* the ndimensional ball centered at zero with measure IE* I = lEI. If lEI = 00, then we define E" = Rn. Let I be a measurable function on E. The spherically symmetric rearrangement of I is the function I; on E* defined by the equality

where the vn-measure is of the unit n-dimensional ball. The function I: is equimeasurable with I/(z)1, possesses spherical symmetry U:(zt) = I:(Z2), if Iz1l = IZ21), and monotonically decreases with increasing Izl·

Rearrangements appeared first in the nineteenth century, in the works of J. Steiner [St] and H.A. Schwarz [Sc], where the symmetrizations of sets and functions were studied for the first time. However, the systematic treatment of the rearrangements of functions and sequences began much later in the works of G.H. Hardy and J .E. Littlewood of the late twenties on fractional integrals and maximal functions.

The value of using rearrangements is due to their extremal properties .. The most important of these are various variational properties. The investigations of the changes of some variational functionals (curve lengths, surface areas) under symmetrizations were begun by J. Steiner and H.A. Schwarz. These investigations were continued in the book of G. P6lya and G. Szego [PS] (for functionals depending on gradients) and later by many other authors.

PROGRESS IN APPROXIMATION THEORY (A.A. Gonchar and E.B. Saff, OOs.), ©Springer-Verlag (1992) 333-352. 333

334 V.1. Kolyada.

The variational character also has various "difference" type functionals (in particular, moduli of continuity). The first results about minimal properties of such functionals for the rearrangements were obtained in the mid seventies in the works of A. Garsia and E. Rodemich [GR], P. Oswald [02], and I. Wik [WI, W2].

In Section 2 of this paper, we study rearrangements of differentiable functions and functions from Sobolev spaces. We give some generalizations of the P6lya-Szego estimates [PS] concerning the gradient of the symmetric rearrangement. In particular, we prove the following multiplicative inequality (1 ~ Pi < 00, i = 1, ... , n,p = n(I/Pl + ... + I/Pn)-l)

Estimates for the LP-moduli of continuity of the rearrangements are given in Section 3. The problems considered there were first raised by P.L. Ul'yanov [U]. In the one-dimensional case, P. Oswald [02] and I. Wik[Wl] independently have proven that for f E LP [0,1] (1 ~ P < 00)

Wp(ri 0) ~ 2wpUi 0), 0 ~ 0 ~ 1/2.

Further, for functions of several variables, P. Oswald [01] has proved, in particular, the following statement: if f E Lip (al,"" aniP)t (a, E (0,1],1 ~p< 00), then

Wp(ri 0) ~ COOl·, a* = (t ail)_l J=l

The main result of this paper is that the following stronger statement holds: if f E Lip(al, ... , anip) and v is the number of those of ai which are equal to one (v ~ 1), then

100 (r a • WpU*it»9dt/t < 00, (J = p/a*v.

This result is sharp. In particular, it is impossible to reduce the value of (J. The above estimates have a close connection with embedding theorems.

We shall use the following elementary properties of rearrangements.

Lemma 1.1 If f E L(E) then

sup l lf(.r)ld.r = t r(r)dr, 0 < t ~ lEI. eCE,lel=t e Jo

For a proof, see [BS, p. 53].

tThe definition of the multidimensional Lipschitz class is given in Section 3.

On the Differential Properties of the Rearrangements of Functions 335

Lemm.a 1.2 Suppose that the sequence {fn(xH of measurable functions converges almost everywhere on E to the function f(x). Then f~(t) -+ /*(t) at every point of continuity of /*.

See [CR, p. 30].

Lemm.a 1.3 Let f,g E lJ'(E), 1 $ p < 00; then

[lEI [ Jo I/*(t) - g*(t)IPdt $ JE If(x) - g(x)IPdx.

This lemma is also known (see, for example, [K05]).

2 Rearrangements of Differentiable Functions

We begin with the one-dimensional case and first observe that even for positive functions f E Coo [0,1], their rearrangement may turn out to be nondifferentiable at some points. However, it is easy to see that the rearrangement of any absolutely continuous function is also absolutely continuous. Moreover, the following statement is true [K05]:

Theorem 2.1 Let f be an absolutely continuous function on [0,1]. Then

(/*')*(t) $ (J')*(t), ° $ t $ 1. (2.1)

Corollary 2.2 If f is an absolutely continuous function on [0,1] then for any p E (0,00)

(2.2)

Inequality (2.2) was proved by G. Duff [Dl, D3] and J.V. Ryff [R] using other methods. G. Duff [D3] has also shown that for p ~ 1 this inequality may be strengthened: if f E C1 [0,1], then for any p ~ 1

where n(y) is the cardinality of the level set (7'" = {x : If(x)1 = y}. For p E (0,1) the sign of inequality (2.3) reverses.

336 V.I. Kolyada

A simple method for proving proof of (2.3) is given in [Ko5]. Using this method, we shall prove the following result:

Theorem 2.3 Let I E C l [0,1] and ,p(z) = 1/'(z)l/n(l/(z)l). Then lor all t E [0,1],

(2.4)

Proof. As has been shown in [Ko5, p. 66], for almost all t E [0,1] 11*' (t)1 :::;

<p'(t) where <p(t) = { ,p(z)dz, Et = {z : I/(z)1 > I*(tn. From this, it lEi

follows that for any interval (a,f3) C [0,1]

l Plr'(t)ldt:::; { ,p(z)dz. a lE~\E ..

Let Ea,p = {z : 1*(f3) < I/(z)1 < I*(a)}. It is clear that IEa,pl :::; 13 - a and 1 ,p(z)dz = 1 ,p(z)dz

E~\E.. E ... ~

If the measure ofthe set Ur(a) is positive, then n(l/(z)1) = 00 on this set. Suppose now that G C [0,1] is a finite or countable union of pairwise

disjoint intervals: G = Uk(ak, 13k). Then,

10 Ir' (u)ldu:::; L ,p(z)dz,

where H = Uk Ea/o,p/o, Since IHI :::; Lk(f3k -ak) = IGI, we have (by Lemma 1.1)

10 Ir' (t)ldt :::; lolGI ,p* (t)dt

for any open set G C [0,1]. From this (2.4) follows. • The following lemma has been proved by G.H. Hardy, J.E. Littlewood,

and G. P6lya [HLP, theorem 249],[BS, p. 88].

Lemma 2.4 Let I and 9 be nonnegative, nonincreasing functions on [0,00). If

foX g(t)dt :::; foX f(t)dt, z E [0,00),

then for any convex increasing on [0,00), the function <p with <p(O+) = 0 has the relation:

{+oo {+oo 10 <p(g(t»dt :::; 10 <p(f(t»dt.


Corollary 2.5 II IE G1[O, 1], then lor any convex increasing on [0, +00), the function I(' with 1('(0+) = 0 has the relation

11 1('(11*' (t)l)dt $11 1('(1/'(z)l/n(l/(z)l»dz. (2.5)

This statement was also proved by G. Duff [D2] using other methods. Let us return to inequality (2.1). From this inequality we obtain, for

symmetric rearrangements the inequality

(/:)*(t) $ 2(/')*(t), 0 < t $ 1. (2.6)

Moreover, by (2.3) under the additional condition 1(0) = 1(1), we have the estimate

1I/:'lIp $ II I' lip , 1 $ p < 00.

This is the one-dimensional case of the P6lya-Szego principle. Now let us consider the multidimensional case.

Let G be an open set in Rn and 1 $ p < 00. Denote by W;(G) the Sobolev space of all functions 1 E V(G) for which there exist every distributional derivative 81 18zi E V(G), with the norm

II/lIw:(G) := II/IILP(G) + IIIV/IIILP(G).

Lemma 2.6 Let G be an open connected set in Rn and 1 E W;(G) (1 $ p < 00). Then the rearrangement r is absolutely continuous on any interval (6, IGI) (0 < 6 < IGI).

This lemma (under some conditions on G) was proved by V.S. Klimov [K] (another proof is given in [K05]).

o For an open set G in R n and 1 $ p < 00, let G~G) denote the set of all

continuously differentiable functions with the compact support in G. o

Theorem 2.7 Suppose that G is an open set in R n and 1 E G1(G). Then, for any convex increasing on [0,00), the function I(' with 1('(0+) = 0 satisfies:

f I('(lVI:(z)l)dz $ f 1('(IV/(z)l)dz. t lGo lG (2.7)

I It is easy to see that inequality (2.7) is equivalent to the estimate

11~ f o l('(t1-1/nll*' (t)l)dt $ lG I('(XnIVf(z)l)dz (2.8)

where Xn is the isoperimetric constant, Xn = (nvn)-l/n.

lThis theorem expresses the P6lya-Szego principle which has numerous and important applications in mathematical physics, embedding theory, and geometric measure theory.

338 V.1. Kolyada

A proof of Theorem 2.7 in the three-dimensional case was given in the monograph [PS]. Later more exact proofs and generalizations of this theorem were obtained by many authors (see [K05]).

An elementary proof of Theorem 2.7 for <pet) = tP , which is based on approximation by polygonal functions, is given in [K05]. Here we show that by the same ideas yield more general results.

A function I which is continuous on an n-dimensional polyhedron Q is said to be polygonal, if Q may be divided onto the finite number of nsimplexes with pairwise disjoint interiors so that the restriction of I to each of those simplexes is linear.

The following elementary statement has been proved in [K05]. Let I be a polygonal function on Q and Et := {z E Q : I/(z)1 > ret)} and 0'; := {z E Q : I/(z)1 = ret)}. Denote A by the set of all t E [0, IQI1 for which 10'; I > o. Then almost everywhere on [0, IQI] \ A

11*' (t)ls(o-;) :5 ! (Lt IV/(z)ldZ) ,

where s(O';) is the (n -1)-dimensional volume of 0';. Suppose now that I(z) = 0 on the boundary 8Q. Then 8Et C 0';, 0 <

t < IQI. By the isoperimetric inequality [H, Ch. 5], we have

IEll-l/n :5 Xn s(8Et ) :5 Xns(O';).

If r' (t) exists and r' (t) #: 0 then IEtl = t. Thus, almost everywhere on [O,IQI1,

h(t) := 11*' (t)ltl- l/n :5 Xn! (Lt IV/(z)ldZ) .

From this it follows as above, that for all 0 < t < IQI

1t h*(u)du:5 Xn 1t(IV/D*(u)dU. (2.9)

Now we shall extend this inequality to a wider class of functions. Denote by Wp~G) (1:5 p < 00, G an open set in Rn) the closure of the

o set C~ G) in the space Wi (G).

Theorem 2.8 Let I E Wl(G) (G an open set in Rn). Then lor all t E (O,IGI),

1t (IV/;I)*(u)du $1t (IV/I)*(u)du. (2.10)

Proof. Let e > O. It is easy to see that there exists a sequence {fl:} of continuous functions on Rn such that


(i) lie is a polygonal function on some polyhedron QIe C G and supp /Jc C QIe; (ii) lim lle(z) = I(z) almost everywhere on G;

Ie_co

(iii) IIIV(11e - J)IIIL(G) < e. Fix t E (0, IGD and let E be the union of a finite number of pairwise

disjoint segments [al,,81], ... , [a.,,8.] from (0, IGI) with lEI < t. Ifh (h > 0) is sufficiently small so that lEI + sh ~ t and the segments [ai,,8i + h] are disjoint, then by Lemma 2.6, inequality (2.9), and condition (iii) we have:

Applying Lemma 1.2 and the Fatou theorem we get

From this it follows that the same inequality holds for any measurable set E with lEI ~ t. This, along with the Lemma 1.1 and arbitrariness of e, yields (2.9) (with h(u) = Ir'(u)lu1- 1/ n ) for a given function I E Wp1(G). It remains to observe that inequalities (2.9) and (2.10) are equivalent. •

It follows from Theorem 2.8 and Lemma 2.4 that for any function I E W1(G), inequality (2.7) holds.

Remark 1. "Pointwise" inequalities of the type (IVI:I)*(t) ~ c(IV/I)*(t) fail to hold for n ~ 2.

Remark 2. It is essential in Theorem 2.8 that I "vanishes" on the boundary in a certain sense. For an arbitrary function from wl(G), this theorem is not true. However in the case G = R n , (2.10) holds for any function

o IE Wf(Rn) (C,Rn) is dense in W;(Rn». Moreover it would be sufficient to prove Theorem 2.8 just for G = R n (indeed, if G is an open set and

IE Wp'G), then

i{z) = { I(z),

0,

zEG

z EcG

340 V.I. Kolyada

belongs to Wi-eRn)). At the same time it is obvious from the proof of Theorem 2.8 that one can get it under the weaker conditions on the behavior of I and IV'II for large Ixl·

Remark 3. Let In be the unit cube, IE wl(In), [t] := min(t, 1 - t) and h(t):= [tF-I/nlr'(t)l. Then

1t h*(u)du 5 Cn 1t (1V'/1)*(u)du.

This inequality easily follows from Lemma 2.1 of [K05]. Indeed by virtue of this lemma,

h(t) 5 Cn ~ (It 1V'/(x)ldx), Et = {x : I/(x)1 > r(t)}·

Now let us consider the anisotropic class L;l •.... P,. with Pi E [1,00) (i = 1, ... , n). A locally summable function on R n I belongs to the space L;l ....• P .. if its distribution function A/(Y) < 00, Y > 0, and in addition I has distributional derivatives ol/oxi E LPi(Rn) (i = 1, ... , n).

Theorem 2.9 Let I E L;l •.... P,., Pi E [1,00) (i = 1, ... ,n) and P := n(l/PI + ... + l/Pn)-I. Then

Proof. For x E Rn let Xi = (Xl"", Xi-I, Xi+l, ... , Xn) (i = 1, ... , n) and let L5:i be the straight line passing through the point (Xl, ... , Xi_I, 0, Xi+l, ... , xn), perpendicular to the hyperplane Hi = {x : Xi = o}.

Fix t E (0,00) such that r'(t) exists and r'(t) f; O. Let Et := {x : I/(x)1 > ret)}; then IEtl = t. We take Et C Et of type Fq with the same measure. Denote by E}i) the orthogonal projection of Et onto the hyperplane Hi' The sets E}i) are measurable in Rn-l.

Let s}i) be a set of all Xi E E}i) such that the set Et n Lti is measurable

in Rand ml(Et n Lt ;} > O. It is easy to see that S?) are measurable in Rn-I. Let Ai(t) = mn_l (S}i)). Then by the Loomis-Whitney inequality [H, p. 162]

n

II Ai(t) ~ tn-I. (2.12) i=l

Since ret) > 0 and r is continuous, there exists 6 > 0 such that r(t + 6) > O. For almost all Xi E S}i) the sets Et+6 n L5:i have finite, onedimensional measures. Further (see [M, p.13]), there exists a set Q~i) C S}i)


of the same (n - I)-dimensional measure such that after a corresponding change of the function 1 on a set of n-dimensional measure zero for any Xi E Q~i) the following properties hold: 1) I( x) is locally absolutely continuous in Xi; 2) ml(Et n L;t;} and ml(Et +6 n L;t,} < 00.

It is clear that for any Xi E Q~i) and any h E (0,6) there exist at least two disjoint bounded intervals such that at the endpoints of these intervals, I/(x)1 takes the values J*(t) and J*(t+h) and in their interiors J*(t+h) < I/(x)1 < J*(t).

For fixed Xi we denote by h,;t; the union of these intervals. We have

J*(t) - J*(t + h) ~ ~ lA.I:; lal/axi(x)ldxi.

Integrating over Q~i) and using Holder's inequality, we get (IEt+h \Et I ~ h):

Ai(t)[J*(t) - J*(t + h)] ~! [ 1%1. (X)I dx 2 1 Et+A \Et x,

From this,

Ai(t)IJ*' (t)1 ~ ~(cpHt))I/P;, CPi(t) := h, I :~ (x)r; dx,

and by (2.12),

tn-IIJ*'(t)1 ~ 2~ n(cpi(t))I/P;. i=l

Let qi = npi/p. Then ql 1 + ... + q;;l = 1. Thus, using Holder's inequality, we have

( [00 )I/P 10 (t l - lIn IJ*' (t)IP dt)

1 n ([00 ) l/np; 1 ( n ) lIn ~ 2"g 10 cpHt)dt =2" glial/aXillp;

• A measurable function Ion a set E C Rn belongs to the Lorentz space

Lqp(E) (1 ~ p,q < 00), if

( I~ )l~

1I/1I;p= 1 (tl/qJ*(t)Ydt/t <00.

It is clear that Lqp C Lq for 1 ~ P ~ q.

342 V.1. Kolyada

Corollary 2.10. Let 1 ~ Pi < 00 (i = 1, ... , n) and P := n(1/P1 + ... + 1/Pn)-1 < n. Then

Lp1, ... ,p" C Lqp(Rn), q = np/(n - p).

Indeed, if 1 E Lp1, ... ,p" then 1*(+00) = 0, and by Hardy's inequality (see [HLP, theorem 330]) and (2.11) we get

100 (t 1/ q r(t»)pdt/t = 100 (t 1/ q 100 Ir' (u)ldu)Pdt/t

,; c 1,00 ,pl. + p-llr' ('W <It ,; d (g 110 f/ oz, lip; )""

In particular, Lp1, ... ,p,. C L'I' q = np/(n - p). We observe that this result is known (for Pi> 1)[BIN, ch. 3].

3 Moduli of Continuity of Rearrangements

The modulus of continuity of a 'function 1 E LP[O, 1](1 ~ P < 00) is defined by

wpU; 6) = sup f - I/(z) - I(z + h)IPdz , 0 ~ 6 ~ 1. ( 1 h ) IIp

09~6 10 In connection with investigations on embedding theory, P.L. Ul'yanov

[U] posed the problem to estimate the modulus of continuity of the rearrangement 1* by means of the modulus of continuity of I. For the case P = 1 he obtained the estimate W1 U*; 6) ~ 9W1 U; 6).

Later P. Oswald [02] and, independently, I. Wik [WI] proved the following theorem.

Theorem 3.1. For any function 1 E LP[O, 1] (1 ~ P < 00)

106 ~(r;t)dt ~ 106 w:U;t)dt, 0 ~ 6 ~ 1/2. (3.1)

From (3.1) it follows that


It is unknown what the sharp value of the constant at the right hand side is.

Here we consider some questions concerning the V-modulus of continuity of rearrangements of functions of several variables. To avoid technical difficulties we consider functions defined on all Rn.

If f E V(Rn) (1 $ p < 00), then the function

Wp(fj01, ... ,On):= sup (' If(Z)-f(Z+h) IP dZ)l /P OSh.9. JR-

(0 $ Oi < 00, i = 1, ..• , n)

is called a modulus of continuity of f. If Oi = 0 for i =F j and OJ = 0 (0 $ 0 < 00,1 $ j $ n) then we obtain a partial modulus of continuity w¥)(fj 0) = wp(fj 01, ... , On).

The function

will be called a reduced modulus of continuity of f (see [Ko3]). P. Oswald [01] has proved the following theorem.

Theorem 3.2 For "any function f E V(Rn) (1 $ p < 00)

wp(rjo) $ cnwp(fjo), 0 $ 0 < 00. (3.2)

The proof of this theorem is based on combinatorial methods and is rather complicated. A simple proof is given in [Ko5].

For a function f E V(Rn) we define the isotropic modulus of continuity wp(fj 0) by wp(fj 0) := wp(fj 0, ... ,0). It is clear the wp(fj 0) $ wp(fj olIn). Thus for any function f E V(Rn) (1 $ p < 00)

(3.3)

As was mentioned above, the interest in estimates for the modulus of continuity of rearrangements arises in connection with embeddings. It turned out in this connection that the estimates (3.2) and (3.3) are not exact for functions with sufficiently great smoothness.

The following theorem gives a strengthening of inequality (3.3).

Theorem 3.3 Let f E V(Rn) (1 $ p < 00, n ~ 2)j then for any 0 < 0 < 00,

(3.4)

344 V.I. Kolyada

For P = 1, n = 2, this theorem was proved by P. Oswald [01]; in the general case, it was proved by other methods in [K05].

If IE Wi(Rn) (1 :5 P < 00) then (see [Ni, p. 166])

wp(f;6)/6:5 nIlIV/IIIP· Therefore we get

Corollary 3.4 II IE Wi(Rn) (1 :5 P < 00) then /* E B;/n(R+) (n ~ 2).

Recall that the Besov space B;,(~)(B;p == B;) for 0 < r < 1 is defined by

B;,(R+) = {I E V(~) : 100 (rrwp(f;t»'dt/t < oo}. We remark that (see [K05])

B;/n C Lpq., q. = np/(n - p), 1:5 p < n.

Thus estimate (3.4) implies the exact embedding theorems for the Sobolev spaces.

It may be shown that the estimate (3.4) is sharp in the following sense: if 1 :5 p < 00, n ~ 2 and w(6) is a moduli of continuity', then there exists a function I E V(Rn) such that wp (f; 6) = O( w( 6» and for any 6 E [0, 00)

(OO rp1nW:(/*;t)dt/t ~ 6-PwP(6). J6 ..

For the proof it is sufficient to use the function from Lemma 2.4 of [K04] (see also [K03, Ko1]).

Let us return to the estimate (3.2). In the general case, we don't know a strengthening ofthis estimate (analogous to Theorem 3.3). However we can obtain the solution of this problem for Lipschitz functions.

Let 0 < Cl:i :5 1 (i = 1, ... ,n) and 1 :5 p < 00. We denote by Lip(Cl:l, ... , Cl:n;p) the class of all functions IE VeRn) such that

wfj>(f; 6) :5 cmin(6a ;, 1) (j = 1, ... , n; 0:5 6 < 00). (3.5)

Let Na,p(f) be the smallest constant c for which inequality (3.5) holds for any j = 1, ... ,n and 6 E [0,00).

We observe that for 1 < p < 00 and 0 < Cl:i :5 1

.c;l ..... a .. C Lip(Cl:l, ... , Cl:n;p)

where .c;l ..... a .. is a fractional Sobolev-Liouville space [Ni, ch.9]; equality holds if and only if all Cl:i = 1. On the other hand,

Lip(Cl:l. ... , Cl:n;p) C H;l ..... a .. (0 < Cl:i :5 1, 1:5 p < 00);

'See for example [KoS].


here H;l ..... a .. is the Nikolskn class [Ni, ch.4] and the equality holds if and only if all ai < 1. Let a" = (all + ... +a;l )-1 < lip and q" = pI(l-a"p). Then for all p :5 q < q", H;l ..... a .. c Lq(Rn ), but for q = q" this embedding fails to hold. On the other hand, for all 0 < ai :5 1 and 1 < p < 00,

(see [Ni, ch.9]). The following question arises: for what values of ai is there an embedding

with the limiting exponent

Lip(at, ... ,an;p) C Lq.(Rn )? (3.6)

The following has been proved in [Ko2].

Theorem 3.5 Let 1 :5 p < 00, 0 < ai :5 1, and a" < lip. A necessary and sufficient condition for the embedding (3.6) to hold is that the following conditions are satisfied: 1) at least one ai is equal to one; 2) if ail' ... ,ai. are the numbers ai that are less then one, then

We return to the modulus of continuity of rearrangements. By virtue of Theorem 3.2, for any function f E Lip(al, ... , anjp),

(3.7)

that is, r E Lip(a"jp) == B;;'. Clearly, Theorem 3.5 is impossible to deduce from this.

We prove that the inequality (3.7) may be strengthened.

Theorem 3.6 Let al, ... ,an E (0,1] (n ~ 2) and" be the number of all ai that are equal to one (0 :5 " :5 n). If f E Lip(al, ... , an;p) (1:5 p < 00), then r E B;;(R,.), where

a" = (~1..) -1, 0 = L (0 = 00, if" cO). L..J a· a"" ;=1 J

Proof. In the cases" = 0 and " = n, the statement of theorem follows correspondingly from (3.2) and (3.4). Let 1 :5 " < n and a1 = ... = all = 1, ai < 1 for i = " + 1, ... , n. We write z = (z', Zll) (z ERn) where z, = (Xl, ... ,XII) and Zll = (XIIH, .. · ,Xn).

346 V.1. Kolyada

Let Na,p(f) = 1. By virtue of Lemma 1.3, we may suppose that 1 is continuous. Further, in the case 1 < p < 00, for almost all z" E Rn-", the function 1 belongs to Wj(RtI) (in z') and

(3.8)

We may also suppose that the same is true for p = 1. In the other case, it is sufficient to consider the function

- 1 l XI +<7 lxv+<7 1<7(x)=- ... l(tl, ... ,t",X"+1,···'Xn) dt1··· dt"

(T" Xl Xv '

and then using Lemma 1.3, take a limit as (T -+ O. For fixed x" E R n -", denote by f*(e, x") the nonincreasing rearrange

ment of the function I(x',x") in z'. Then for fixed e E (0,00), denote by gee, TJ) the nonincreasing rearrangement of the function f*(e, x") in x", g(e,TJ) = [f*(e,·)]*(TJ). The function g(e,TJ) is nonincreasing in every variable and equimeasurable with I/(z)1 (see [BD.

It is easy to show using Lemma 1.3 and Theorem 3.2 that the modulus of continuity of 9 in the variable TJ satisfies the condition

w(2)(g·6) < c min(6fJ 1) P , _ n " ( )

-1 n 1

(3= L-1:=,,+1 al:

(3.9)

Further let

pet) := (I t de 100 (elg«e, TJ)I)P dTJ) l/p

We may suppose that pet) > 0 for t > O. Next, we have

[00 r p/" pP(t)dt/t :::; ~ [00 dTJ [00 Ige(e, TJ)lPe(1-1/")de. Jo p Jo Jo

By virtue of Lemma 1.3, for almost all e E (0,00),

On the other hand, by Theorem 2.9, for almost all x" E Rn-"

Thus, by (3.8)

( [00 ) l/p " Jo rp/"pp(t)dt/t :::;c~lI/~jllp:::;c'. o 1=1

(3.10)


Let ctl(t) := min { r : JJ(r) = min[(t/r)/J, In,

ct2(t) := t/ctl(t) and w(t) = JJ(ctl(t)).

Fix 6 > ° and denote

1 111+a2(cS) <p(z, y) := ct2(6) II 9(Z,71)d71, (z, y) E (R+)2.

By Lemma 1.3 and inequality (3.9),

wp(g"; 6) :5 wp(<p"; 6) + 211g - <pllp :5 wp(<p"; 6)

+ 2w~2)(g;ct2(6)):5 wp(<p*;6) + cw(6). (3.11)

Let us estimate wp( <p*; 6). We have by Hardy's inequality

w,. (,,'; 6) ~ l' U" I,," (. )Id. r <It

+ 6P 100 l<p'" (u)IPdu (3.12)

:5 Cp [16 uPI<p*'(u)IPdu+6P 100

1<P*'(U)IPduj.

Let Ef := {(z, y) E (R+)2 : <p(z,y) > <p"(t)}. Next, denote by Al the set of all t E (0,6) such that the projection E}1) of Et onto the z-axis has the one-dimensional measure ml(Ep») ~ ctl(t). For any z E E}l) (t E AI) and any h > 0, there exists a line segment It,h,:r: such that (z, y) E Et+h \ Ef

for all y E It,h,:r: and

<p"(t) - <p"(t + h) :51 l<p~(z, y)ldy. If.A,s

Integrating over E}l) and using the Holder inequality, we obtain

From this, for almost all tEAl

348 V.I. Kolyada

Thus by (3.9)

f tPlcp*'(t)IPdt $ ~(c5)J' f Icp~(z,y)IPdzdy JAl J(~)2

$ [w~2)(9i Q2(c5»]P $ cW"(c5). (3.14)

Now let A2 := (O,c5)\A1• Since QI(t)Q2(t) = t and ml(Ep»ml(E~2» ~ t (where E~2) is the projection of Et onto the y-axis), then ml(E~2» > Q2(t) for all t E A2. For any y E E~2) and any h > 0, there exists a line segment Jt,h,y such that

cp*(t) - cp*(t + h) = 1 Icp~(z,y)ldz, (t E A2). J t ,",11

Let St(Y) be the length of the section of Et which intersects the straight line which is parallel to the z-axis and passes across the point (0, y). It is clear that St(Y) coincides with the left end of Jt,h,y. Therefore

St(y)[cp*(t) - cp*(t + h)] $1 zlcp~(z,y)ldz. Ji,.,JI

Integrating over E~2) and taking into account that f St(y)dy = IEtl = t JE(2)

t we get, a.e. on A2 ,

tlcp*'(t)1 $ [~ (J Lt zPlcp~(z,Y)IPdZdY) rIP. (3.15)

Let r:= SUpA2i then IE~I)1 $ QI(r) and

f tPlcp*' (t)IPdt $ J' f zPlcp~(z,y)IPdzdy JA2 JE~

Thus (see (3.14»,

16 uPlcp*' (u)IPdu $ cW"(c5).

Now, let BI := {t E [15,00): ml(E~I» ~ QI(t)}. As above, for almost all t E Bl, the estimate (3.13) holds. Therefore,


~ [W~2)(g;a2(6»]P ~ ctiJP(6).

Further, for almost all t E B2 == [6,00) \ B l , we have the estimate (see (3.15»

l<p*' (t)IP ~ r P p'(t),

where p(t) := J fe, xPI<p~(x,y)IPdxdy. Since for all t E B2, ml(E}I» <

al(t), we have

p(t) ~ la1(t) xPdx 100 1<p~(x,y)IPdy

~ foa1(t) xPdx 1000 Ig~(x,Y)IPdy = JlP(al(t» = UJP(t), t E B2.

From this, it is easy to derive the estimate

Thus, we have (see (3.11) and (3.12»

~(g*; 6) ~ c[tiJP(o) + oP 100 r P -1 tiJP(t)dt].

Therefore, by Hardy's inequality

It remains to estimate the last integral. Further, it is easy to see that we only have to consider the integral

J = 11[o-a O w(o)]Bd6j6

(because w(6) ~ 1 for all 6.) Let

60 := 1, 61:+1 := min{6 : w(6) = ~W(6k)}, k = 1,2, ....

Then, we have by (3.12) (w(6) ~ (6jal(6»P)

J = f: f6 k r p/ lI - l [w(t)]p(1+1/PII)dt k=O J6 k +1

00

< C ~)W(6k)]P(1+1/PII)6kP/1I k=O

350 V.1. Kolyada

00

< c L W"(6k)[a1(6k)]-P/v k=O

::; c' [~W"(6k) {(a1(bk»-P/v - (a1(bk_1»-P/v} + W"(I)a~P/V(I)l

< C" [f: iiI'(6k) l a1(6"-1) r P/v- 1dt + W"(I)a~P/V(I)l

k=1 al(6,,)

::; c'" laOO I'p(t)rp/ v - 1dt ::; c.

The theorem is proved. • The following easily follows from Theorem 3.6.

Corollary 3.7 Let ai E (0,1] (i = 1, ... , nj n ~ 2), 1 ::; p < 00, a* < lip, q = pI(I- a*p), and 8 = pla*v, where v is a number of all ai which are equal to one, 1::; v::; n. Then Lip(al, ... ,anjp) ~ Lq8(Rn).

For 1 < p < 00 this statement was proved by Yu. V. Netrusov [Ne] by other methods.

Theorem 3.5 follows from Corollary 3.7. In conclusion, we observe without proofthat Theorem 3.6 is sharp in the

following sense: Let 1 ::; p < 00, ai E (0,1], and v be a number of ai which are equal to one, v ~ 1. Further, let w( 6) be a moduli of continuity such that·

laOO [r a • w(tW dtlt < 00 (8 = pla*v).

Then there exists a function f E Lip( a1, ... , an j p) such that wp (/* j t) ~ w(t).

References

[BS] Bennett, C. and Sharpley, R., Interpolation of Operators, New York: Acad. Press, 1988.

[BIN] Besov, O.V., Il'in, V.P., and Nikolskii, S.M., Integral representations of functions and embedding theorems, "Nauka," Moscow, 1975j English transl.: Vols. 1 & 2, Wiley, 1979.

[B] Blosinski, A.P., Multivariate rearrangements and Banach function spaces with mixed norms, Trans. Amer. Math. Soc., 263, No. 1(1981), 149-167.


[CR] Chong, K.M. and Rice, N.M., Equimeasurable rearrangements of functions, Queen's papers in pure and appl. math., 1971, N28.

[D1] Duff, G., Differences, derivatives and decreasing rearrangements, Can. J. Math., 19, No. 6(1967), 1153-1179.

[D2] Duff, G., A general integral inequality for the derivative of an equimeasurable rearrangement, Can. J. Math., 28, No. 4(1976), 793-804.

[D3] Duff, G., Integral inequalities for equimeasurable rearrangements, Can. J. Math., 22, No. 2(1970), 408-430.

[GR] Garsia, A.M. and Rodemich, E., Monotonicity of certainfunctionals under rearrangement, Ann. Inst. Fourier, 24, No. 2(1974), 67-116.

[H] Hadwiger, H., Vor/esungen uber Inhalt, Oberflache, und Isoperimetrie, Heidelberg: Springer-Verlag, 1957.

[HLP] Hardy, G.H., Littlewood, J.E., and P6lya, G., Inequalities, Camb. Univ. Press, Cambridge, 1934 (2nd ed., 1952).

[K] Klimov, V.s., Embedding theorems for Orlicz spaces and its applications to the boundary problems, Sibirsk. Math. Zh., 13, No. 2(1972), 334-338.

[Ko1] Kolyada, V.I., Estimates of rearrangements and embedding theorems, Math. Sb., 136, No. 1(1988), 3-23; English transl.: Math. USSR Sbornik, 64, No. 1(1989), 1-21.

[Ko2] Kolyada, V.I., On embedding H;l, ... ,W" classes, Math. Sb., 127, No. 3(1985), 352-383.

[Ko3] Kolyada, V.I., On embedding in the classes <pel), Izv. Akad. Nauk. SSSR, Ser. Mat., 39(1975), 418-437; English transl.: Math. USSR Izv., 9(1975).

[K04] Kolyada, V.I., On relations between moduli of continuity in different metrics, Tr. Math. Inst. Steklov, 181(1988); English transl.: Proceedings of Steklov Inst. Math., 1989, Issue 4, 127-148.

[Ko5] Kolyada, V.I., Rearrangements of functions and embedding theorems, Uspehi Math. Nauk., 44, No. 5(1989), 61-95 (Russian).

[M] Maz'ya, V.G., Sobolev spaces, Leningrad, 1985.

[Ne] Netrusov, Yu. V., Embedding theorems of Lizorkin-Triebel spaces, Zapiski nauchin. semin. LOMI, 159(1987), 103-112.

[Ni] Nikolskii, S.M., Approximation of functions of several variables and embedding theorems, 2nd rev. ang. e. "Nauka", Moscow, 1977.

352 V.1. Kolyada

[01] Oswald, P., Moduli of continuity of equimeasurable functions and approximation of functions by algebraic polynomials in LP, Candidate's dissertation, Odessa State Univ., Odessa, 1978 (Russian).

[02] Oswald, P., On moduli of continuity of equimeasurable functions in the classes <p(L), Math. Zametki 17(1975), 231-244; English transl.: Math. Notes, 17(1975).

[PS] P6lya, G. and Szego, G., Isoperimetric inequalities in mathematical physics, Princeton, 1951.

[R] Ryff, J .V., Measure preserving transformations and rearrangements, Math. Anal. and Appl., 31, No. 2(1970), 449-458.

[Sc] Schwarz, B.A., Gesammelte Abhandlungen, Vol. 2, Berlin: Springer, 1880.

[St] Steiner, J., Gesammelte Werke, Vol. 2, Berlin: Reimer, 1882.

[U] Ul'yanov, P.L., Embedding of certain classes H;, Izv. Akad. Nauk. SSSR Ser. Math. 32(1968),649-686; English transl.: Math. USSR Izv. 2(1988).

[WI] Wik, I. The non-increasing rearrangement as external function, Preprint: Univ. Uniea, Dept. of Math., No. 2(1974).

[W2] Wik, I. Symmetric rearrangement of functions and sets In R n I

Preprint: Univ. Umea, Dept. of Math., No. 1(1977).

V. Kolyada LOMI Odessa UKRAINE

A Class of I.M. Vinogradov's Series and Its Applications in Harmonic Analysis

K.I. Oskolkov

1 Introduction

The present paper is a survey of the author's recent research in the onedimensional trigonometric series of the type

L: j(n)e27r'(nrxr+··+nxd. (1.1) n

In (1.1), r is an integer 2: 2; X r , ... ,Xl real variables, A, and j(n) (n = 0, ±1, ... ) denote the Fourier coefficients of a complex-valued function f( x) of one real variable x, periodic of, say, period 1 and Lebesgueintegrable over the period:

j(n) = 11 f(x)e-27rinXdx, n = 0, ±1, ....

Just to illustrate the various fields of applications of such series, we confine ourselves to the case r = 2 and start from the Propositions 1-3 below, which at first sight may seem to be distant from each other in their nature, although they are not. (We postpone exact references to the literature in this section; they will be given later in the main text.)

Proposition 1. The Lebesgue constants of the quadratic spectrum have exact logarithmic order of growth:

(1.2)

By a spectrum, we mean a sequence K = {kn} (n = 1,2, ... ) of distinct integers not necessarily monotonic in general; the quadratic spectrum is just the sequence K = {kn } with kn = n2 • Furthermore, given a spectrum K denote by C(K) the subspace of the space C of continuous periodic functions g(x), g(x + 1) == g(x), whose Fourier coefficients vanish outside of the sequence K:

"


354 K.I. Oskolkov

and equip C(K) with the usual Chebyshev norm of the space C: IIgll = max2: Ig(z)l. Then the N-th Lebesgue constant of the spectrum K is defined as the norm of the operator of the N -th partial sum acting in C( K):

(Note that this definition is dependent on the original ordering of the sequence K = {kn } ).

Proposition 2. Let f(z) be a function with bounded total variation over the period, i.e. var {I; [0, I)} < 00. Then the solution of the Cauchy initial value problem for the Schroedinger equation

81/J .821/J &i =, 8z2' 1/J(z, 0) = f(z) (1.4)

exists in the class of regular functions 1/J(z, t) bounded on the whole real plane (z, t).

The solution to (1.4) is understood in the Schwartz generalized sense, i.e. as a functional over the appropriate space of test-functions. Using the Fourier method of separation of variables, that solution can be formally represented as the series

1/J(z,t) "" L:i(n)e2l1"i(2l1"n2t+n 2:) (1.5) n

which is clearly of the form (1.1).

Proposition 3. Let a and q be relative primes, with q > O. Then the following estimates are valid for the incomplete Gaussian sums

(1.6)

where c > 0 is an absolute constant.

Recall that the moduli of the complete Gaussian sums satisfy (for odd q) the classical relationships due to K.F. Gauss

~ (211"ian2) L."exp =..;q «a,q) = 1, q == l(mod2». n=l q

(1.7)

Therefore, (1.6) means that the operator of partial sums is bounded on the set of Gaussian sums.

We will attach the necessary historical comments and quotations to each of these assertions in due course, and now only stress that all of them have a common foundation. All of them follow from

A Class of I.M. Vinogradov's Series 355

Proposition 4. The discrete Hilbert transforms

are bounded uniformly in N = 1,2, ... , on the real plane E2 = {(Z2' Z1)} :

sup sup IHN(z2, z1)1 < 00, (1.8) N (:l:2,:l:l)eE2

and the pointwise limit

(1.9)

ensts everywhere on E2.

(As a matter of fact, both Propositions 2 and 3 are equivalent to the boundedness ofthefundion H(Z2' Z1».

A complete analogue of Proposition 4 is valid also in the case of real algebraic polynomials of degree r ~ 3 in the imaginary exponent (see Proposition 11 below).

Since the proof of these assertions essentially requires the method of exponential sums due to I.M. Vinogradov, the author found it proper to call (1.1) Vinogmdov's series, or shortly, V-series. The value of its sum, whenever it is properly defined in one sense or another, will be denoted V(fi z,., ... , Z1) and, due to the formal identity

V(jiO, ... ,O,z) = Ei(n)e2l1'in:l: "" fez), n

will be called V-continuation (of r-th degree) of the function f. In this terminology, the sum of the series

(1.10)

= lim " N ..... oo L..J O<lnl:5N

is the V-continuation of r-th degree of the function

e 2l1'in:l: 00 sin 21rnz 1 fez) = v.p. E -2 -. =" = -2 - {z},

1r1n L..J1 1rn n¢O n=

(z =F 0, ±1, ... ; {z} denotes the fractional part of z), Le. the V-continuation of the Bernoulli kernel of the first order.

356 K.1. Oskolkov

Section 2 below contains a description of the background which has led the author to consider the discrete Hilbert transforms of the above type as well as its generalizations. It should be emphasized that the nontrivial results on discrete Hilbert transforms gave rise to the author's interest in series (1.1) and subsequently in the applications to the time-dependent Schroedinger type equations. (We will present a comparatively detailed outline ofthe proof ofthe assertion concerning discrete Hilbert transforms.)

We will also discuss the fun'ctional properties of the continuous (integral) analogues of these transforms. This is useful for applications to the Schroedinger equations.

The applications are given in Section a. In particular it turns out that the solutions to the Cauchy initial value problem of Schroedinger type equations with periodical initial data exhibit very funny functional features. Namely, they provide an explicit model for the now fashionable area in mathematical physics - the so-called "Quantum Chaos."

2 Discrete Hilbert Transforms, Spectra of Uniform Convergence, and the Growth Order of Fourier Sums Almost Everywhere

Let M, N be positive integers, OM denote the (real) M-dimensional unit cube ofthe vectors ~ = (ZM,"" Zl), satisfying 0 ::; Zm ::; 1. Consider the following discrete Hilbert transforms of the imaginary exponentials:

M e2riNz,,"m H = Hn(M,N,~,y) = ~ 05' - L...J n-m+ .

m=l

where n = 1, ... ,M, Y E OM. The obvious estimate for these quantities is

M 1 IHI::; L I _ 051 ::; 2logM + 0(1).

m=l n m+.

On the other hand, if we choose, for example,

N = aM, Zn = a-n , Ym = 0.5. am - M ,

(2.1)

(2.2)

(2.a)

we can easily see that for n ::; m the product 2N ZnYm = am - n is an odd number, and thus e2riNz,.lI ... = -1, while for n > m that product is very small, and the exponent is close to +1. Hence for all n = 1, ... , M the Hilbert transforms Hn with N, ~, and Y as in (2.a), satisfy the reverse inequality -

IHnl ~ logM + 0(1) (M -+ 00). (2.4)


Even this simple consideration reveals the Diophantine nature of the transforms (2.1). The following two very interesting problems in the theory of one-dimensional trigonometric series led us to the transforms Hn (cf. [26]).

ProbleDl 1. What is the exact estimate of the almost everywhere order of growth of the sequence of partial sums of one-dimensional Fourier series?

The two known basic results here are: A. The estimate from above, due to H. Lebesgue (cf. [45, v. 1, Ch. 2, Theorem 11.9]). If fez) is Lebesgue integrable over the period [0,1]' I.e. f E L1 and

Sn(/,z)= E j(m)e27rimlt:, n=I,2, ... Iml$n

its Fourier sums, then for almost every z

Sn(f, x) = o(Iogn) (n -+ 00). (2.5)

B. Estimate from below for the possible growth order. This is a quantitative version of the famous result due to A.N. Kolmogorov [18] (see also [17]) on the existence of almost everywhere (and everywhere) divergent trigonometric Fourier series. Namely (cf. [5] and also [36]), given an arbitrary sequence of positive numbers {en}, en -+ 0 (n -+ 00), there exists such a function f E L1 that for almost every z

(2.6)

There is an obvious gap between log n in the estimate (2.5) from above and log log n in the result (2.6). To somehow fill in this gap seems to the author to be one of the most interesting problems in the theory of one dimensional Fourier series which remains open af~er the great theorem of L. Carleson [4] on the convergence a.e. of trigonometric Fourier series from L2, and its subsequent generalizations for the spaces IJ', p> 1 in [14], et al.

ProbleDl 2. How to characterize the spectra of uniform convergence, and in particular, how dense can these spectra be?

A sequence of distinct positive integers 1\; = {kn } (for the definitions, see the Introduction, after Proposition 1) is called the spectrum of uniform converyence (SUC), if each Fourier series

g(z) ,.... E 9k,.e27rik .. 1t:

n

of the class C(1\;) (Le., assuming that g(z) is continuous) converges uniformly.

358 K.1. Oskolkov

The characterization problem, i.e. how to judge whether or not a given sequence I:, = {kn }, n = 1,2, ... , of distinct integers is a SUC, was posed by P.L. Ul'yanov in [38]. Due to the Banach-Steinhaus theorem, a criterion (albeit impractical) for a sequence K to be SUC is the boundedness of the corresponding Lebesgue constants LN(I:,) (see (1.3)), i.e. I:, E SUC iff LN(I:,) = 0(1) (N ~ 00). The principal difficulty here consists precisely of obtaining explicit estimates of the quantities LN(I:,) in terms of the sequence I:, (for a survey of some results in these directions, the reader is referred to [9] and [25]).

We briefly list the results known to the author on the study of SUC. First of all, if the sequence I:, = {km } satisfies the lacunarity condition

of Hadamard, i.e. if inf{kn+dkn : n = 1,2, ... } > 1, then it is a Sidon sequence (see [45, Ch. 6, 6, 1]), i.e. the Fourier series of every bounded function with harmonics from I:, converges absolutely and, a fortriori, uniformly. Thus, every Sidon sequence, of course, is SUC. Lacunary sequences in the sense of Hadamard have low density:

dN(I:,) = L: = O{logN) (N ~ 00). (2.7) n:l: .. $N

It was shown by S.B. Stechkin [31], that (2.7) is necessary for I:, to be a Sidon sequence.

A very interesting construction of SUC of somewhat higher density was obtained by A. Figa-Talamanca [8]. Namely, take a sequence of positive integers C = {in}, satisfying inf{in+l/in : n = 1,2, ... } ~ 3. Then the sequence JC of the form

I:,=C+C={k:k=in+im, n<m},

indexed in increasing order, is a SUC. In particular, if we take in = 3n, then the estimate dN(I:,) ~ a log2 N, N = 1,2, ... , holds for the density of 1:" where a is a positive constant. The papers [6] and [27] are devoted to the further development of this result; it follows from them that for any natural number s there exist SUC whose density can be estimated from below as

dN(I:,) ~ a, {log NY (a, > 0).

As far as the author knows, the problem of existence of still denser SUC remains open. At the same time, as follows from [9], the method of construction of SUC applied in [6] and [27] cannot be extended to the case of increasing s. We also note the paper [37], devoted to the study of certain properties of SUC of a general nature.

Now we turn to the results of the negative character, i.e. showing that a given sequence I:, is not a spectrum of uniform convergence.

By the classical result of du Bois-Reymond (see [45, Ch. 8, 1, 1]) neither the whole sequence of integers n = 0, ±1, ... , nor the sequence of positive


integers in the natural ordering is SUC. This easily implies that neither an infinite arithmetic sequence, nor any increasing sequence IC containing arithmetic sequence of ali arbitrary length can be a SUC.

Moreover, there exist no rearrangements of the whole set of integers or natural numbers (or arithmetic sequences) that could improve the order of corresponding Lebesgue constants LN(IC), which is 10gN. This follows from the validity of the Littlewood's conjecture, proved by S.V. Konyagin [19] and B. Smith, et al. in [29]: there is an absolute constant a > 0 such that for any sequence of distinct integers n1, n2, .. . the following estimate is valid

(2.8) 11 Ee27rin;o: dz ~ alogN. o ;=1

However, the solution to the Ul'yanov's problem on SUC has not been known even for the classical sequences {n2},{n3 }, ••• or more general, for polynomial spectra, i.e. for sequences of the form kn = P(n), n = 1,2, ... , where P(,\) = a,.,\r + .. ·+a1'\+aO is an algebraic polynomial (ar , ... , a1, ao are fixed positive integers). The answer was obtained in [25].

Proposition 5. Let r ~ 2 and IC = {P(n)} be a polynomial spectrum. Then the following estimate from below is valid for its Lebesgue constants:

where a r is a strictly positive quantity depending only on r = deg P. In particular, IC is not a spectrum of uniform convergence.

Although Proposition 11 below (proved in [2]) implies (see Corollary 2) a better, sharp, estimate of the Lebesgue constants for polynomial spectra, we present here the completely elementary proof of (2.9), which does not require any preliminaries.

Lemma 1. Let IC = {km }, n = 1,2, ... , be a spectrum (not necessarily polynomia~, M and N be positive integers with M < N, and let

1~lnl~M n

(2.10)

TN,M(IC) = IIFN,MII = m,:x IFN,M(IC,z)l· Then for the Lebesgue constants of IC the following estimates are valid

{ 10gM } LN(IC) ~ max TN,M(IC) : M < N , (2.11)

and in particular

(2.12)

360 K.I. Oskolkov

(Relations of the form A A between positive quantities A and B mean, here and below, that A :s cB, where c is a positive absolute constant. If A :s crB or A :s cr,.B, etc., where the positive factors c are finite and depend only on the indicated parameters, the notation A <r,a B will be used).

Lemma 2. Let Pr , r = 0,1, ... , denote the set of algebraic polynomials of degree r with real coefficients and, for N = 1,2, ... , let

l$lnl$N

Then, for r = 1,2, ... ,

eiP(n) --j

n z(N,r) = sup IhN(P)I.

PE1'r

z(N, r)< (log(N + l»l-ar , with E:r = 2-r+1 . (2.13) "r

It is obvious that the polynomials FN,M(K"Z) belong to the class C(K,) and that they are some discrete Hilbert transforms ofthe type (2.1). Furthermore, they are the appropriate analogues, corresponding to the given spectrum K" of the well-known Fejer polynomials (see [45, Chapter 8, 1])

n (2.14)

The latter have been used many times by many authors to construct examples of continuous functions (with no spectral restrictions) possessing "bad" sequences of Fourier sums.

To verify (2.11), it suffices to recall that FN,M E C(K,) , IIFN,Mlloo = 'TN,M(K,)j furthermore,

and (2.11) follows.

M 1 =:L;>logM,

n=l

The idea is to apply (2.11), (2.12) to obtain estimates of Lebesgue constants LN(K,) from below, in particular, to show that K, is not a SUC. It suffices to non trivially estimate'the uniform norm 'T of the Fejer polynomials F.

Certainly, this is not always possible, but as Lemma 2 shows, the idea works at least in the case of polynomial spectra. Since the estimate (2.13) does not depend on the coefficients of the polynomials, and for each fixed z and N, zP( n + N) is a polynomial in n with real coefficients of the same degree rasp, the estimate (2.9) is an immediate corollary of (2.12) and (2.13).


In turn, (2.13) follows by "squaring out" and by induction in the degree r of the polynomials in the exponent. In this context, the proof (and also the character of the reducing factor (log N)-Er) can be regarded as going back to the investigations of Gauss and Weyl in exponential sums (see [39, Russian pp. 6-8, English pp. 183-185]).

Indeed, we have

ei(P(n)-P(m»

nm

n~ = n~m (! -~) (n;fm).

This, after the introduction of the summation variable v = n - m, gives the estimate

IhN(P)1 2 $2 L ~ L cos(P(n+;)-P(n)) +3 L :2 +R. 1:$1"I:$N 1:$lnl:$N 1:$lnl:$N

(2.15) Furthermore, the error term R.in (2.15), corresponding to the changing of domains of summation, trivially does not exceed

The double sum on the right is bounded by an absolute constant i.e. R ~ 1. Indeed, consider the function l/zy in the triangle t::. = {O < z $ 2, IZ-ll < y $ I}. This function is integrable in the (improper) Riemannian sense over t::.:

11 dzdy 12 1 (11 dY) 12 (1 1) -- = - - dz - -log -- dz < 00 I:;. zy 0 z 1:1>-11 Y - 0 z Iz - 11 '

and the above double sum is exactly the Darboux-approximant to the integral. Thus, it follows from (2.14) that

IhN(P)1 2 ~ E IhN(P,,)1 + 1,

l:$I"I:$N Ivl (2.16)

where for fixed v, 1 $ Ivl $ N, p,,(n) = P(n + v) - P(n). If P E Pr , then for each fixed v, P,,(n) is a polynomial of a reduced degree with respect to the variable n, namely P E .Pr-1. Hence, (2.13) follows from (2.16) by induction in r from the estimate

z2(N, r + 1) ~ z(N, r) logN + 1. (2.17)

362 K.I. Oskolkov

(Note that we can start the induction from the trivial case of the polynomials of degree r = 0, i.e. just constants, when obviously hN(P) == 0. Then it follows from (2.17) that z(N,l) <: 1, which is not quite so trivial thing already - it means the uniform boundedness of the ordinary Fejer polynomials FN,M(Z) in (2.14), etc.).

The essence of the above considerations is that in order to prove a given sequence K, is not sue, it suffices to establish nontrivial estimates of the appropriate Hilbert transforms of the type (2.1) from above (see Lemma 1). Moreover, for that purpose, nontrivial estimates are needed not for all Hn. Rare subsequences would also be sufficient.

Now we return to Problem 1 of the almost everywhere order of growth of the sequence of partial sums of trigonometric Fourier series of the class L1. The aim is to show that, within the framework of the Kolmogorov's construction ([17] and [18]), the desire to somehow improve (2.6) leads to the necessity of estimating Hn from below. More accurately, one needs to find "bad" 1l., Y so that, given an M, the quantity IHn(M, N,1l.. y)1 in (2.1) can be estimated as > log M from below for all or at least a "majority" of n's.

The problem is that this should be done for N = N (M) growing along with M as slowly as possible.

The idea of Kolmogorov's construction is to consider (generalized) functions of the type

1 M fez) = M L: cS(z - zm)

m=1

(2.18)

where cS(z) denotes the I-periodic Dirac delta function, and the points Zm are "nearly equidistant" on the period. The latter can be understood as, say,

m+Om Zm= M m=l, ... ,M, (2.19)

Instead of the delta function, Fejer kernel of large order were considered in [17] and [18]. The form (2.18) was proposed by E.M. Stein in [32]. In that work, consideration of Diophantine nature, namely the well known Kronecker's theorem, was applied for the firs~ time in search of the appropriate Om's.

Given a positive integer Ie, denote by Dk(Z) the Dirichlet kernel of order Ie and by Sk(f, z) the Ie-th Fourier sum of the function fez):

D · ( ) - 1 2 ~ 2 l _ sin 211'( Ie + i)z k Z - + L..Jcos 11' Z - . ,

l=1 S1D1I'Z

Sk(f, z) = E j(l)e2f(il~, Ie = 0, 1, .... Ill~k


Then for I(z) as in (2.18), we have

1 M S.(I, z) = M L D.(z - zm)

m=1

= ..!... t sin 211"(k + 1/2)(z - zm) M m=1 sin 1I"(z - zm) .

Now we choose k to be multiples of M, and let kn = kn(z) = inM for Z

belonging to the middle-half ofthe interval (zn, zn+1) which we denote by W n • Then we readily see that the absolute value of S. can be estimated from below on a subset w~ of Wn of measure ~ (1/2)lwn 1 as follows:

(2.20)

At this point, as in [32], it is clear how Kronecker's theorem works: one needs to find in and (Jm such that the Hilbert transform on the right of (2.20) is large for all n, say of the largest possible order logM. By Kronecker's theorem, this can be achieved whenever (Jm's are linearly independent over the field ofrational numbers. On the whole, what is needed here is the Diophantine approximation oUhe jump matrix (sign (n - m», n, m = 1, ... , M: find (Jm, -1/2 < (Jm :$ 1/2 and positive integers in such that the difference

e2l1"il .. ' ... _ sign (n - m)

is "small," and the numbers max{in : n ~ M} grow as slow as possible. An explicit example of such approximation is provided by (2.3), and the result (2.6) easily follows.

In particular, the above approach enables us to obtain the following result (cf. [26, Theorem 4]).

Proposition 6. lithe sequence JC = {kn } satisfies the Hadamard condition, I.e.

inf{kn+t/kn : n = 1,2, ... } > 1 (2.21)

and e = {en} is a sequence 01 positive numbers, tending to 0 as n - 00, then there is a function I E L1 such that lor almost all Z

limsup IS ... (I,z)1 = +00. n_oo enlogn

(2.22)

This is a quantitative variation of a result due to R.P. Gosselin [10], who proved that for each fixed increasing sequence JC = {kn } of natural numbers there is a function I E L1 such that the subsequence of partial sums {S ... (I, z)} diverges unboundedly almost everywhere. Due to A.N.

364 K.I. Oskolkov

Kolmogorov's theorem [16], Fourier-Lebesgue series are convergent in measure. Thus, for each fixed IE Ll there is a subsequence {SA:.(/,z)} of its Fourier sums, which converges a.e. This subsequence certainly depends on I, since the above result of Gosselin shows that there is no subsequence of a.e. convergence, universal for the whole class Ll.

We mention also the following result from [26].

Proposition 7. Let IC = {kn } be an arbitrary increasing sequence ofpositive integers and let fELl. Then the following estimate is valid for almost all z:

SA:.(/,z) = o(logn) (n - 00). (2.23)

Clearly (2.23) generalizes the initial estimate (2.5) due to Lebesgue and also a result of R.A. Hunt [13], who proved that if IC = {kn } satisfies the Hadamard condition (2.21), then

SA: .. (/,z) = o(log log kn ) (n - 00) (2.24)

almost everywhere. However, the idea of the proof of (2.23) in [26] does not essentially differ from that of (2.24) in Hunt's [13]. Its background is the exponential estimate of the distribution of the values of the conjugate function j outside of the "twice enlarged" maximal set of f. This fact is due to R.A. Hunt [13]. We remark that (2.23) positively answers the question put by V. Totik in a private conversation with the author at the International Conference on Constructive Function Theory in Varna (Bulgaria, June 1981).

Still, the main Problems 1 and 2 remain open, and application of the generalized Fejer polynomials (2.10) to Problem 2, and Kolmogorov's functions (2.18) to Problem 1 shows a peculiar duality between them. This duality can be quantitatively expressed in a multiplicative form as a sort of "uncertainty principle," cf. [26, Section 4].

The following result of provisional character illuminates this duality.

Proposition 8. If there ezists a monotonic spectrum of uniform convergence with polynomial density, i.e. (cf. (2.7» such that

log n = O(logdn(lC» (n - 00), (2.25)

then the Lebesgue estimate (2.5) cannot be improved: for each sequence of positive numbers g == {gn} tending to zero as n - 00 there ezists a function fELl such that for almost all z

1. ISn(/, z)1 + lID sup = 00. n_oo gn logn

Conversely, if (2.5) can be improved on the whole class Ll, say, if the conjecture

Sn(/,z) = o(log log n) (n - 00) (2.26)


a.e. is true, then there exists no SUC of polynomial density and (2.26) implies that for each K, = {kn} satisfying (2.25) the corresponding Lebesgue constants (cf. (1.3» can be estimated from below as follows:

I· . f £n(K,) log log n 0 lIDlD > . n-oo logn

We conclude this Section by a few open problems, dealing with the above topics.

Problem 3. Is there any connection between the uniform (mod 1) distribution property of sequences K, = {kn } of positive integers and their property not to be spectra of uniform convergence?

In the above, we mean that K, possesses the uniform (mod 1) distribution property (UDP), if for each irrational number 0 the sequence of fractional parts (knO), n = 1,2, ... is uniformly distributed on [0,1] in the classical sense. That is, for each interval (a, {3) C [0, 1] the number of (knO) belonging to (a,{3) is asymptotically proportional to its length.

Due to the famous result of H. Weyl [44], all polynomial spectra possess the UDP. On the other hand, as it was shown above (Proposition 5), polynomial sequences are not SUC, and moreover, their Lebesgue constants have the exact order of growth log N - the same as in the classical case of all positive integers (see Section 3, Corollary 2). In connection with these facts and the validity of Littlewood's conjecture (see (2.8», which means that there are no rearrangements of the natural order of the set of positive integers which can decrease the order of growth of the Lebesgue constants, the following problem seems to be of some interest.

Problem 4. Is it possible, using rearmngements, to improve the order of growth of the Lebesgue constants of polynomial spectra? In particular, is it true that for each rearmngement {kn } of the set of positive integers the Lebesgue constants of the correspondingly rearmnged squares {k~} satisfy the estimate of Littlewood's type

£N({k~}) > logN?

There is also another aspect of spectral Lebesgue constants which seems not to be touched seriously as yet. Namely, what can be said about the 'norms of operators of Fourier sums on the classes of functions with spectral restrictions, as acting from £1 into £1? We formulate only one particular case of the wide circle ofthese problems. Let K, = {kn } be some spectrum and V(K,) be the set of Lebesgue measurable functions integrable in the pth power whose Fourier coefficients are identically zero outside of K" with the usual £P norm 1I·lIp. Furthermore, let

£N(K, : £1 -+ £1) = sup { t It .. e21ri1c .. 1I: : f E £1(K,), IIfllt $; I} n=1 1

be the corresponding £1 -+ £1 Lebesgue constants.

366 K.I. Oskolkov

Problem 5. Let JC be a polynomial spectrum of exact degree r ~ 2. Is it true that the corresponding L1 ..-+ L1 Lebesgue constants are bounded?

The author is rather inclined to think that this is so. Moreover, it seems very likely that for each polynomial spectrum JC of exact degree r ~ 2 there is a p = p( r) > 1 such that the following imbedding holds true:

(This is similar to the result of W. Rudin: The set JC = {n2 } is not an 84 system i.e. the embedding L2({n2}) C L4 does not hold.)

3 Local and Global Properties of V-continuations

In what follows, Er denotes the real r-dimensional space of vectors 1l. = (zr, ... ,Zl).

Furthermore, we denote by'Pr (r = 1,2, ... ) the class of algebraic polynomials P(~) of degree r with real coefficients and zero constant term (cf. Lemma 2). The following notation will also be adopted:

P(~) = P(~,~ = ~rzr + ... + ~Zl for ~ = (zr,'" ,Zl) E E r ,

to emphasize the dependence of the polynomial P E 'Pr on its coefficients. Clearly, the operator of V-continuation

V: f(z),... L:J(n)e2lfin~ (z EEl) n

-+ V(f,~ ..... L: J(n)e2lfiP(n,:.> (1l. E N) (3.1) n

is unitary, i.e. it preserves the L2-norm of the initial function fez) (say, in the variable Z1, while the "senior" variables Zr," ., Z2 are fixed).

The following observations show that the scale of the spaces U, 1 ~ p ~ 00, is too rough to feel the difference between V-continuations and general unitary operators on the set of Fourier series, i.e. transformations of the type

n n

where e = {en} is an arbitrary sequence of complex numbers with lenl = 1. In the fundamental paper [12], G.H. Hardy and J .E. Littlewood proved

in particular, that if t is a fixed irrational number with bounded partial


quotients (say, t can be any quadratic surd, for example, t = ",'2), then uniformly in the variable z, the following estimate is valid:

It e211'im2f. e211'in:l:I = O(vIn) (n - 00).

m=1

(3.3)

Thus, the L2 and Loo-norms ofthese trigonometric polynomials are equivalent. It follows from (3.3) that if 0 < Q < 1, then the function

(3.4)

belongs to the class Lip Q (in z), i.e.l/(z)-l(y)1 = O(lz-yIQ) (Iz-YI- 0). However, by the definition,

00 e211'in:l:

V(fjt,z)"" E n1/ Ha ' n=1

which is essentially worse than the initial function in neighborhood of the origin z = O.

Furthermore, consider the function

This (cf. [45, Ch. 5, 2]) I belongs to all U for p < 2 (but, of course, not to L2). The V-continuation of I(z) cannot be defined as an ordinary function. In fact, it was proved in [12] that neither of the two series

is Fourier series (they are non-summable by Cesaro or Abel-Poisson methods at every irrational point t). Moreover, it can be proved that lor almost all Jized t, the V -series 01 I,

00

V(fjt, z) ,... E e211'i(n2Hn:l:) Iv'n (3.5) n=1

is divergent in measure on the period z E [0,1) and thus is not a Fourier series.

With this, we finish the discussion of the "bad" properties of V-continua.tions (more details can be found in [21] and [22]j for the proof of the divergence result of (3.5), see [22]) and turn to the "good" ones. The main difficulty with the discrete V-continuation (3.1) lies in the fact that if we

368 K.I. Oskolkov

try to represent this linear operator as a (formal) convolution of the initial function f with some kernel (Green function), i.e. in the form of the integral operator

V(fjZr, ... ,Z2,Zt}-- fal K(zr, ... ,Z2,ZI-e)f(e)de

then we easily see that the Fourier expansion of K has the form

K(... ... ... ) ""' e2lri(nr"r+···+n"1) "'r,···,"'2,"'1 -- L..J . (3.6) n

Even in the case of r = 2 the series on the right of (3.6) (elliptic O-series) is known not to be summable by any regular method of summation, unless Z2 is a rational number (cf. [12]). The finite partial sums ofthe series (3.6), namely

n

W. (... ... ) - ""' e 2l1"i(m r"r+··+mzl) n "'r,···, "'1 - L..J '

m=1

are called H. Weyl's sums. I.M. Vinogradov (cf. [39]) thoroughly investigated the behavior of these sums as a function of r + 1 variables: n : n = 1,2, ... , and ~ : ~ = (zr, ... , zt) E Er. The method of this investigation is known as "Vinogradov's method of exponential sums." Rational approximations of ~ play the main role in that method.

We will call a point Y = (Yr, ... , Y2, yt) E E r rational if its coordinates Y6 are of the form

_ a 6 _ A6 Y6 - - - -, s= 1,2, ... ,r,

q q6 (3.7)

where q6, q are positive integers, a6, A6 are integers, A6 is co-prime with q6 for s = 2, ... , r (but not necessarily for s = 1), and q = [q2, ... , qr] is the least common multiple of q2, ... , qr (note that ql = q). For a fixed q, denote by Rr(q) to be the set of such pointsj the union of Rr(q) over q = 1,2, ... , will be called the set of rational points and is denoted by Rr.

Furthermore, given a rational point y, say y E Rr(q) as in (3.7), we relate to it the following two sums (cf. (3.7»): -

S (aqr , ... , aq1 ) = S{J!) = ~ t e2l1"iP(n,!)

q n=1

= ~ t e2lri(nrtJr+··+ntJl)/q,

q n=1

(3.8)


q-l

= _1_ E e2l1"i(n r 4 r +··+n41)/q cot~. 2qi n=l q

The sums S(y) are called (normalized) complete rational sums or Gaussian sums.

The quantities H(i) obviously are finite discrete Hilbert transforms of the exponentials; we will discuss them later.

In the case r = 2, the complete sums S(y) were introduced by K.F. Gauss. In particular, the moduli of G.aussian sums are defined by the following

classical formulas: if y = (i, i) E R2(q), then

(3.9i)

1 (~~) 1- 1 + (_1)4Q+h S , - flr:: ' q q y~q

·f . Q q I q IS even, = '2. (3.9ii)

Note that the arguments of the Gaussian sums are distributed in a rather complicated manner, which is one of the very mysterious aspects in the analytic number theory. For example, if q is a prime number, then

S (~, 0) = ~ t exp(27rian2 jq) = (~) ~, q q n=l q yq

(3.10)

where (ajq) denotes the so-called Legendre symbol, whose value is +1 or -1, depending on whether a is a quadratic residue or nonresidue modq.

As for the complete rational sums of higher degree r = 3,4, ... , they are still more complicated. Anyhow, their absolute values satisfy L.K. Hua's estimate (for the proof, see [15] and [30)) if J!. E K(q), then

IS(J!.}I~ q-l/r. r

(3.11)

These definitions and preliminaries being made, we start by the following simple lemma, which contains an identity for the V-continuations as functions of the linear variable Xl, while the "senior" variables are fixed rational numbers.

Lemma 3. Let yO = (Yr, ... , Y2, 0) E Rr(q) (cf. (3.7), (3.8» and let, for n = 0,±1, ... ,

1l.n = (Yr, ... , Y2, %) . Then for real Xl, the V -continuation of f can be represented as the discrete convolution:

(3.12)

370 K.I. Oskolkov

The identity (3.12) is proved by checking it for the functions I(x) e2rivz, v=O,±I, ... (cf. [21]).

Relation (3.12) can be interpreted in the following way. IT the senior variables are rational, we can sum up the series on the right hand side of (3.6) (say, by (e, I)-means) and the "Green function" K of the V-continuation is defined in this case as

K(Vr, ... ,V2,X1) = tS(Jt)6 (Xl -~), n=l q

(3.13)

where 6 denotes the I-periodic Dirac delta-function. As it was already stated above, we cannot assign any definite values to K in other situations. This is one of the reasons for the difficulties with the discrete Vcontinuations.

However, if we integrate the series (3.6) formally (termwise) only once in the variable Xl which gives birth to the series

e2ri(nrzr+··+nzl)

H(xr, ... ,xd"'" L 2 . , 7nn n;tO

we get a regular function. The meaning of H for the V-continuations is clear from the following (formal) identity:

V(f;xr , ... , X2, Xl) "'" 1(0) + 11 H(xr , ... , X2, Xl - e) dl(e)· (3.14)

Thus, H is the kernel of the representation of the V-continuations as Stiltjes-convolution. The properties of H, ensuring in particular, existence of the Stiltjes integral (3.14) are contained in the Proposition 11 below.

First we need a description of the continuous analogue of H - the function, defined as the following improper integral:

100 e2ri(>.rZr+···+>'Zl) 1 G(xr , .. . , Xl) = p.v. 2 'A dA = lim ....

-00 71"' :::::!. e'I<A (3.15)

Given an interval w on (0,00) and vector.! = (xr , ... , Xl) E Er, let

I.!I

peA) P(A,~) = A-r Xr + ... + ..\X1; P.(A) = P(..\, I~I);

[(p,e) = i foe e 2riP(>')dA (e > 0),

= d;\. 1 e2riP(>.,£>

1>.IEw A


Proposition 9. (i) The quantities G(w, z) are bounded uniformly in ~ E E r

and intervals we (0,00)

sup sup IG(w,~)1 ~ r. wc(O,oo)~EE'

(3.16)

(ii) If w = (e, A) and g -+ +0, A -+ 00, the limit G~) of these quantities exists everywhere in Er. G(~) can also be represented as an absolutely convergent integral. Let for e > 0

Then

Moreover, for each g > 0

sup [00 ,\-ll.1('\,~ld'\ ~ r. ~EE' Jo

(3.17)

(3.18)

Proof. The boundedness and convergence results of the integral G and more general integral oscillatory Hilbert transforms are not new. They are due to E.M. Stein and S. Wainger [34] (cf. also [33], [35], [40], [41], [42]), where oscillatory integrals are treated in the context of the theory of operators in LP. For the polynomial case, we prove a bit more: (3.17), (3.18), and also investigate the local behavior of G(~) in the neighborhood of ~ = O. For that purpose, the following estimate of oscillatory integral due to I.M. Vinogradov (for the proof, cf. [39, Ch. 2, Lemma 4]) is needed.

Lemma 4. The following estimate is valid

(3.19)

Thus (3.20)

In connection with (3.19) we note that in fact this estimate is of "Chebyshevian nature." Indeed, the following property of algebraic polynomials is crucial. If a polynomial has at least one big coefficient, then it cannot be small on a set of large measure. The estimate of the measure in terms of that "big coefficienf' is achiev~d just by the Lagrange interpolation formula, cf. [39] .

372 K.I. Oskolkov

Moreover, (3.20) immediately follows from (3.19) and the definition of p •.

Lemma 5. Let P,Q E P r and P 'I- 0, Q 'I- 0, R = P - Q,

(XJ e2lriP().) _ e2lriQ().)

u(P, Q) = p.v. Jo A dA. (3.21)

Then the integral is convergent, and the following identity is valid

u(P, Q) = (XJ I(P, A) - I(Q, A) dA, Jo A

(3.22)

where the integral on the right is absolutely convergent and

100 II(P, A) ~ I(Q,A)l dA ~ J~~ (R.(e) + rp.-I/r(e) + rQ;I/r(e»).

(3.23)

Proof. For 0 < e < A < 00, integration by parts gives:

= I(P,A)-I(p,e) + lAA-11(P,A)dA.

Using (3.20) and the elementary inequality I sin ul ~ lui (1m u = 0), we obtain:

II(P,A) - I(Q,A)I ~ min (R.(A),P.-1/r(A) + Q;l/r(A») (A> 0),

and it is easy to see that

l e A-I R.(A)dA ~ R.(e); iOO A-I P.- 1/ r (A)dA ~ rp.-I(e); (e> 0).

Thus, for each e > 0

which completes the proof of (3.23). Proposition 9 is an immediate corollary of Lemma 5. Indeed, in this case

we just take peA) = P(A,~), Q(A) = PC-A). Then we obviously have p.(e) == Q.(e), R(A) = peA) - P( -A) = 2P-(A), where P-(A) denotes the odd part of P, and thus R.(e) ~ 2P.(e); furthermore

:l(A ) = I(P, A) - I(Q, A) ,~ 2ri'

A Class of I.M. Vinogra:dov's Series 373

and thus, by (3.23)

roo 1.1(A,~)ldA~rinf (p-(e)+p.- 1/ r (e»). (3.24) Jo A e>o·

We may assume that P';(A) ¢. 0, since in the opposite case we have G == O. Thus P.-(O) = 0 and, as e increases, so do p.-(e) and p.(e), and there is a point e, in which p-(e) = 1 (since p.-(e) -+ 00) as e -+ 00. This completes the proof of (3.17). The proof of (3.18) follows the same lines.

Remark 1. It immediately follows from Proposition 9 that if P( A) is an odd real polynomial, say of degree r = 2s + 1, then the integral

roo sin P(A) dA Jo A

(3.25)

is convergent, and its absolute value can be estimated as ~ r. One can also easily see that the oddness of P is nonessential in case of this integral. To check it, one should take Q(A) ~ -P(A) in Lemma 4, and apply the same arguments as above.

The first part of the assertions, concerning odd polynomials, has an analogue for the discrete series (see Proposition 11 and Corollary 3). However, the situation with the sine-series of even polynomials is essentially different. The series f: sin2:n2 x

n=l

was investigated by G.H. Hardy and J .E. Littlewood in [12]. It has an everywhere dense and uncountable set of points where it diverges. For example, x = 2/3 is a bad point.

Remark 2. Keeping in mind the applications, we specify here the two special cases, when the function G can be expressed in terms of other known special functions.

(i) If r = 2, G(X2' xt} coincides with the appropriate incomplete Fresnel integral:

(3.26)

(ii) For r = 3, the trace G(X3' 0, Xl) of the function G(X3, X2, Xl) is expressed in terms of the integral of Airy's function Ai(A):

374 K.1. Oskolkov

(3.27)

where 1 (Xl (t3 )

Ai(~) =;: Jo cos "3 +t~ tIt.

(To prove these relations, just differentiate the integrals defining G with respect to ZI. The differentiation under the sign of integral is lawful, since the differentiated integrals converge uniformly on compact subsets of the half-spaces Z2 > 0, Z2 < 0, or resp., Z3 > 0, Z3 < 0.)

In general, the function G( Zr, ... , ZI) is very much "wrinkled" in the neighborhood of ~ = 0 (see Proposition 10, (iii), (iv». It possesses the following symmetry and homogeneity properties:

G( -~ = -OW, G(Q) = 0 (3.28)

where 0 means the complex conjugate of G;

G(zrtr, ... ,ZIt) = G(zr, ... ,zl)signt (t EEl). (3.29)

A description of other global and local properties, which is useful in applications to Schroedinger equations, is contained in the following assertion. (For the proof and more details, see [21]).

Proposition 10. (i) The function GW is Holder continuous everywhere in E'" ezcept for the origin ~ = .Q..

(ii) In each of the open hal/-spaces Zr > 0 and Zr < 0 of the space E'", G is analytic in ~.

(iii) Let 81 take on the odd values on [1, r] and S2 all integer values of this segment; furthermore,

Then IG(~I <: g(~). (3.30)

r

(iv) Let r be an odd number, zr =F 0,

A Clus of I.M. Vinogradov's Series 375

Then

I signzr I G(.~) - 2;- ~ g.(~). (3.31)

Assertions (iii) and (iv) can be used to distinguish between the manifolds in the neighborhood of ~ = 11, along which GW is continuous and resp., discontinuous, as ~ --+ 11. Namely, let us call a nonempty set FeE'" Gregular at a point Zo if ~ is its limit point, and G~ - ~) --+ 0, as ~ --+ ~ and~ E F.IfG(~-~) f+ 0 as~ --+ ~ and~ E F we call F G-nonregularat Zoo Furthermore, a function V(2D, defined in the neighborhood of ~ E Er, is said to be G-continuous at ~ whenever

VW --+ V~) as ~ --+ ~ and G~ - ~) --+ o. Some conditions of G-regularity and, resp., G-nonregularity can be derived from (iii), (iv): if (see (3.30»

g(~ - ~) --+ 0, (~ E F, ~ --+ ~),

then F is G-regular at~. On the other hand, if r is odd and

g.~ -~) --+ 0, ~ E F, ~ --+ zo),

then F is G-nonregular at ~ (see (3.31».

Corollary 1. Let F be a subset of a straight line in E'" passing through the origin and not lying on the hyperplane Zr = 0, and let ~ = 0 be a limit point of F. Then F is G-regular if r is even and G-nonregular if r is odd.

Qualitatively, the condition gW --+ 0 ~ E F, ~ --+ Q) being sufficient for G-regularity, means that in a neighborhood of the origin the set F possesses "dominating even coordinates." Thus, if r = 2 the condition

(3.32)

is necessary and sufficient (cf. (3.26) and" resp., (3.30» for G-regularity of F at ~ = 11. In particular, all the rectilinear rays, with Z2 :F 0, entering the origin, are good. Furthermore, if r = 3, the following conditions ~ = (zs, Z2, Zl) E ES)

Izdlz21-1/2 + IZsIIz21-s/2 --+ 0 (~E F, ~ --+!l)

and IZlllzsl-1/S + IZ211zsl-2/S --+ 0 (.~ E F, *- --+ 0)

are sufficient for G-regularity and, resp., G-nonregularity of F at the origin. We also note that G-regular sets exist also in any odd subspace of di

mension ~ 2 in Er i.e. such that Z2 = Z4 = ... = 0 (cf. [24]). By the above corollary, such subspaces are very "poor" in the sense G-regular sets at

376 K.I. Oskolkov

2!0 = Q: each of such sets is necessarily rather twisted - it cannot contain at all any rectilinear rays entering the origin. For example, such is the trace G(za,O,zl) (cf. (3.27» ofthe function G(Za,Z2,Zt). In this case G-regular sets are concentrated in "sharp horns" around the cubic parabola

(3.33)

where e is the (real) root of the equation

1000 sin2'11"(~a +e~) d~ = 0.

(This equation possesses at least one negative root; the uniqueness is not clear to the author).

Now we return to the discrete case. Given a polynomial P(~,.a:.) = ~r Zr + ... + ~Zl E Pr , introduce the following quantities:

e211'iP(n,£>

2'11"in

N = .!.. ~ e211'iP(n,£>

N L.J ' n=l

Proposition 11. (i) The quantities HNW are bounded uniformly in N = 1,2, ... and.a:. E E":

(3.34)

(ii) As N -+ 00, the limit

exists everywhere in E" . Moreover, the appropriate symmetric A bel transformation makes the series defining H(.a:.) converge absolutely;

H( ) = ~ Tn(.a:.) £ L.J n +l'

n=l

~ Tn(z) where sup L.J --=-1 < 00,

1!,EEr n=l n + (3.35)

and Tn (.a:.) -+ ° (n -+ 00) at each point £ E E" . Moreover, for each e > ° (3.36)


(iii) At each irrational point of Er, i. e. ~ E Er \ Ii! the function H is continuous. Furthe1'fJlore, H(~) is discontinuous at all those (and only those) points lI. E Ii! , where the corresponding complete rational sum S0!) is nonzero. In the neighborhood of each of these points, the following asymptotic formula is valid (cf. (3.15»

H{J!.+ ~ - H(lI.) = S(lI.)GW + 0(1) (,! - ill (3.37)

and in particular HW-GW-O ~-O). (3.38)

Therefore, H(gz) is G-continuous everywhere in E r and continuous in ordinary sense also at the points y E Ii!, where the complete rational sums S(JJJ are zero. -

Following the lines of [2], we prove here the assertions (i) and (ii) (for (iii), cf. [21]).

Remark 3. An independent proof of assertion (i), (also using Vinogradov's method) was found by E.M. Stein and S. Wainger in 1988 (personal communication). The author uses this opportunity to thank Professor Stein and Professor Wainger for their invitation, hospitality, and useful discussion of the subject during his visit to the USA in the spring of 1990.

First of all, in the sums HN(~) we sum over positive n. Then for a given vector ~ = (zr,"" Z1) E W, setting

* (* *) ~ = Zr"'" Z1 , where z: = (-1Yz.,

we see that

(3.39)

(3.40)

It is plain, but will be essential for us (cf. (3.53» that if y E Rr , then the corresponding complete sums at y and y coincide: -- .....

(3.41)

(This immediately follows from (3.39), since along with n, -n runs through the same complete system of residues mod (q) in the definition of S.

Obviously ITn(~)1 ~ 1/1r, and it follows from (3.40) that to establish (i), (ii) it suffices to prove the second of the relations (3.35) and the pointwise convergence Tn(~) - O(n - 00; ~ E ~).

The latter fact follows from (3.39) and (3.41). Indeed, if the point ~ is rational and in Rr, say in Ii!(q), then both P(n,~) and P(n,~*) are periodic in n of period q, and it is easy to see that

378 K.I. Oskolkov

and thus T,.(~) -+ 0 (n -+ 00) by (3.41). On the other hand, if ~ E Er \ Rr, then both s,.(~) and S,.(!:*) tend to 0 as n -+ 00, due to a theorem of H. Weyl [44].

Given a rational point Il E K denote by q(W the denominator of its representation (3.7).

In accordance with Vinogradov's method (it is convenient to use this method from the exposition [1]), for a given natural n we split E r into two classes, relative to the approximation of its points!: by the rational points y. To the first class, which we denote (FC),., we allot those points ~ E E r

;hich admit the representation

(3.42)

All the remaining points of E r we allot to the second class (SC),., i.e. (SC),. = Er \ (FC),.. If n is sufficiently large, say

n > no = 210 = 1024 (3.43)

and ~ E (FC),., then the representation (3.42) is unique. Indeed, assume the contrary. Then there are two distinct points y1 and y2 in K such that, due to (3.42), for the denominators q1 = q(y1), q; = q(y2), and the vectors of errors ~1 = (z;, ... , zf), ~2 = (z;, ... , zf) we have -

max (ql,q2) ~ nO.3, max (l"z:l, Iz;1) ~ nO.3- 6 , (8 = 1, ... ,r). (3.44)

By the assumption y1 :I y2, we see that there exists 8 with 1 ~ 8 ~ r such that - -

_1_ ~ la! _ a~ 1= Iz: - z;1 ~ 2nO.3- 6 •

q1q2 q1 q2 (3.45)

Since 8 2: 1, it follows from (3.45) that q1 q2 2: 0.5nO.7 , and this, under the condition (3.43), contradicts the first of the estimates (3.44), which implies q1q2 ~ nO.6. Everywhere in the sequel we shall assume that the natural numbers n, m and N are larger than no = 1024j for the smaller values it is sufficient to use the trivial estimate IT,.(~)I ~ 1/7r.

Given!: E E r , let

N1(~) = {n : n > no, ~ E (FC),.}, N2(~) = {n : n > no, ~ E (SC),.}.

If N1 (!:) is nonempty, denote Y(!:) = {y1, y2, ... } the collection of distinct rational points y in the representation (3.42), taken successively under the increase of the natural number n on the set N1(~). Note that if ~ ERr, then this collection is finite. We furthermore set qj = q(t), ~j = ~ - t and for a fixed j, denote by Wj (~) the longest segment of the set of integers n > no on which the vector Il E K defined by (3.42) remains unchanged and coincides with t. Obviously, we have

N1(~) = UWj(~)j Wj(~) () -Wk(~) = 0, t:l Ilk (j:l k). (3.46) j


As above (cf. (3.45», let us show that the denominators qj are also distinct and grow very fast:

(3.47)

Indeed, let n and m be natural numbers with n E Wj(~), m E Wj+l(~). Since (cf. (3.46» vi "I t+1, as in (3.45) we see that for some s, 1 ~ s ~ r,

In view of (3.42),

1 ..+1 -- ~ Iz~ -z~ I. qjqj+l

q . < nO.3 n~ < mO.3 • J _ , "2,1+1 - ,

Izjl < nO.3-. < n-O.7 . Izj+1 1 < m-O. 7 8 __ , 6 _ ,

so that, taking into account that m > n, we obtain from (3.48):

hence (3.47) follows.

O 5 0.7 > 0 5 7/3. 9.jqj+1 ~ . n _. qj ,

(3.48)

Now we present (in the form of lemmas) those estimates on exponential sums of H. Weyl, which are used in the proof.

Lemma 6. (Vinogradov [39]). If r ~ 3, n E N2(~)' i.e. for n under consideration the point x belongs to the second class (SC)n, then

ISn~)1 ~ n-P, p = (8r2(logr + 1.5 log log r + 4.2»-1. (3.49) r

For the proof, see [1, Lemma 7]. (Note that for our purpose the value of p is not essential- it would suffice to have some p = p(r) > 0).

Lemma 7. (Vinogradov [39]). Let n E Nl (~), i.e. for n under consideration the point lZ. belongs to the first class (FC)n, and let y and!. be defined by (3.42) and q = q(y). Then the following estimate (asymptotic formula) is valid -

(3.50)

where

and

For the proof, see [1, Lemma 6]. The following estimates are immediate corollaries of (3.49), (3.50), (3.39),

and (3.41) (cf. also Proposition 9),

(3.51)

380 K.1. Oskolkov

ITn(~) - S(lLj)3(n'~j)l~ n-O.7 (n E Nl(~)'

and the latter estimate can obviously be substituted by

I Tn(~ _ S(.,.)3(>"~i) I ~ n-1.7 n+l \l!..J >. r

(3.52)

if>. E [n, n + 1], n E Wj(~). Now we apply Hua's estimate (3.11) and also (3.17) to obtain the following estimate for ~ in the first class:

(3.53)

nEwj(£)

Sin~e by (3.48) the denominators qj grow very rapidly, the assertion (3.35) (and also 3.36), cf. (3.18)) follow from the estimates (3.51), (3.53).

Remark 4. If y = (T,···, 7) E. K(q), then the sum of the series, defining H, can be represented ill the form of finite Hilbert transform as in (3.8).

Corollary 2. The Lebesgue constants of any polynomial spectrum have the exact order of growth 10gN as N -+ 00.

This is an immediate consequence of (3.34) and Lemma 1.

Corollary 3 (also see Remark 1). (i) Let Q(>.) be an odd algebraic polynomial with real coefficients. Then the series

t sinQ(n)

n=l n

is convergent and the sequence of its partial sums is bounded by a number, which depends only on the degree of Q.

In particular, if r is an odd number and t is a real variable, then the sum of the series

(3.54)

is everywhere bounded and continuous at each irrational t. On the other hand, if t is rational, say, t = a/q, where q = 1,2, ... and a = 0, ±1, ... , (a, q) = 1, and if the corresponding complete rational sum

S ( a) 1 ~ (2?rianr) 1 ~ 2?ranr r - = - L...Jexp --- = - L...Jcos--

q q n=l q q n=l q (3.55)


is non-zero, then b has a jump at t:

!~ (b (~ + T) - b (~ ) ) sign T = 21r Sr (~) . (3.56)

(In connection with (3.55), it should be noted that the sin-part of the sums Sr is always zero, if r is odd, cf. (3.41)).

A good set of concrete examples, when the complete sums can be found explicitly, is when q is a power ~ 2 of prime number p, and r:t 0 (modp). Namely,

Sr (~) = -2-1 , S = 2,3, .... P p8-(3.57)

In any case, the set of points, where b(t) is discontinuous, is everywhere dense on the period.

Relation (3.56) follows from the assertion (iii) and property (3.31) of the function G(~).

Corollary 4. The oscillatory Hilbert matrix

eiP(n-m) Hnm=----, n-m

n,m = O,±l,... (n::f= m),

where P is an algebraic polynomial with real coefficients, is a bounded operator £2 _£2, whose norm can be estimated independently of the coefficients olP.

In other words, the linear transformation

eiP(m-n) a - b: bn = L am, n = O,±l, ... ,

m;tn m-n

possesses the property:

where the factor Cr is finite and depends only on the degree r of P.

(3.58i)

Next we consider local and global properties of V-continuations of more general functions. Let VN(J,~) denote symmetric partial sums of the Vseries of I:

Also let

VN(J,~) = L j(n)e21tiP(n,!IJ (N = 1,2, ... ). Inl$N

Tn(J,~) = n- 1 L Imlj(m)e21tiP(m,!IJ (n = 1,2, ... ) Iml$n

382 K.1. Oskolkov

and denote by BV the class of all complex-valued periodic, of period 1, functions fez), possessing bounded variation on the period [0,1). Let v(l) = var (I; [0, 1» + sup(lfl; [0, 1».

Proposition 12. Let fEB V. Then: (i) The sequence VN(J,~is uniformly bounded:

and its pointwise limit V(J,~) exists everywhere in Er. The function V(J, ~ can be represented as a sum of everywhere absolutely convergent series

V(J,~ = 1(0) + f: Tn(J,~ n=1 n + 1

(ii) The set of discontinuities of V(J'~ in V is countable. If, in addition to the main condition f E BV, we require the continuity

of f, then V(J, -z) is conti~uous everywhere in V. (iii) At each point ~ E V, where V(I,~ is discontinuous, it is still Gcontinuous, and in particular it is G-continuous everywhere in V. Furthermore, V(J,~ is continuous at each point ~ E V, where at least one of the senior coordinates Zr, ... ,X2 is irrational.

This assertion is essentially a corollary from Proposition 11, which corresponds to the basic case of V -continuation of the Bernoulli kernel of the first order:

H~)=V(B,~,

cf. also (3.4). For details of the proof, in particular, connected with (3.14), see [21].

Corollary 5. Let w be an interval on the real axis, of the length Iwl $ 1 and let the points Jt ' n = 0, ±1, ... , be the same as in Lemma 3. Then

L S(Jl') <: 1. n:qnEw r

(3.58ii)

This assertion is a consequence of the identity (3.12) and proposition 12 - ir. the capacity of f one should take the characteristic function of w. The estimate (3.58ii) shows that S0t) essentially interfere, since we also have

q q

L s0t) = L IS{Jt)12 = 1 n=1 n=1

(cf. also (3.11».


The case r = 2 is special, and in addition to (3.12) there are other identities for the V -continuations of the second degree. They allow, in particular, to obtain estimates for the incomplete Gaussian sums (see Proposition 3 in the Introduction and Proposition 14 below). For the proofs, see [23].

Lemma 8. Let f(z) be square-integrable over the period [0,1) and let Z2, Zl be real numbers, Z2 i= O. Then as A -+ 00,

e7ri/ 41 e-7ri'>'~f('\Z2+Zt)ci'\- E j(n)e"'i(n~"h2n.,.) -+ O. (3.60) I'>'I:S;A n:ln"~I:S;A

In particular, if f(z) is of bounded variation over the period, then both the integral and the series converge in Cauchy principle value sense and

p.v. e7ri/ 4 I: e-7ri'>'~ f('\Z2 + zt)d'\ = (3.61)

2 = p.v. 2: j(n)e27ri(n~"~+2n"d = V(fj Z; ,zt).

n

(This can be interpreted as a variance of Poisson Summation formula).

Lemma 9. Let q be an odd natural number, Jl. = (i, i) E R2(q), a' the unique (mod q) solution of the congruence aa' == 1 (mod q), and

y' = _ (4a)', (2a)'b) . - q q

Furthermore, let f(z), for all real z, be equal to the sum of its Fourier series, which is assumed to converge everywhere. Then

! t f (~) exp (21ri(an2 + bn») = S(JL)V(fj'i). q n=l q q

(3.62)

In particular, if 0 < a ::; 1 and

W (a) _" (21rian2) a--L..Jexp , q O:S;n:S;qa q

S ( a) _ 1 ~ (21rian2) - -- L..Jexp , q q n=l q

then

(3.63)

In accordance with the assumption on f(z), in the sum Wa the first summand and the last one, are taken with the factor 1/2 if aq is an integer. The bar in H means the complex conjugate of H. For details concerning even denominators see [23].

384 K.I. Oskolkov

The identity (3.62) explains the interrelation between the estimate (1.6) of Proposition 3 and Proposition 4, which is the special case of Proposition 11 when r = 2. In fact, it follows from (3.62), that the boundedness of the function H(:l:2' :1:1) (at least on the set R2) is equivalent to the estimate (1.6) of incomplete Gaussian sums.

Furthermore, (3.60) and (3.61) imply that

~ ~ f (~) exp (21ri;n2) = S (~) exp (:i) p.v.l: e-ri~2 f(>.e)d>',

(3.64) where e is an arbitrary (real) solution of the congruence

(3.65)

In particular, for odd q, the incomplete Gaussian sums

(a ) ~ (21rian2) W -,W = L..J exp , q neqw q

where w is an interval on the real axis of length Iwl ~ 1, can be expressed as the product of the complete sum and the Fresnel integral over appropriate periodic interval:

w (~jw) = qS (~) e ri/ 4 p.v.l e-ri~2 d>.. q q ~~ew

(3.66)

In (3.65), e is an arbitrary solution of (3.64) and

W = {y : y = k +:1:, k = 0, ±1, ... ,:I: E w}.

As for the estimate (1.6) itself, it is due to G.H. Hardy and J .E. Littlewood [12]. Although it has not been explicitly emphasized in [12], it is a corollary of iterative application of the approximate functional equation which was discovered in that paper for the sums

n

W~(:l:2' :1:1) = L: e2ri(m2a:2+2ma:l).

m=1

Those iterations, based on continued fractions of :1:2, were carried out in [12, pp. 212-213]. E.C. Titchmarch commented on [12] in [11] and presented (cf. [11, pp. 113-114] a more detailed estimate of incomplete Gaussian sums in terms of denominators of continued fractions for :1:2. That comment implies (1.6) as a particular case. We note also [7], devoted to the asymptotical formula ofI.M. Vinogradov's type (cf. Lemma 7) for the sums W~ ; [7] contains


the proof of (1.6), using iterations of the above mentioned functional equation, which is also derived in [7]. On Figure 1, the biography of Gaussian sum is plotted, corresponding to i = 5l0~. Namely, the complex numbers

~ (21ri ·11· m2) Zn = ~exp 503 ,n= 1, ... ,503,

are computed in succession and plotted on the plane (x,y), x = Rez, y = Imz. The author is indebted to Ivonne Nagel, Irene Tyuleneva, Robert Sharpley, and Sherman Riemenschneider for computation and plotting of this and forthcoming graphs.

Remark 5. E.M. Stein showed the author how to derive the boundedness of the finite transforms HN(X2' Xl) and thus the boundedness of H(X2' Xl) and also (1.6), using L. Carleson's theorem [4] on a.e. convergence of Fourier series. Namely, the discrete version of the strong type (2.2)-estimate for the operator of maximal Hilbert transform

I ameimB I An a = max () BEE' ~ m- n+0.5 '

is sufficient and this fact was proved in [20]. However, this argument seems to work only for r = 2. Thus, it is natural to also try the maximal Hilbert transforms

a eiP(m) I An(a) = A}:")(a) = sup L m ,

PE'Pr m - n + 0.5 m

where for each fixed n the sup is taken over all algebraic polynomials of degree r, with real coefficients. (If it is true that A(r) is of strong type (2.2), that would mean a generalization of the result on uniform boundedness of HN of degree r + 1).

Given a q = 1,2, ... and a point!!. = (i, i) E R2(q), denote by N(!!.) the following rectangular neighborhood of!!.:

In the case of r = 2, the assertion (iii) of Proposition 11 and Proposition 12 can be complemented as follows (for the proof, see [23]).

Proposition 13. (i) Let q = 1,2, ... , !!. = (Y1, Y2) E R2(q), and let ~ = (Z2' zt), ( = «(2, (t) E E2 be such that the points ~ = Y +!. and ~' = Y + ( belong to-N(!!.). Then the following asymptotic formul; holds true: - -

1 (H(!!. +~) - H(!!. + Q) - S(!f.}(G(~) - G(~))I <

~ ql/2(lz2I l / 2 + 1(211/2 + IZ1 - (11),

(3.67)

386 K.I. Oskolkov

where G(&:.) = G(Z2, zt} denotes the incomplete Fresnel integral (3.26)

G(Z2, zt} = sign Zle¥ 1~ e-1rU2signz2dA (e = Iz1l (2Iz21)-1/2, Z2 #: 0);

G(O, zt} = ! sign Zl (Zl #: 0); G(O,O) = o.

(ii) For each fixed X2 the function H(X2, Xl) is of weakly bounded 2-variation over the period [0,1) and this property is uniform in X2:

supwar2(H(X2,·) : [0,1» < 00. (3.68) 1&2

In (ii), we made use of the following definition. Let h(x) be some complexvalued and bounded function of the real variable X on an interval w, and let

osc(h,w) = sup{lh(x) - h(y)1 : x,y E w}.

Let -y > 1 and consider collections n = {w",} of nonoverlap ping subintervals w'" ofw. If

sup L osc-Y(h,w",) < 00,

n k

the function is said to be of (strongly) bounded -y-variation on Wj we use the notation var-y(h,w) for the value of the sup at the righthand side. Furthermore, fix a collection n and a positive e, and count the number (denoted by card) of those Wk En, for which oSC(h,Wk) > e. If

supsupe-Ycard{wk En, oSC(h,Wk) > e} < 00 n ">0

we say that h is of weakly bounded -y-variation on w, and denote war-y( h, w) the value of the double sup ab9ve. The notion of (strong) -y-variation was introduced by N. Wiener, and usefulness of -y-variations in Fourier analysis has been thoroughly studied, cf. e.g. [3, Ch. 4]. It is easy to see that for 1::5-y<6

war-y(h,w) ::5 var-y(h,w);

so that if, for example, h is of weakly bounded 2-variation, then it is of strongly bounded -y-variation for each -y > 2.

Corollary 6. Let n = {Wk} be an arbitrary set of nonoverlapping intervals on [0,1); q = 1,2, ... ; a = 0, ±1, ... , (a, q) = 1. Then the following estimate holds true for the incomplete Gaussian sums which correspond to the intervals Wk:

(3.69)

A Class of 1.M. Vinogradov's Series 387

(w (~'W) = n~ e~) . This follows from (3.67) and the identities, expressing the incomplete sums in terms of H(Z:2' z:I), cf. (3.62) and [23] for the case of even q.

Remark 6. One cannot take strong instead of weak 2-variation in (3.67), and it can also be checked that

q-l 2 2 ([fq ffl) 211"in Ie Ie + 1 E E exp (--) ::> qlogq, WI< = 2' -2- . 1<=1 nEqwlo q q q

Corollary 7. Let f(z:) be of bounded ordinary variation on the period, (i.e. f e BV, or Vall(f, [0,1) < (0). Then for each "'{ > 2 and fixed X2, the Vcontinuation V(fj Z:2, Z:l) is of strongly bounded "'{-variation in the variable Z:l and this property is uniform in Z:2:

sup var'Y(V(fj Z:2, .), [0,1» < 00 ("'{ > 2). (3.70) "':I

4 Applications to Schroedinger Type Equations and Quantum Chaos

Let D"" D t denote the differential operators

1 8 D"'=-2 ·-8 ' 11"' z:

1 8 Dt = 211"i at'

where z: and t are real variables (z: - one dimensional space coordinate and t - time).

We discuss applications of V-series to investigation of global and local properties of the solutions to Cauchy initial value problem for linear Schroedinger type equations with periodic initial data.

The applications are straightforward, if the coefficients of the space differential operator are constant or depend only on time t.

Namely, let ar(t), ar_l(t), ... , al(t), ao(t) be some functions defined on an interval [-T,71, T> 0, integrable on [-T,71, and let ar(t), .. . , al(t) take on only real values (ao(t) may be complex-valued). Introduce the differential operator in the space variable x

and consider the following Cauchy problem with respect to the unknown function 't" = 't"(z:, t):

Dt't" = L(t, Dt)'t", 1J(z:,0) = f(z:). (4.1)

388 K.I. Oskolkov

In (4.1) the initial function is complex-valued periodic (of period 1) and Lebesgue integrable over the period [0,1).

Then using Fourier method of separation of variables, we easily see that the (generalized) solution to (4.1) is represented by the series

W(:c,t)....., Ei(n)e2lrin.,. e2lri(nrAr(t)+ .. +nAl(t)+Ao(t» , (4.2) n

with

A.(r) = iT a.(t)dt (s=O, ... ,r).

It follows that the solution to (4.1) is in fact the trace of the general Vcontinuation (of degree r) of the initial function f on the two dimensional manifold in E'" corresponding to the coefficients a of the space operator L(t, D.,):

e-2lriAO(t)W(:C, t) = V(J; :Cr,···,:Cd IA(."t), (4.3)

A(:c, t) = {(:cr , ... ,:cd : :Cr = Ar(t), ... ,:C2 = A2(t), Xl = X + Al(t)}.

Therefore, the following assertion is an immediate corollary of Proposition 12 in the previous section. (Remember that we keep the most important restriction, that the coefficients al (t), ... ,ar(t) should be real-valued).

Proposition 14. If the initial function f(x) of the problem (4.1) is of bounded variation over the period [0,1), then the generalized solution w(x, t) exists in the class of regular functions in the strip It I ~ T, Ixl < 00, and moreover

sup sup le-2lriAo(t)W(x, t)1 ~ Cr(sup If I + varl(J, [0, 1))), (4.4) ItlET .,

where the factor Cr depends only on the order r of the operator L. If, in addition, f is continuous, then W(x, t) is everywhere continuous.

Remark 7. It is easy to understand what conditions should be imposed on the initial function f(x), to ensure the existence ofthe classical solution to (4.1). It follows from the Proposition 14 that such a sufficient condition is: the r-th derivative f(r)(x) should be a continuous function of bounded variation over the period.

Consider the following two examples of the problems (4.1) corresponding to the equations

(4.5)

and oW 1 03W 7ft = - 411"2 ox3 •

(4.6)

(4.5) is a time dependent Schroedinger equation of a free particle, and (4.6) a degenerated Korteweg-de Vries equation.

A Class of 1M. Vinogradov's Series 389

The solutions are defined in these cases, resp., by the series

"I)(z, t) "" 2: j(n)e2I1"i(n2f+nll:), (4.5i) n

that is, the V-series of second degree of the initial function, and

"I)(z, t) "" 2: j(n)e2I1"i(n3f+nll:), (4.6i) n

the trace of the V-series of third degree on the plane Z2 = 0. Figures 2 Re, 21m, and 3 illustrate the 3-dimensional graphs of the so

lutions to resp., (4.5) and (4.6) in the case, when the initial function fez) is the Bernoulli kernel of the first order:

1 e211"inll:

fez) = 2 - {z} = p.v. 2: -2-·-· n'jll!O 1!"1n

In this case, the solution to (4.5) is

e2I1"i(n2t+nll:)

"I)(z, t) = H(t, z) = p.v. 2: 2. , ?ran

n'jll!O

(4.7)

( 4.5ii)

and Fig.2 Re corresponds to the real, while Fig.2 1m corresponds to the imaginary part of "I).

Furthermore, the solution to the equation (4.6) with the same initial function equals

e2I1"i(n3t+nll:)

"I)(z, t) = H(t, 0, z) = p.v. 2: 2. = n'jll!O ?ran

(4.6ii)

= E sin2?r(n3t + nz), n=l ?rn

and one can get an impression of its graph from Fig. 3. The values are computed at the rational points (t,z) = (i,i), where q = 97 and a,b = 0, ... , q - 1 using the second of the relations (3.8).

In both cases, the pictures look rather chaotic and oscillatory, although the initial function (4.7) has only one discontinuity of the first kind on the period. This is not surprising since both of the functions "I) are traces of the function H, whose character is very complicated, due to the assertion (iii) of Proposition 11.

The laws which regulate this chaos follow the general asymptotic formula (3.37) (in the case of Schroedinger equation (4.5), a more detailed description follows (3.67)). Anyhow, the most striking feature of this chaos is the so-called property of self-similarity, which also follows from (3.37),

390 K.I. Oskolkov

(3.38). Namely, the increments of the function H (and also the solutions W) in the neighborhoods of all rational points reproduce those of H in the neighborhood of the origin. This reproduction is scaled by the values of the corresponding complete rational sums S0i):

H0!. +~) - H0!.) = S0!.}H(~) + 0(1) (~-+ .0.),

H(&J = G(Z) + 0(1) (~ -+ .0.).

(4.7ii)

In other words there are two constituent parts in this chaos: 1) the "messy" one, for which the sums S(y) are responsible and 2) the "regular" one, controlled by the special function G(&:) , cf.(3.15) and also (3.26), (3.27). However, there is a considerable difference in dependence on the time t of (4.5ii) and (4.6ii) and more general of the solution (4.5i) and (4.6i) with the initial function f of bounded variation, which can also possess discontinuities of the first kind. (In the latter case, we make a natural assumption that f(x) = 0.5(1(x - 0) + f(x + 0) for all x. Below we assume that f is of bounded variation over the period.)

1. Schroedinger equation of a free particle (4.5)

(li) The solution 'J!(x, t) to (4.5) is continuous in both variables at all points (x,t) with irrational t.

(Iii) If f(x) has at least one discontinuity on the period, then each straight line of the form (x, t), where t is a fixed rational number and x E [0,1), necessarily contains discontinuities ofw(x, t).

(liii) The trace of w(x, t) on each straight line, nonparallel to the x-axis, is continuous. In particular, for each fixed x, the evolution of w( x, t) is continuous in t, and the initial condition f(x) can be understood as pointwise limit relation, which is true everywhere on the initial straight line t = 0 as long as nontangential approaches are applied. Thus, there is no chaos in the time-dependence.

2. Degenerated Korteweg - de Vries equation (4.6) Here, the conclusions (li) and (Iii) are still valid, but in contrast to (liii), all straight lines, even parallel to the time-axis x = 0, are "dangerous." For example, the solution (4.6ii) for x = 0 is represented by the series

- the particular case of the functions (3.54) (r = 3). It has an everywhere dense set of discontinuities ofthe first kind and evolution in time is chaotic; the property of self-similarity of 'J!(O, t) is also there, cf. (3.56).

Remark 8. We note that chaotic features of the solutions to quantummechanical equations are being very intensively studied, cf. e.g. [28]. In


these investigations, some classical functions in analytic number theory, such as Riemann zeta-function, also appear. Namely, D.M. Wardlaw and W. Jaworski [43] study the explicit connections which exist between the zeros of the zeta-function from one side and the scattering matrix and time delay, from the other, for the system which consists of a quantum particle moving on a two-dimensional surface of constant negative curvature.

From what was said above, it follows that quantum chaos, in particular, self-similarity is a typical property of the solution to even the simplest evolutionary equations of Schroedinger type like (4.5) and (4.6) whenever the initial function is periodic.

In conclusion, we briefly sketch a more complicated situation, where the coefficients of the space operator L in the Cauchy problem (4.1) depend on both variables t and z and are periodic in z. (For details, cf. [22]). Consider the following differential operator

L(D:c) = L(z,t, D:c) = D; + ar_l(z,t)D;-l + ... + ao(z,t)

where the coefficients a,(z,t) are complex-valued sufficiently smooth functions in a strip It I ~ T, z E (-00,00) and periodic in z:

a,(z + 1, t) == a.(z, t).

(Note that here, the leading coefficient in L equals 1). For the operator L, the same Cauchy problem as in (4.1) is posed: find the function w(z,t), satisfying the relations

Dtll = L(z,t,D:c)W, lI(z, 0) = fez) (f(z+ 1) == f(z». (4.8)

Certainly as above, we are looking for a generalized solution to (4.8), and the existence problem is not at all trivial.

In this new situation the direct Fourier method of separation of variables does not work, and a proper substitute must be found. The existence problem, as well as the definition of what actually a Schroedinger type equation is, essentially depend on certain (in general, non-linear) momentum conditions (cf. (4.13» which should be imposed on the coefficients. The idea is to somehow "approximate the equation (4.8)" by another one, in which the coefficients of the new space differential operator depend only on the time t, i.e. by an equation of the type (4.1). This is done using asymptotical methods, which can be briefly described as follows.

Let D;l denote the "inverse" operator to D:c: if a(z,t) is periodic in z, of period 1, and its Fourier expansion in z is

a(z, t) ,... L a(n, t)e211'in:c,

n

then define

392 K.I. Oskolkov

Let h(z, t) = e-(,.D.,)-ltJr_1(..:,t), w(z, t) = h(z, t).p(z, t)

where .p(z, t) is the new unknown function. Then (4.8) is reduced to the new problem for .p of the same type:

Dt.p = L(z,t,D,,:).p, tfJ(z,O) = l(z) (4.8i)

(with a new initial function 1(z». It can be checked that the coefficient a,.-1 of the modified differential operator does not depend on the space variable z and equals just the mean value of a,._l(z,t) over the period

So without any loss of generality at this point, we can assume that the coefficient a,._l is in fact a function only of t . Let for n = 0, ±1, ...

en(z, t) = e2 ... in..: . e2 ... i(nr Ar(t)+ .. +nA1(t)+Ao(t», (4.9i)

where

A.(t) = 1t a.( T)dT, s = 0, ... , r,

with a,.(T) == 1, a,._l(T) == a,._l(T) and let Fo(z,t) == 1,

Fn(z,t) = en(z,t) (1 + {h~,t) + ... + fJ,.~~~~,t»)

(4.9ii)

( 4.9iii)

and the functions a, fJ do not depend on n and are to be determined. If we apply the differential operator (Dt - L(z, t, D..:» to the function

Fn(z, t) (note that a,.-l depends on t only), we get the expression of the form

(Dt - L(z,t,D..:»Fn(z,t) = (4.9)

( ) ( ,.-1 () () 1-1 (z, t) 11-,.(Z, t») =en z,t n 1,.-1 z,t +···+10 z,t + + ... + 1 ' n n"-

i.e. in the general case positive powers of n are also present in this expansion.

The "asymptotical idea" is to find the functions a,._2(t), ... , ao(t) depending only on t and the functions fJ1(Z, t), . .. ,fJ,.-l (z, t) periodic in z : fJj(z + 1,t) == fJj(z,t), in such a way that in (4.9) all nonnegative powers of n are identically zero, that is

1,._1(Z,t) = 1,._2(Z,t) = ... = 10(Z,t) == 0. (4.10)

Clearly, (4.10) is a system of equations with respect to the unknown functions a,._2(t), ... ,ao(t) and fJ1(Z,t), ... ,fJ,._1(Z,t). That this system is solvable and the solutions a,._2(t), ... , ao(t) are unique, is shown in [22].

A Class of 1M. Vinogradov's Series 393

After this step, the solution to (4.8i) is (roughly speaking) searched in the form ofthe expansion with respect to the system {Fn}:

<p(Z, t) '" LYn(t)Fn(z, t). (4.11) n

(In fact, a modified system {Fn} is applied, defined by Fn(z,t) = en(z,t), if Inl ::; N, and Fn(z,t) = Fn(z,t), for Inl > N, where N is a sufficiently large number.)

For this schedule to work, it is necessary to see that {Fn} is a representation system, namely Riesz basis in the variable z for each fixed t. Due to the definition (4.9) for this one must first of all implore, that the integmls Ar(t), Ar-1 (t), ... , A1 (t) be real-valued (Ao(t) may admit complex values). Thus we require that the functions a r -1 (t), ... , a1 (t) must be real-valued:

Ima.(t) : 0, s = 1, ... , r - 1. (4.12)

This system of equations is exactly the condition that L(z, t, DII:) is an operator of Schroedinger type. For the operator of order r = 2,

L(z, t, DII:) = D~ + a1(z, t)DII: + ao(z, t),

the conditions (4.12) are reduced to the single and simple requirement that

The corresponding a, f3 are defined by the relations

ao(t) = 11 ao(z, t)dz + ~ ( a~(t) -11 a~(z, t)dZ) ,

Ao(t) = 1t ao( T)dT;

f31(Z, t) = iD;laHz,t)+i(a1(Z, t)-a1(t»-~D;lao(z, t)-iD;2 Dta1(Z, t).

For the operator of the third order

L(z,t, DII:) = D! + a2(z,t)D~ + a1(z,t)DII: + ao(z,t)

the conditions (4.12) are equivalent to the following two requirements

Im{11 a2(z,t)dZ} :0, Im{1\a~(z,t)-3a1(z,t»dz} :0,

and note that the second one is non-linear.

394 K.1. Oskolkov

In this way, the Cauchy problem (4.8i) is reduced to the following one, posed for an infinite linear system of ordinary differential equations:

00

fin = -21ri :L: r m,n(t)Ym; Yn(O) = in (n = 0, ±1, ... ), m=-oo

where r n,m(t) and resp. in, are defined as the coefficients of the following expansions (cf. (4.10), (4.8i), (4.9»

00

(Dt-L(z,t,D,:»Fn(z,t) =:L: rn,m(t)Fm(z,t), <p(z) = :L:imFm(z,O). m=-oo m

The above approach works whenever the conditions (4.12) are fulfilled, and using it, one can explicitly extract the "main" terms of the solution to (4.8). Namely, the solution <p(z, t) to (4.8) in the case of initial function I from £2 can be represented in the form of two summands:

(4.13)

On the right of (4.13), the first summand ("main" term) is obviously of the form (4.3), i.e. it is the trace of V-continuation on the manifold A(z,t) corresponding to the integrals (4.9ii). This term plays the role 01 a "camer 01 singularities" olthe solution: the second summand 6(z, t) in the representation (4.13) is smoother, than the main term. In any case, if I E L2, one can assert that 6(z,t) is continuous in z in the strip It I ~ T (for more details, cf. [22, theorem 1]). In particular, theorem (4.13) of equiconvergence type, and the above results on G-regular and G-nonregular manifolds and general properties of V-continuations make it possible to judge on the boundedness and continuity; to foresee the chaotic or nonchaotic features of the solutions.

For a general introduction to the field of quantum chaos, the author is indebted to Professor D. Offin.


q=503, c=2, 3=11 30

25

20

15

10

5

0

-5 -15 -10 -5 10

Figure 1

396 K.I. Oskolkov

real,q=97, rotated 45 ,elevation 0

Figure 2. Real Part


imag,q=97, rotated 45 elevation 30

imag,q=97, rotated 45 elevation 0

Figure 2. Imaginary Part

398 K.I. Oskolkov

cubic,q=97

cubic,q=97 ,rotated 9O,elevation 30

Figure 3


References

[1]

[2]

[3]

[4]

[5]

[6]

[1]

[8]

[9]

G.I. Arkhipov, On the Hilbert-Kamke Problem, Izv. Akad. Nauk SSSR, Ser Mat., 48(1984), 3-52j English transl. in Math. USSR Izv., 24(1985).

G.I. Arkhipov and K.I. Oskolkov, On a special trigonometric series and its applications, Matern Sbornik, 134(176) (1987), N2j Engl. transl. in Math. USSR Sbornik, 62(1989), Nl, 145-155.

N.K. Bari, A Treatise on Trigonometric Series, v. 1, (Pergamon Press, N.Y.), 1964.

L. Carleson, On convergence and growth of partial sums of Fourier series, Acta Math., 116(1966), 133-157.

Yung-Ming Chen, A remarkable divergent Fourier series, Proc. Japan Acad., 38(1962), 239-244.

L. de Michele and P.M. Soardi, Uniform convergence of lacunary Fourier series, Colloq. Math., 36(1976), 285-287.

H. Fiedler, W. Jurkat, and O. Koerner, Asymptotic expansion offinite theta series, Acta Arithmetica, 32(1977), 129-146.

A. Figa-Talamanca, An example in the theory of lacunary Fourier series, Boll. Un. Mat. Ital. (4),3(1970),375-378.

J.E. Fournier and L. Pigno, Analytic and arithmetic properties of thin sets, Pacific J. Math., 105(1983), 115-141.

[10] R.P. Gosselin, On the divergence of Fourier series, Proc. Amer. Math. Soc., 9(1958), 278-282.

[11] G.H. Hardy, Collected Papers of G.H. Hardy (Oxford: Clarendon Press), 1966, v. 1.

[12] G.H. Hardy and J .E. Littlewood, Some problems of Diophantine approximation. 11. The trigonometrical series associated with the elliptic (J-functions, Acta Math., 37(1914), 193-238.

[13] R.A. Hunt, An estimate of the conjugate function, Studia Math., 44(1972), 371-377.

[14] R.A. Hunt, On the convergence of Fourier series. Orthogonal Expansions and Their Continuous Analogues. (Proc. Conf. Edwardsville, Ill. (1967», 235-255. Southern Ill. Univ. Press, Carbondale, Ill., (1968).

[15] Chen Jing-run, On Professor Hua's estimate of exponential sums, Sci. Sinica, 20 (1977), 711-719.

400 K.1. Oskolkov

[16] A.N. Kolmogorov, Sur les fonctions hannoniques conjugees et les series de Fourier, Fund. Math., 7(1925), 24-29.

[17] A.N. Kolmogorov, Une serie de Fourier Lebesgue divergente partout, C.R. Acad. Sci Paris, 183(1926), 1327-1328.

[18] A.N. Kolmogorov, Une serie de Fourier-Lebesgue divergente presque partout, Fund. Math., 4(1923),324-328.

[19] S.V. Konyagin, On Littlewood's conjecture, Izv. Akad. Nauk SSSR Ser. Mat., 45(1981), 243-265.

[20] E. Makai, On the summability of the Fourier series of L2 integrable functions. IV. Acta Math. Ac. Sci Hung., 20(1969), 383-391.

[21] K.I. Oskolkov, I.M. Vinogradov series and integrals and their applications, Trudy Mat. lost Steklov, 190(1989), 186-221.

[22] K.I. Oskolkov, I.M. Vinogradov's series in the Cauchy problem for Schroedinger type equations, Trudy Mat. lost. Steklov, 200(1991) (in print).

[23] K.I. Oskolkov, On functional properties of incomplete Gaussian sums, Canad. J. Math., 43(1991), No.1, 182-212.

[24] K.I. Oskolkov, On properties of a class of Vinogradov series, Doklady Acad. Nauk SSSR, 300(1988), N4, 737-741; Engl. transl. in Soviet Math. Dokl., 37(1988), N3.

[25] K.I. Oskolkov, On spectra of unifonn convergence, Dokl. Akad. Nauk SSSR, 288(1986), Nl; Engl. transl. in Soviet Math. Dokl, 33(1986), N3, 616-620.

[26] K.I. Oskolkov, Subsequences of Fourier sums of integrable functions, Trudy Mat. lost. Steklov, 167(1985), 239-360; Engl. transl. in Proc. Steklov Inst. Math. 1986, N2 (167).

[27] L. Pedemonte, Sets of unifonn convefYJence, Colloq. Math., 33(1975), 123-132.

[28] N. Saito and Y. Aizawa, editors, Progress of Theoretical Physics, Supplement, No. 98,1989. New trends in chaotic dynamics of Hamiltonian systems, Kyoto University, Japan.

[29] B. Smith, O.C. McGehee, L. Pigo, Hardy's inequality and Ll_nonn of exponential sums, Ann. Math. 113(1981), N3, 613-618.

[30] S.B. Stechkin, Estimate of a complete rational trigonometric sum, Trudy Mat. lost. Steklov, 143(1977),188-207; English transl. in Proc. Steklov lost. Math. 1980, Nl (143).


[31] S.B. Stechkin, On absolute convergence of Fourier series. III, Izv. Akad. Nauk SSSR Ser. Mat., 20(1956),385-412. (Russian).

[32] E.M. Stein, On limits of sequences of operators, Ann. Math., 74(1961), 140-170.

[33] E.M. Stein, Oscillatory integrals in Fourier analysis, In: Beijing Lectures in Harmonic Analysis, Princeton Univ. Press, Princeton, N.J., (1986), 307-355.

[34] E.M. Stein and S. Wainger, The estimation of an integral arising in multiplier transformations, Studia Math., 35(1970), 101-104.

[35] E.M. Stein and S. Wainger, Problems in harmonic analysis related to curvature, Bull. Amer. Math. Soc., 84(1978), 1239-1295.

[36] V. Totik, On the divergence of Fourier series, Publ. Math. Debrecen, 29(1982), 251-264.

[37] G. Travaglini, Some properties of UC-sets, Boll. Un. Math. Ital. B(5), v. 15 (1978), 275-284.

[38] P.L. Ul'yanov, Some questions in the theory of orthogonal and biorthogonal series, Izv. Akad. Nauk Azerbaidzhan. SSR Ser. Fiz.Tekhn. Math. Nauk (1965), no. 6, 11-13 (Russian).

[39] I.M. Vinogradov, The Method of Trigonometric Sums in Number Theory, 2nd ed. "Nauka," Moscow 1980; English transl. in his Selected Works, Springer Verlag, 1985.

[40] S. Wainger, Applications of Fourier transforms to averages over lower dimensional sets, Proc. Symposia in Pure Math., 35 (part 1), 1979, 85-94.

[41] S. Wainger, Averages and singular integrals over lower dimensional sets, In: Beijing Lectures in Harmonic Analysis, Princeton Univ. Press, Princeton, N.J., (1986), 357-421.

[42] S. Wainger, On certain aspects of differentiation theory, Topics in modern harmonic analysis (Proc. Seminar Torino and Milano, MayJune 1982), 42(Roma, 1983), 667-706.

[43] D.M. Wardlaw and W. Jaworski, Time delay, resonances, Riemann zeros and Chaos in a model quatum scattering system, J. Phys. A: Math. Gen., 22(1989), 3561-3575.

[44] H. Weyl, tiber die Gleichverteilung der Zahlen mod Eins, Math. Ann., 77(1915/16), 313-352.

402 K.1. Oskolkov

[45] A. Zygmund, Trigonometric Series, 2nd rev. ed., v. 1, Cambridge Univ. Press. 1959.

K.1. Oskolkov Department of Math. & Statistics Jeffrey Hall Queen's University Kingston, Ontario CANADA K7L 3N6

A Lower Bound for the de Bruijn-N ewman Constant A. II

T.S. Norfolk A. Ruttan R.S. Varga*

ABSTRACT A new constructive method is given here for determining lower bounds for the de Bruijn-Newman constant A, which is related to the Riemann Hypothesis. This method depends on directly tracking real and nonreal zeros of an entire function F>.(z), where .A < 0, instead of finding, as was previously done, nonreal zeros of associated Jensen polynomials. We apply this new method to obtain the new lower bound for A,

-0.385 < A,

which improves previous published lower bounds of -50 and -5.

1 Introduction

The purposes of this paper are i) to give a new constructive method for finding lower bounds for the de Bruijn-N ewman constant A, which is related to the Riemann Hypothesis, and ii) to apply this method to obtain a new lower bound for A. This new lower bound (to be given below) is the best constructive lower bound for A known to us at this time.

By way of background, in Csordas, Norfolk, and Varga [4], the entire function H>.(x) was defined by

(,\ E R), (1.1)

where

00

(t) := I:(2n41r2e9t - 3n21reSt ) exp( _n21re4t ) (0 $ t < 00). (1.2) n=l

It is known (cf. P6lya [12] or Csordas, Norfolk, and Varga [3, Theorem A))

"Research supported by the National Science Foundation. AMS (MOS) subject classification: 30DIO, 30D15, 65E05j CR:G1.m.


404 T.S. Norfolk, A. Ruttan, R.S. Varga

that 0 (t E R); (1.3)

iii) for any e > 0, lim <p(n)(t) exp[('Ir - e)e4t] = 0 (n = 0,1, ... ). t_oo

It was also shown in [4, Appendix A] that, for each A E R, H)..(x), as defined in (1.1), is an entire function of order 1 and of maximal type (i.e., its type, (T).., satisfies (T).. = +00).

For the choice A = 0, the function Ho(x) is related to the Riemann e-function through the following identity:

e (i) 18 = Ho(x), (1.4)

where the Riemann e-function, in turn, is related to the Riemann (-function through

e(iz) = 4 (z2 -~) 'Ir- z / 2- 1/ 4 r (~ +~) ( (z + 4) . (1.5)

It is known (cf. Henrici [6, p. 305]) that the Riemann Hypothesis is equivalent to the statement that all the zeros of e(z) are real, which implies from (1.4) that the Riemann Hypothesis is equivalent to the statement that all zeros of Ho(x) are real.

Next, two results of de Bruijn [2] in 1950 established that

Ii) H)..(x) has only real zeros for A ~ 1/2, and

ii) if H)..(x) has only real zeros for some real A, then

H)..I(x) also has only real zeros for any A' ~ A.

(1.6)

In particular, it follows from (1.6ii) that if the Riemann Hypothesis is true, then H)..(x) must possess only real zeros for any A ~ O. In 1976, C.M. Newman [10] showed that there exists a real number A, satisfying -00 < A ~ 1/2, such that

{i) H)..(x) has only real zeros when A ~ A, and

(1.7) ii) H).. (x) has some nonreal zeros when A < A.

This constant A has been called in [4] the de Bruijn-Newman constant. Since the Riemann Hypothesis is equivalent to Ho(x) having all its zeros real, then from (1.7i), the truth of the Riemann Hypothesis would imply

A Lower Bound for the de Bruijn-Newman Constant A. II 405

that A ::5 O. (Interestingly, Newman [10] makes the complementary conjecture that A ~ 0.) Because of the connection of this constant A to the Riemann Hypothesis, there is an obvious interest in determining upper and lower bounds for A. A constructive lower bound, -50 < A, was first given in [4] in 1988. Subsequently, te Riele [14] has given strong numerical evidence that -5 < A. Our object here is to report on recent research activity in finding improved lower bounds for A.

Returning to H>.(z) of (1. 1), we see, on expanding cos(zt) and integrating termwise, that the Maclauren expansion for H>.(z) is given by

(A E R),

where

(m = 0, 1, ... ).

On setting Z = _z2 in (1.8), the function F>.(z) is then defined by

so that

~ bm(A)Zm F>.(z) := ~ (2m)! (A E R),

(A E R).

(1.8)

(1.9)

(1.10)

(1.11)

Since H>.(z) is an entire function of order one, it follows from (1.11) that F>.(z) is an entire function of order 1/2. Hence, for each real A, F>.(z) necessarily has (cf. Boas [1, p. 24]) infinitely many zeros. Moreover, it follows from (1.7) that

{i) F>.(z) has only real zeros when A ~ A, and

ii) F>.(z) has some nonreal zeros when A < A. (1.12)

The constructive·method used in [4], for finding lower bounds for the de Bruijn-Newman constant A, can be described as follows. With the moments of (1.9), define the m-th Jensen polynomialfor F>.(z) by

. ._ ~(m) h(A). k! A: Gm(t, A) .-~ k (2k)! t (m=I,2, ... ). (1.13)

It was shown in Proposition 1 of [4] that if, for some real i and some positive integer m, Gm(t; i) possessed a nonreal zero, then

(1.14)


In [4], each of the exact moments {bm ( -50)}~=o was approximated by the Romberg integration method with a relative accuracy of at least 60 significant digits, thereby producing the approximate moments {,8m(-50)}~=o, and the associated approximate Jensen polynomial (cf. (1.13», namely

16 ~

~(16) f3J:(-50)k! J: 916(tj -50) := ~ k (2k)! t,

was shown to possess a nonreal zero. Then, using a perturbation argument of Ostrowski (cf. [4, Proposition 2]), it was rigorously shown that G16(tj -50) also possessed a nonreal zero, so that from (1.14), -50 < A.

Further use of t~is Jensen polynomial method subsequently produced for us the (unpublished) lower bounds for A of Table 1. (All entries in the tables which follow are truncated to 3 decimal digits.)

TABLE 1

~ degree m digits required complex zero of Gm(tj~)

-100 10 12 -453.840 + i 9.703

-50 16 12 -220.919 + i 7.092

-20 41 18 -111.065 + i 1.322

-15 56 20 -79.834 + i 0.282

-12 75 20 -59.204 + i 0.536

-10 97 21 -45.530 + i 0.156

-8 142 21 -30.993 + i 0.124

By means of an improved perturbation argument, far fewer total significant digits (than that suggested in [4]) were actually required, in the computa.tion of the moments {bm(~)}~=o, to produce guaranteed lower bounds for A. This is indicated in column 3 of Table 1. The second column of Table 1 gives the smallest degree m for which the Jensen polynomial Gm{tj ~), considered as a function of m, possessed nonreal zeros. The entries in this second column of Table 1 show an alarming increase in this smallest degree as ~ increases to O. To underscore this, te Riele [14], using this Jensen polynomial method but with a modification involving Sturm sequences, has recently reported strong numerical evidence for the lower bound:

-5<A, (1.15)

A Lower Bound for -the de Bruijn-Newman Constant A. II 407

based on a Jensen polynomial of degree 406, where 250 significant digits were used in the associated computations! The results of te Riele and Table 1 seem to indicate that further improvements in lower bounds for A, using this Jensen polynomial method, would require lengthy calculations involving great precision.

2 Tracking Zeros of FA (Z )

We propose here a new method for determining lower bounds for A, based on directly tracking particular pairs of zeros of F),(z), as a function of A. We begin by noting that F),(z) of (1.10) can be expressed, in analogy with (1.1), in integral form as

F),(z) = 100 e),t

2 <r>(t) cosh(tv'z )dt (A E R). (2.1)

Now suppose, for AO real, that Z(AO) is some simple zero of F),o(z), so that z( A) remains a simple zero of F), (z) in some small real interval in A containing AO in its interior. In this interval, F),(Z(A)) == 0 so that, with the definition of bm(A) of (1.9),

F),(Z(A)) == 0 = 100 e),t

2 <r>(t) cosh (tv Z(A) ) dt

(2.2)

On differentiating (2.2) with respect to A, we obtain

Because the final sum above is nonzero (as Z(A) is assumed to be a simple zero), then solving for dZ(A)jdA yields

dZ(A) E:_obm+1(A)(Z(A))m j(2m)!

-;rr:- = - E:=o(m + l)bm+1(A)(Z(A))m j(2m + 2)!· (2.3)

Thus, with accurate estimates of {bm(A)}~=o and with asymptotic estimates for {bm(A)}~=N+l' accurate estimates of dZ(A)jdA can be obtained from (2.3).


It is also important to note that replacing e.\t2 by its Maclauren expansion and integrating termwise in (1.9), gives

(m = 0,1, ... ; ~ E R). (2.4)

Hence, for ~ small and negative, one needs from (2.4) to compute only one extended table of high-precision moments {bm(O)}~=o, from which the moments {bm(~)}~~o can be directly estimated from (2.4), where N' < N. (We remark that the choice of N' depends on m,~,N, and the desired accuracy; cf. [11].)

In our applications described below, our extended table of high-precision moments was {bm(O)}~~~, where each moment was computed (on a SUN 3/80 computer in the Department of Mathematics and Computer Science at Kent State University) to an accuracy of 360 significant digits, using basically the trapezoidal rule with a sufficiently fine mesh. (This will be discussed in detail in §3.)

It is well known that considerable numerical effort has been given to the problem of studying the (nontrivial) nonreal zeros of the Riemann (function in the critical strip 0 < Re z < 1. In 1986, van de Lune, te Riele, and Winter [8] impressively showed that all 1,500,000,001(=: T) nonreal zeros of (z), in the subset of the critical strip defined by

o < 1m z < 545,439,823.215 ... ,

lie on Re z = 1/2 and are simple. Expressing these zeros as

{ pn := 4 + i1'n}T n=1

(where 0 < 1'1 < ... < 1'T), (2.5)

it follows from (1.4), (1.5), and (1.11) that

(n = 1,2, ... ,T) (2.6)

are then the consecutive T largest ~ne~ative and simple) zeros of Fo(z) of (1.10). From the tabulation of hn}n~~ 0, accurate to 28 significant digits, given in te Riele [13], one can easily determine from (2.6) accurate estimates of {Zn(O)}~~~oo.

In Table 2, we give the values {zn(0)}~;1' along with their differences and the derivative dZn(O)/d~, determined from (2.3).

It turns out, as is indicated in Table 2, that certain pairs of these known consecutive real zeros zn(O) and Zn+1(O) of Fo(z) are quite close and give promise of producing, as ~ decreases from 0, nonreal conjugate complex zeros zn(~) and Zn+1(A) of F.\(z). (We recall from (1.12) that if a real i is such that zn(i) and zn+l(i) are nonreal zeros, then i < A.) In Table 3, we give the associated pairs, zn(O) and Zn+l(O), on which we concentrated.


TABLE 2

n zn(O) := -4"Y~ dzn(O)/d>' zn(O) - Zn+l(O)

1 -799.161 +32.771 +968.542

2 -1,767.704 +63.486 +734.467

3 -2,502.171 +58.608 +1,200.520

4 -3,702.692 +116.274 +636.180

5 -4,338.873 +61.317 +1,312.010

6 -5,650.883 +126.089 +1,046.483

7 -6,697.366 +140.344 +811.574

8 -7,508.941 +77.875 +1,709.036

9 -9,217.978 +230.133 +691.759

10 -9,909.737 +84.666 +1,313.682

11 -11,223.419 +138.447 +1,521.295

12 -12,744.715 +199.889 +1,343.571

13 -14,088.286 +267.284 +713.734

14 -14,802.021 +35.196 +2,156.552

In column 3 of Table 3, we again give dzn(O)/d>', determined from (2.3). We note, because ofthe difference in signs of dZ34 (O)/d>' and dZ35(0)/d>' in Table 3, that the last pair of zeros, Z34(0) and Z35(0), are tending toward one another as >. decreases from 0, i.e., these two zeros are attracted to each other. In Table 4, we show how Z34(>'), dZ34(>.)/d>., Z35(>'), and dZ35(>')/d>' change with decreasing values of >.. Table 4 suggests that not only are Z34(>') and Z35(>') tending toward one another, but also that dz34(>.)/d>. and dZ35(>')/d>' are respectively tending to +00 and -00.

The actual tracking of the pair of zeros {Z34(>') and Z35(>')} generates interesting geometrical results! In Figure 1, we have graphed the 21 pairs of zeros

{Z34 (-[0.04]j) and Z35 (-[0.04]j)} ;~o . (2.7)


TABLE 3

n Zn(O) dzn(O)/d>' zn(O) - Zn+1(O)

4 ~3,702.692 +116.274 +636.180

5 -4,338.873 +61.317 -9 9,217.978 +230.133 +691.759

10 -9,909.737 +84.666 -13 -14,088.286 +267.284 +713.734

14 -14,802.021 +35.196 -19 -22,924.800 +414.348 +880.504

20 -23,805.305 +140.940 -24 -30,572.714 +392.063 +975.518

25 -31,548.232 +44.267 -27 -35,835.507 +465.401 +929.206

28 -36,764.714 +26.826 -34 -49,310.231 +877".835 +753.526

35 -50,063.757 -26.626 -

We see from Figure 1 that the pair of zeros Z34(>') and Z35(>') of (2.7) start out as real distinct zeros which move toward one another. These zeros then

meet, forming a real double zero of FA(Z) when>. == -0.38, and then this

pair of zeros bifurcates into two nonreal conjugate complex numbers which follow, as >. decreases, a parabolic-like trajectory in the complex plane when>. $ -0.40. Because FA(Z) apparently has, from Figure 1, nonreal zeros when>. $ -0.40, it would appear from (1.12) that -0.40 is a lower bound for A, i.e., .,

-0.40 <: A. (2.8)


TABLE 4

~ Z34(~) dZ3'(.~) Z35(~) dZ35(~)

d~ d~

-0.30 -49,633.457 +1,489.525 -49,997.614 -626.913

-0.31 -49,648.703 +1,561.893 -49,990.996 -698.909

-0.32 -49,664.748 +1,650.191 -49,983.583 -786.835

-0.33 -49,681.783 +1,761.399 -49,975.183 -897.671

-0.34 -49,700.092 +1,907.715 -49,965.513 -1,043.617

-0.35 -49,720.131 +2,112.957 -49,954.117 -1,248.488

Our task is to rigorously establish in §3 the following slightly improved form of (2.8), namely

Theorem 1. If A is the de Bruijn-Newman constant, then

-0.385 < A. (2.9)

We remark that each of the pairs of zeros, Zn (0) and Zn+1 (0), of Table 3 did similarly give rise, via this new tracking method, to a lower bound for A, and the best such lower bound (coming from tracking the pair Z34(~) and Z35(~» is the result of (2.9). These results are summarized in Table 5, where the final column in Table 5 gives the largest value of A (to three decimal digits) for which zn(A) and zn+l(A) were nonreal complex conjugate numbers, and for which 11m zn(~)1 ~ 1.

Our primary interest here has been to introduce a new method for obtaining rigorous lower bounds for A, and to show, with a moderate amount of computing effort, that this method does produce improved lower bounds for A. We are confident that further improved lower bounds for A can be similarly numerically obtained for this tracking method applied to particular pairs of zeros, zn(~) and Zn+1(~)' with n > 34, as ~ decreases from 0, but at the expense of more computer time.

3 Proof of Theorem 1

This section consists first of a 'brief discussion on how high-precision numerical approximations of the moments bm(~) of (1.9) can be determined, and this is followed by a perturbation analysis which is used to rigorously


TABLE 5

n zn(O) := -4'Y~ largest value of A for which Zn(A) and Zn+l(A) are nonreal

4 -3,702.692 -3.955

5 -4,338.873

9 9,217.987 -1.878

10 -9,909.737

13 -14,088.286 -1.286

14 -14,802.021

19 -22,924.800 -1.276

20 -23,805.305

24 -30,572.714 -1.144

25 -31,548.232

27 -35,835.507 -0.882

28 -36,764.714

34 -49,310.231 -0.385

35 -50,063.757

show that FA(Z) has a nonreal zero when A = -0.385. We remark that the complete details (which are lengthy and rather tedious) for producing high-precision approximation of the moments bm(A) are given in Norfolk, Ruttan, and Varga [11].

To begin, our first step Was to determine high-precision floating-point numbers {Pm(O)}~~~ which approximate the moments {bm(O)}~~~, where (cf. (1.9»

(m = 0,1, ... ). (3.1)

Fortunately, because the integrand in (3.1) is from (1.3i) an even function which is analytic in the strip 11m zi < ?r/8 for each m ~ 0, it follows from

A Lower Bound for the de Bruijn-Newma.n Consta.nt A. II 413

the work of Mr.,rtensen [9] and Kress [7] that the familiar trapezoidal rule approximation (on a uniform mesh of size h) of bm(O), defined by

Tm(h) := h { ~ [t 2m cll(t)]t=0 + ~ (kh)2mcll(kh)} (m = 0,1, ... ),

(3.2) converges exponentially rapidly to bm(O) as h decreases to 0, i.e., (cf. [7, Thm. 2.2 with p = 0]),

ITm(h) - bm(O)1 ~ e;:~(:;/~) 100 I(s + ia)2mcll(s + ia)1 ds, (3.3)

for any a with 0 < a < 7r/8 = 0.39269 ... , where the path of integration in (3.3) is the nonnegative real axis. From (1.2), it directly follows that the integrand in (3.3) is bounded above by

00

(s2+a2)mL: (2n47r2e9t+3n27re5t)exp(_n27re48cos4a) (s~O), (3.4) n=l

and on specifically choosing

1 (In32) & := 4 arccos a;- = 0.29855 ... «7r/8), (3.5)

an easily computed upper bound, 1(&; m), for the integral in (3.3) can be found, so that

(3.6)

is an upper bound for the error in the trapezoidal approximation of bm(O). (Further details are given in [11]).

Next, we observe that the exact trapezoidal rule approximation, Tm(h), involves an infinite sum in its definition in (3.2), and, in addition, there is an infinite sum in the definition of cIl(t) in (1.2), which is used in each term ofTm(h). In our actual computations of approximations of bm(O), the sum in (3.2) was summed only for k ~ 21h because of the exponential decay (cf. (1.3iii)) of cIl(t) for large t > 0, and only the first sixteen terms of the infinite sum defining cIl(t) in (1.2) were used to approximate cIl(t). (An upper bound of the sum of the remaining terms of cIl(t) is constructively given in [3, eq. (4.6)].) These two errors, introduced into the computation of the trapezoidal rule Tm(h), can again be constructively bounded above, and the details are again given in [11]. In this way, the approximations {.8m(0)}~~~ to the moments {bm(O)}~~~ were determined, each with a computable error. Finally, from the approximate moments {.Bm(O)}~~~, the moments (cf. (2.4))

.Bm(>") := I: .Bm+i.~O)>..i i=O J.

(m = 0,1, ... ,550) (3.7)

414 T.S. Norfolk, A. Ruttan, R.S. Va.rga

were determined. All floating-point calculations were performed with 360 significant digits of accuracy, and, based on the error estimate outlined above, the approximate moments {Pm(-0.385)}~~0 are each accurate to 314 significant digits (cf. (3.11».

For the perturbation analysis to show that F>o. (z) has a nonreal zero when A = -0.385, we begin by establishing the following known, but useful, result. (We remark that Lemma 1 is a special case of a more general result given in Henrici [5, p. 454].)

Lemma 1. Let p(z) be a complez polynomial 0/ degree n. 1/ P'(zo) =1= 0, then the disk

{z: Iz - zol ~ nlP(zo)I/lp'(zo)l} (3.8)

contains at least one zero o/p(z).

Proof. As the result of Lemma 1 is obvious if p(zo) = 0, assume p(zo) =1= 0 and write p(z) = I'n~=l(z-(I:)' where the (I:'S are the zeros ofp(z). Taking the logarithmic derivative of p( z) and evaluating the result at the point Zo gives

p'(zo) ~ 1 p(zo) = ~ Zo -(I:.

On taking absolute values in the above expression, then

1P'(zo)I t 1 n Ip(zo) I ~ 1:=1 Izo - (I:I ~ minl:Sl::Snlzo - (I:I '

and rewriting this inequality directly gives (3.8). • Our next result is also elementary in nature.

Lemma 2. Given the complez number Zo, assume that /(z) := E~o aizi is analytic in the disk Iz - zol < R ~ 00. For each positive integer N, set PN(Z) := Ef=o ai zi , and write PN(Z) =: Ef=o ci(z - zo)i, where ci := ci(Nj zo). Assume that there ezist a positive integer N and positive real numbers OtN, 6 (with 0 < 6 < 1), and T (with 0 < T < R), such that

i)

ii)

iii)

iv)

v)

vi)

0=1= Cl(:= p~(zo»,

T> Nlcol/lcll, N 3 L: Icilrl ~ "2lclIT, i=O

OtN (Izol + T) ~ 6 < 1, and

(1/(1 - 6» [OtN (Izol + T)]N+l ~ llcdT,


where strict inequality holds in iv) or vi). Then, fez) also has at least one zero in Iz - zol < T.

Proof. To begin, assumption iii) implies, from Lemma 1, that PN(Z) has at least one zero in the disk Iz - Zo I < T. On the circle Iz - Zo I = T, we have from i), v) and vi) that

00 00

If(z) - PN(z)1 = I: ajzil ~ I: [aw(lzol + T)]i j=N+1 j=N+l

= [cxN(lzol + T)]N+1 < [cxN(lzol + T)]N+l 1 - cxN(lzol + T) - 1 - 6

~ ICll! 2

Since (3/2)lclIT - Ef=o ICjlTj ~ 0 from iv), the above inequality implies that

If(z)-PN(z)1 < Ic~IT + {~';I' -f,1<;1,J} = IClIT-~ ICjl"'; j¢l

N

~ II: Cj(z - zoY 1=: IpN(z)l, j=O

and, since strict inequality by assumption holds in either iv) or vi), then

If(z) - PN(Z) I < IPN(Z)I·

But this inequality implies, on applying Rouche's theorem on

Iz - zol = T, that fez) and PN(Z) have the same number of zeros in Ix - zol < T. Consequently, fez) has at least one zero in Iz - zol < T. •

The next result, which reduces to the result of Lemma 2 (when PN(Z) == PN(Z», is an easy consequence of the proof of Lemma 2.

00 • Lemma 3. Given the complex number Zo, assume that fez) := Ej=o ajzJ is analytic in the disk Iz - Zo I < R ~ 00. For each positive integer N, set

N .. N . PN(Z) := Ej=o ajzJ, and wnte PN(Z) =: Ej=o Cj(z - zo)', where Cj := cj(Nj zo). Assume that there exist a positive integer N, an approximation

polynomial PN(Z) := Ef=o Cj(z - zoY to PN(Z), positive real numbers CXN, e, 6 (with 0 < 6 < 1), and T (with 0 < T < R) such that


i) OtN ~ sup la; 11/; ;>N

ii) Ic; - c; I < e (j = 0,1, ... ,N),

ii')

iii)

iv)

v)

vi)

Ict! > e, ICoI +e

r>Nr I ' C1 -e N

I: (lc;1 + e) ri $ ~ (lcll- e)r, ;=0 OtN (Izol + r) $ /j < 1, and

6 [OtN(lzol + r)]N+1 $ ! (lcll- e) r,

with strict inequality holding in iv) or vi). Then, fez) has at least one zero in Iz - zol < r.

Proof. With the hypothesis above, it is elementary to verify (by the triangle inequality) that PN(Z) and fez) satisfy all the hypotheses of Lemma 2; hence, fez) has at least one zero in Iz - zol < r. •

With Lemma 3, we come to the

Proof of Theorem 1. Let fez) be the entire function

00

F,\(z) := I: bm(~)/(2m)! zm, m=O

where ~ is defined by ~:= -0.385,

and define the complex number Zo by

Re zo = -4.985226399929367054457428908808825137

17835429943591222950674598866282510463

328370182827604591192414841702633722~4

1m zo = 1.323062852274493439584297961431473867

46981439309032787945567788558243812285

140059530997087566968399990919977650~1

(3.9)

(3.10)


Now, with N := 550, the numbers ,8m(j), which approximate the moments bm (j), were determined so that

(m = 0,1, ... ,550). (3.11)

From this, the polynomials P550(Z) := L~~o bm(j)zm /(2m)! and P550(Z) :=

L~~o ,8m(j)zm /(2m)!, re-expanded as Ps50(Z) = L~~o cm(z - zo)m and Ps50(Z) = L~~o cm(z - zo)m, can be verified to satisfy

ICm - eml ~ e = 1.10-217 (m = 0,1, ... ,550). (3.12)

Then, with 6 := 9/10, and with T := 1.10-5 , an application of Lemma 3 to F).(z) shows that F).(z) has at least one zero in Iz - zol < T. But since (cf. (3.10)) 1m Zo = 13.2306 ... and since T = 1 . 10-5 , it is geometrically evident that this zero of F). (z) in the disk Iz - Zo I < T is necessarily nonreal.

Thus, from (1.7), j = 0.385 < A, the desired result of (2.9). •

References [1] R.P. Boas, Entire Functions, Academic Press, Inc., New York, 1954.

[2] N.G. de Bruijn, The roots of trigonometric integrals, Duke Math. J. 17(1950), 197-226.

[3] G. Csordas, T.S. Norfolk, and R.S. Varga, The Riemann Hypothesis and the Turan inequalities, Trans. Amer. Math. Soc. 296(1986), 521-541.

[4] G. Csordas, T.S. Norfolk, and R.S. Varga, A lower bound for the de Bruijn-Newman constant A, Numer. Math. 52(1988), 483-497.

[5] P. Henrici, Applied and Computational Complex Analysis, vol. 1, Wiley & Sons, New York, 1974.

[6] P. Henrici, Applied and Computational Complex Analysis, vol. 2., Wiley & Sons, New York, 1977.

[7] R. Kress, On the general Hermite cardinal interpolation, Math. Compo 26(1972), 925-933.

[8] J. van de Lune, H.J.J. te Riele, and D.T. Winter, On the zeros of the Riemann zeta-function in the critical strip. IV, Math. Compo 46(1986), 667-681.

[9] E. Martensen, Zur numerischen Auswertung uneigentlicher Integrale, Z. Angew. Math. Mech. 48(1968), T83-T85, MR 41 #1221.


[10] C.M. Newman, Fourier transforms with only real zeros, Proc. Amer. Math. Soc. 61(1976), 245-25l.

[11] T.S. Norfolk, A. Ruttan, and R.S. Varga, A detailed numerical examination of the tracking of zeros of F),(z) to produce lower bounds for the de Bruijn-Newman constant A, Technical Report of the Institute for Computational Mathematics, 1990, Kent State University, Kent, OH 44242.

[12] G. P6lya, Uber die algebraisch-funktionen Untersuchungen von J.L. W. V. Jensen, Kg!. Danske Vid Sel. Math.-Fys. Medd. 7(1927), 3-33.

[13] H.J.J. te Riele, Tables of the first J 5000 zeros of the Riemann zeta function to 28 significant digits, and related quantities, Report Number NW67/69 of the Mathematisch Centrum, Amsterdam, 1979.

[14] H.J.J. te Riele, A new lower bound for the de Bruijn-Newman constant, Numer. Math. (to appear).

T.S. Norfolk, A. Ruttan, and R.S. Varga Department of Math. & Computer Science Kent State University Kent, Ohio 44242 USA

On the Denseness of "Weighted Incomplete Approximations

P. Borwein* E.B. SafF

ABSTRACT For a given weight function w(x) on an interval [a, b], we study the generalized Weierstrass problem of determining the class of functions f E C[ a, b] that are uniform limits of weighted polynomials of the form {wn(x )Pn(X nr', where Pn is a polynomial of degree at most n. For a special class of weights, we show that the problem can be solved by knowing the denseness interval of the alternation points for the associated Chebyshev polynomials.

1 Introduction

In the asymptotic analysis of orthogonal polynomials with respect to an exponential weight of the form w(x) = exp(-lxIQ), a> 1, on R= (-00,00), an important step is to determine the class of functions f continuous on R that are uniform limits of weighted polynomials {wnpn}, where Pn E fin (the class of polynomials of degree ~ n), and the power n of w matches the (maximum) degree of the polynomial. For these so-called Freud weights, this problem was solved by Lubinsky and Saff [9] using techniques from potential theory. The analogous problem for weighted polynomials of the form {xn>'Pn(x)} on [0,1]' which are called incomplete polynomials, was raised by G.G. Lorentz and was resolved independently by Saff and Varga [12] and by M. v. Golitschek [3]. Further extensions to Jacobi type weights were obtained by He and Li [6] and He [5].

The above investigations are special cases of the following:

Genemlized Weierstrass Problem: Given a closed set E C R and a weight w : E --+ [0,00), determine necessary and sufficient conditions on f such that f is the uniform limit on E of a sequence of weighted polynomials {wnpn}, Pn E fin, as n --+ 00.

For the case when E is an interval and w(x) = e-Q(x), with Q(x) con-

·The research of this author was supported, in part, by NSERC of Canada. tThe research of this author was supported, in part, by NSF grant DMS-881-

4026.


420 P. Borwein, E.B. Saf[

vex on E, a plausible solution can be described as follows. From potential theoretic considerations, it is known (cf. [10)), that there exists a unique smallest compact interval Sw such that for every n ~ 1 and every Pn E lIn,

where II·IIA denotes the sup norm over the set A. Moreover, if one considers weighted Chebyshev polynomials Tn(z) = wn(z)(zn + ... ) that are defined by the extremal property

then the alternation (extreme) points of Tn are dense in Sw' Based on the above mentioned special cases, the second author has previously made the following

Conjecture. If E C R is a compact interval and w( z) = cQ(z), with Q( z) convex on E, then I E C( E) is the uniform limit on E of a sequence of the form {wnpn}f, Pn E ITn, if and only if I vanishes identically on E \ Sw. (In case E is unbounded, additional assumptions need to be imposed on Q(z) as Izl--+ 00, i E E.)

The aim of the present paper is to show that for a special class of weights w, a proof of the above conjecture follows from the denseness property of the alternation points of the weighted Chebyshev polynomials. Thus we avoid much of the "hard analysis" involved with the potential theoretic arguments used in [9]. However, our technique requires strong assumptions on the weight w and so falls short of proving the general conjecture.

2 An Approximation Lemma

Let Hn :=span{go, ... ,gn}, gi EC[a,b]

be a Chebyshev system on [a, b]. Define Tn, the normalized Chebyshev polynomial for Hn on [a, b], by

n

Tn := Tn,[II,1I] = L Cigi,

i=O

where the Ci are chosen so that IITn 11[11,11] = 1 and so that Tn has exactly n zeros Z1 < ... < Zn in (a, b) and oscillates n + 1 times between ±1 on [a, b]. So defined, Tn exists and. is unique up to multiplication by -1. (See [7, p. 72].) With Zo := a and Zn+1 := b we define the mesh of Tn by

Mn := Mn(Tn : [a, b)):= ~ax IZi - zi-11 1~I~n+1

On the Denseness of Weighted Incomplete Approximations 421

and the mesh of Tn restricted to an interval I := [a,,8] C [a, b] by

where Zk-l := a , Zj+1 := ,8 and Zk < ... < Zj are all the zeros of Tn in (a,,8).

Lemma 1. Assume that

Hn := span{go, ... ,gn}

is a Chebyshev system on [a, b] with associated Chebyshev polynomials Tn.

a) Suppose that each gi E C 1[a,,8] and H~ := span{g~, ... ,g~} is a Chebyshev system on [a,,8] C [a, b]. II I E C[a,,8], then there exists h~ E H~ := span{l,go, ... ,gn} such that

where 6n := Mn(Tn : [a, b])l[o.,I1]. (Here D is a constant that depends only on a and b, and wJ is the modulus o/continuity). b) Suppose further that I E C[a,,8], that I is a closed interval contained in [a, ,8], and that I is constant on [a,,8] \ I. Then there exists h~ E H~ such that

IIh: - 111[0.,11] :5 D'wJ(../~) where 6~ := Mn(Tn : [a, bDlI and D' depends only on a and b.

Proof. The proof of Lemma 1 follows [1] closely where a similar result is proved for Markov systems, but is reworked for current purposes. Note that H~ a Chebyshev system on [a,,8] implies that H~ is a Chebyshev system on [a,,8]. Suppose Sn E H~ is the best uniform approximation from H~ to F on [a, c] U [d,,8], where

{ 0,

F(z) := 1,

Z E [a,c]

Z E [d,,8].

Then we claim the following: A) Sn is monotone on [e, d]. B) IISn - FII[o.,c]U[d,.8] :5 106n/(d - e).

Let" := n + 2 be the size of the Chebyshev system H~. Since Sn is a best approximant to F, there exist T) + 1 points where the maximum error, En, occurs with alternating sign. Suppose m+ 1 of these points Yo < ... < Ym lie in [a, c] and ,,- m of these points Ym+1 < ... < Yf/ lie in [d, ,8]. Then S~ has at least m - 1 zeros in [a, e] (one at each alternation point in [a, e] except possibly at the endpoints a and c). Likewise S~ has at least ,,-m-2 zeros

422 P. Borwein, E.B. Saff

in (d,{J). So S~ has at least '7-3 zeros in (O',e)U(d,{J). Note that this count excludes Ym and Ym+1. Thus S~ has at most one more zero in (a, (J) unless S~ vanishes identically (which is not possible for '7 > 1). Now suppose S~ has a zero (with sign change) on (e, d). Then since there is at most one zero of S~ in (e, d) it cannot be the case that both Ym = e and Ym+l = d with both S~(e) =1= 0 and S~(d) =1= O. (Otherwise sign(Sn(e) - !(e» = sign(Sn(d) - !(d) as a consideration ofthe two cases shows.) But if Ym =1= e or Ym+1 =1= d or S~(e) = 0 or S~(d) = 0, we have accounted for all the zeros of S~ by accounting for the (possibly) one additional zero (either S~ vanishes at e or d or one of Yin or Ym+l is an interior alternation point where S~ vanishes). Thus S~ has no zeros with sign change in (c, d) and claim (A) is proved.

For claim (B) we make the following observation; Let

fn := IIF - Snll[a.c]u[d.~].

Then Dn := fnTn - Sn

has at least m zeros on [a, c] and

D~ := Dn + 1 = 1 + fnTn - Sn

has at least '7 - m - 1 zeros on [d, {J] (counting the possibility of double zeros). Thus D~ has at least '7 - 3 zeros on [a, c] U [d, (J]. Suppose Tn has at least 4 alternations on an interval [6,,,),] C (c,d) and suppose that

Then, because of part (A) and the oscillation of Tn on [6,,,),],

has at least 3 zeros on [a, (J] and hence

has at least 2 zeros on [6,,,),]. This, however, gives ~ E H~ a total of at least '7 - 1 = n + 1 zeros which is impossible. In particular,

on any interval [6, ")'] c (c, d) where Tn has at least 4 alternations. Thus


However, since Sn is a best approximation,

and we deduce claim (B) on comparing these last two inequalities and noting that €n ::; 1/2.

The proof of (a) is now a routine argument which for simplicity we present only on the interval [a,p] := [0,1]. Let

m-l( ('+1) (.)) v(x) := t; I 'm - I ~ Si(X) + 1(0),

where Si(X) E H~ is the best approximant to

(as in claims (A) and (B». Then with (d - c) = l/m we deduce that

and with m := 1/.,;0;:

The proof of part (b) is an obvious modification of the proof of part (a) .

• 3 Weighted Incomplete Approximants

for Special Weights

We restrict our attention to systems of the form

H . { n 1 n n n} n .= span w . ,w X, ••• , W x

where w := w(x) ~ 0, x E [a,b]. Then for a large class of weights w we are guaranteed the existence of. a support set Sw where all the zeros of all the associated Chebyshev polynomials lie. Moreover, whenever Hn satisfies the conditions of Lemma 1 we will be able to conclude that {Hn} is dense in the continuous functions that vanish off of Sw' Denseness, for such I, in this context means that there exists In E Hn,liffin_oo In = I. The basic result we need is (essentially) Corollary 2.5 due to Mhaskar and Saff [10].

424 P. Borwein, E.B. Saf(

Theorem 1. Let E := [a, b] with a, b possibly infinite. Let w(z) = e-Q(~) where, Q(z) is continuous on [a, b], convex on (a, b) and where w(z) ·Izlo as Izl - 00 (when E is unbounded). There exists a smallest compact interval Sw C E with the following properties. a) The Chebyshev polynomials for Hn have all their zeros in Sw. b) The zeros are dense in Sw in the sense that Mn Is .. - 0 as n - 00.

c) If Pn E Un, then IIwnpnlls .. = IIwnPn 111:, n = 0, 1, .... d) If Pn E Un and A is a compact subset of E \ Sw, then

IIwnPnllA = o(lIwnPnllsw)' as n - 00.

The interval Sw is known (cf. [10]) to be the support of the unique probability measure Pw that minimizes the generalized energy integral

I[P]:= J J log[lz - tlw(z)w(t)]-ldp(z)dp(t)

over all probability measures supported on [a, b]. Moreover, for the case when Q(z) is convex on [a, b], the endpoints of the support set Sw = [c*, d*] can be obtained by maximizing the so-called F-/unctional

F(c, d) := log (d - c) _! ill Q(z)dz , 4 11" c V(d - z)(z - c)

over all pairs (c, d) with a $ c < d $ b. This maximum will be attained precisely when c = c* and d = d*, i.e. at the endpoints of Sw. .

Lemma 2. a) Suppose w satisfies the conditions of Theorem 1 on [a, b] and S; = [a, b] \ Sw is nonempty. Let Hn := span{ wn . 1, wnz, . .. ,wnzn }. Suppose that

H* .- {I n 1 n n n} n .-span ,w . ,w z, ... ,W z

and H~ := span{(wn .1)', (wnz)" ... , (wnzn)')

are both Chebyshev systems on the interval [a, b], for all n. Then for every f E C[a, b] that vanishes identically on S; (a collection we denote by Co[SwD, there exists a sequence Pn E Un, with

lim IIwnPn - fll[G 11] = o. n~oo J

(This is referred to as {Hn} being dense in Co[SwD. b) Suppose H: and H~ are Chebyshev systems on Sw (but not necessarily

on [a, bD. Suppose the other assumptions of (a) hold. Then there exists a sequence Pn E Un with


and lim IIwnPn - IliA = 0,

n-+oo

where A is any compact subset 0/ [a, b] \ Sw.

Proof. By Lemma 1 and Theorem 1, IE Co[Sw] is uniformly the limit of elements h: E H~ on [a, b]. We now show that I is actually the limit of elements qn E Hn. If h: --+ /, h: E H~, then we may write

If lanl --+ 00, then qn{z)/an --+ -1 uniformly on [a,b] and we may approximate constants from {Hn }. If lanl f+ 00, then there exists {an.} with an. --+ C I- ±oo. In this case IIqn.lIs .. is uniformly bounded and by Theorem 1, part (d), if A is a compact subset of E \ Sw, then IIqn.IIA --+ 0. From this and the assumption that / == 0 on S~ we deduce that an. --+ 0 and we are done. •

We wish now to record classes of weights which satisfy the conditions of Lemma 2, part (a), because for these weights we can conclude that the weighted incomplete approximants are dense exactly in Co[Sw].

Lemma 3. Suppose wE COO [a, b], w(z) ~ o. I/span{l, wn ·l, ... , wnzm } is a Chebyshev system/or all positive integers nand m, then span{(wnl)', ... , (wnzm ),} is also a Chebyshev system.

Proof. See [7, p. 378].

Lemma 4. Suppose either a) l/w is totally monotone on [a, b] or b) l/w(z) = E:=o (In(z-a)n ,an ~ 0, is convergent on [a, b], where a ~ o. Then w satisfies the conditions 0/ Lemma 3.

Proof. To show that w satisfies the conditions of Lemma 2 it suffices to show that a non-vanishing linear form

1 m . Lm{z):= ~( ) - Ebiz'

W Z i=O

has at most m+ 1 zeros. This follows, in both cases, on differentiating m+ 1 times and observing that (Lm(z»(m+l) has no sign changes in [a,b]. •

This gives us the following result.

Theorem 2. Suppose w satisfies the conditions 0/ Theorem 1 and that either a) w-1 is totally monotone on [a, b] or b) w-1 has a power series expansion at a, convergent on [a, b], with nonnegative coefficients or ( equivalently to (b»


b') w-1 has all derivatives strictly positive on (a, b].

Then {wnpn}, Pn E lIn, is dense in Co [Sw].

Observe that weights of the following form work on any interval [a, b] C [0,00).

a) exp( -zP), P a positive integer; b)zll, 9>0; c) exp( _z6) , 6 E (0,1).

For (c) above, the convexity condition of Theorem 1, doesn't hold. However in the case ~ := [a, b] c [0,00), we can replace the convexity of Q = log(l/w) by the condition that zQ'(z) is strictly increasing on (a, b).

We remark that for the generalized Weierstrass problem von Golitschek, Lorentz, and Makovoz (cf. [4]) have simultaneously but independently obtained results similar to Lemma 2.

4 The Sublinear and Superlinear Cases

For a sequence of positive numbers {An}f , we consider weighted spaces

We expect the following to happen for "decent" nonconstant weights. If An -+ 00, the approximation should be impossible. If An -+ 0, then the whole interval becomes the interval of approximation. If An -+ C > 0, then the approximants should live on the set Swc associated with we. We prove the following.

Theorem 3. Assume w E C[a, b], w ~ 0 and w is nonconstant on [a, b]. Suppose that An -+ 00 as n -+ 00. If there exist wn>'''Pn E Hn(w, An) such that wn>'''Pn -+ f as n -+ 00 uniformly on [a, b], then f == O.

Proof. Suppose wn>'''Pn E Hn(w,An) converges uniformly to f > 0 on [a,.8] C [a, b]. Since w is not constant on [a,.8]' there exist intervals h and 12 contained in [a,.8] with

0< Cl ~ w(z) ~ C2, Z E 11 ,

0< ca ~ w(z) ~ C4 < cl, z E h

for some positive constants cl, C2, ca, C4. Now from the convergence of wn>'''Pn to a strictly positive limit on [a,.8] we deduce the existence of positive constants db d2 , da, d4 so that

(4.1)


and d4 d3

fi:r $ IIPn(z)1I12 $ fi:r' n ~ N2 • C4 .. C3 ..

(4.2)

From (3.1) and Bernstein's inequality we have

IIPn(z)II[o,II] $ ~, n ~ N l , cl ..

(4.3)

for some constant ds . However with (3.2) and the facts that ~n -+ 00 and C4 < Cl this leads to the contradiction that

for some large n. • Theorem 4. Suppose that for n large w>'.. as well as w satisfy the conditions of Lemma 2. a) Iflimn ...... co~n = (J > 0, then {Wn>'''Pn} is dense in Co[S'], where S' is the support associated with w'. b) If limn ...... co ~n = 0, then {I, wn>'''Pn} is dense in C[a, b].

Proof. Part (a) requires knowing that Sw only depends on the nth root asymptotic (cf. [11]) and the rest follows as before.

For part (b) we show that the zeros of the Chebyshev polynomials fill out [a, b] and apply Lemma 1. For this purpose consider functions of the form

where the integer m+k divides n. Observe that since (Fn)(mH)/n converges to (b - z)m(z - a)1: uniformly, Fn behaves like a c5-function. In particular, given I C [a,b] it is possible to construct qn E {wn>'''Pn} , V n ~ N£ so that

IIqn(z)II[o,II]_1 $ E, r:!:fqn(z) ~ 2 and

minqn(z) < -2. IICEI -

In fact, such qn can be constructed having many oscillations of magnitude ~ 2. It now follows, that for n ~ N£ the Chebyshev polynomial, Tn, for {wn>'''Pn} has a zero in I; otherwise Tn would have too many zeros. Thus, M n l(a,II] -+ ° and we can apply Lemma 1, to get denseness. •

5 Remarks

1. The condition in Lemma 2 that H~ be Chebyshev can be weakened to the following condition (as is apparent from the proof of Lemma 1).


Condition. Suppose Sn E H~ is the best approximation to 1 on [a, e] U [d,,8] C Sw where

{ 0, z E [a,e]

I(z) := 1, z E [d,,8] .

Then S~ has at most n + 1 zeros in the interval (ai, b' ) where a' is the first alternation point and b' is the last alternation point of error.

This always holds for the Jacobi weights ZU (1 - z)tI , u, V > 0 because the zeros at 0 and 1 imply the above condition and we deduce that Lemma 2, part (b) holds for these weights. In this case the interval Sw can be given explicitly (cf. [7]):

and Tl = arcsin (l~t~tI) and T2 = arcsin (l~~~tI) . Similar extensions to the generalized Jacobi weights (z - al)Ul(z - a2)UlI ... (z - an)u" exist for similar reasons.

References

[1] P.B. Borwein, Zeros 01 Chebyshev Polynomials in Markov Systems, J. Approx. Theory, 63(1990), 56-64.

[2] P.B. Borwein, Variations on Muntz's Theme, Bull. Canadian Math. Soc., 34(1991), 1-6.

[3] M. v. Golitschek, Approximation by Incomplete Polynomials, J. Approx. Theory, 28(1980), 155-160.

[4] M. v. Golitschek, G.G. Lorentz and Y. Makovoz, Asymptotics 01 weighted polynomials (these Proceedings).

[5] X. He, Weighted Polynomial Approximation and Zeros of Faber Polynomials, Ph.D. Dissertation, University of South Florida, Tampa (1991).

[6] X. He and X. Li, Uniform Convergence of Polynomials Associated with Varying Jacobi Weights, Rocky Mountain Journal, 21(1991), 281-300.

[7] S. Karlin and W.J. Studden, Tchebysheff Systems with Applications in Analysis and Statistics, Wiley, New York, 1966.

[8] A. Kroo and F. Peherstorfer, On the Distribution of Extremal Points of General Chebyshev Polynomials, (to appear).


[9] D.S. Lubinsky and E.B. Saff, Uniform and Mean Approximation by Certain Weighted Polynomials, with Applications, Const. Approx., 4(1988), 21-64.

[10] M.N. Mhaskar and E.B. Saff, Where Does the Sup Norm of a Weighted Polynomial Live?, Constr. Approx., 1(1985), 71-91.

[11] E.B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer-Verlag, (to appear).

[12] E.B. Saff and R.S. Varga, Uniform Approximation by Incomplete Polynomials, Internat. J. Math. Soc., 1(1978), 407-420.

Peter Borwein Department of Mathematics Dalhousie University HalifaJ(, N.S. B3H 4H8 CANADA

E.B. Saff Institute for Constructive Math. Department of Mathematics University of South Florida Tampa, FL 33620 U.S.A.

Asymptotics of "Weighted Polynomials

M. v. Golitschek G.G. Lorentz Y. Makovoz

ABSTRACT We survey recent developments in the theory of the weighted polynomials w(x)n Pn(x), Pn E Pn on a closed set A C R, with a continuous weight w(x) 2: 0 on A. Important questions are: Where are the extreme points of the weighted polynomials distributed on A, in particular the alternation points of the weighted Chebyshev polynomials WnCw,n? Which continuous functions f on A are approximable by the weighted polynomials? How do the polynomials Pn of weighted norm Iiwn Pnlic(A) = 1 behave outside of A? This is based on our own work (in particular, in the first two sections) and on work of Mhaskar and Sa.ff, and others.

1 Introduction, Essential Sets, Weighted Chebyshev Polynomials

Our exposition is sufficiently different from that in existing papers, where many results were often given for special weights, in different form and with different proofs. See for example Gonchar and Rahmanov [4], [5], Kemperman and Lorentz [6], Mhaskar and Saff [12], [13], [14], Rahmanov [15], Saff, Ullman and Varga [16], Saff and Varga [17]. Often we cannot give complete references, but the proofs that we give seem to be new. The notion of minimal essential sets is also new.

Instead of ordinary algebraic polynomials Pn E 'Pn of degree ~ n, we consider the weighted polynomials w(z)n Pn(z) on a (usually compact) set A C R, with a continuous weight w(z) ~ 0 on A. Two reasons can be given for this interest: for w(z) = zu, u> 0, on [0,1] one obtains the class of incomplete polynomials of Lorentz [7] with interesting and important properties, developed by him, by Kemperman, Saff, Varga, v. Golitschek and others. Another reason lies in the theory of orthogonal polynomials Pn(z) on (-00,00) with exponential weight W(z) = e- 1xl". The functions W(z)Pn(z), by the substitution z = n1/O/y, become the weighted polyno, mials w(y)nQn(Y), Qn E 'Pn. The problems that arose proved to be of great importance for the theory of orthogonal polynomials. Investigations

PROGRESS IN APPROXIMATION THEORY (A.A. Gonchar and E.B. Saff, eds.), @Springer-Verlag (1992) 431-45l. 431

432 M. v. Golitschek, G.G. Lorentz, Y. Ma.kovoz

of Saff, Mhaskar, Lubinsky and others led finally to a solution of the famous Freud conjecture, in the papers Lubinsky and Saff [9] and Lubinsky, Mhaskar and Saff [11] and to their investigation of "strong asymptotics" in [10]. A tool in proving Freud's conjecture were results on the distribution of zeros of orthogonal polynomials with weight exp (-lzla) obtained independently by Mhaskar and Saff [12] and Rahmanov [15], (see also Gonchar and Rahmanov [4]).

In the book [8], being written by the present authors, this theory will be also treated, at the level of ''root asymptotics" (behavior of v'IPn(z)1) in the uniform norm. This has been developed by Mhaskar and Saff [12], [13]. However, their work is not easily accessible. The present paper can be viewed as an exposition of a chapter of [8], which treats this theory. (We shall mark with an * theorems whose proof is omitted.)

A closed subset B of A is an essential set for w if

(1.1)

for each weighted polynomial tP := wn Pn , Pn E Pn . One obtains an equivalent definition if one requires that (1.1) is valid only for sufficiently large n, since a positive integer power of any function tP is again a function of the same form and both functions attain their norms at the same points. The space 4>n of all tP = wn Pn of given degree n is linear, but the space 4> of all tP of arbitrary degree is not. However, if tP = wn Pn , tP = wmQm, then atPm + btPn E 4> for all real a, b.

For a polynomial P E Pn , let En(P) denote the set of all maxima of Iwn Pn I in A. Then we have

1. For each P E Pn , E2n(P2) = En(P)j 2. For each P ~ 0 on A, n ~ 2, and each Zo E En(P), there is aPE Pn

with En(P) = {zo}. For the proof, one takes Pn(Z) := Pn(Z) - e(Z - zO)2, with sufficiently small e > O. From these properties we derive the existence of a (unique) minimal essential set Bo for w, which is contained in any other essential set B:

Proposition 1.1. The set Bo exists and is equal to

Bo = U En(P) j (1.2) PeP ..

equivalently, Bo is the set of points Zo E A with the property that for each 6> 0, there is a Pn E Pn with the maximum of Iwn Pnl attained at some point Z1 E A, Izo - z11 < 6.

Example 1.2. For A = [0,1] and w(z) = zU, (1 = 0/(1-0),0 < 0 < 1, the wn Pn are essentially the O-incomplete polynomials, of Lorentz, originally defined to be zk Pn-k(Z), k/n -+ 0, and it is known that Bo = [02,1].

But Bo is not always an interval if A is one:

Asymptotics of Weighted Polynomials 433

Example 1.3. For A = [-1,1] there exists a weight w > 0 and a sequence of disjoint closed intervals Ij, j = 1,2, ... , Ij C A numbered from left to right, so that 12i+1 C Bo, 12i n Bo = 0.

If III 12 are two disjoint intervals, and if 12 is contained in the 6-neighborhood, 6 > 0, of 11 , then by an estimate of Bernstein, IPn(x)1 ~ pnllPnllIl' x E 12, where p := p(I1, 12) > 1 depends only on 6 and the length of 11; moreover, for given It, p -+ 1 for 6 -+ O.

We take a sequence of intervals Ij C A := [-1,1]' j = 1,2, ... beginning with 11 = [-1,0], so that Pj := p(/j, IHI) decreases strictly to 1, and take pj > Pj, pj -+ 1. Let w(x) := 1, x E 12j_ b j = 1,2, ... , w(x) := I/Pj, x E 12j, j = 1,2, ... , w(±I) = 1, we interpolate w linearly outside of the Ij. If Cn is the ordinary Chebyshev polynomial, then IIwnCnllA = 1 for large n. The extreme points of wnCn on each 12j-1 are dense there, and by Proposition 1.1, Bo :J 12j-1. On each 12j, however, for Pn E Pn , Iw(x)nPn(x)1 ~ (pj/pD < 1, ifllwnPnllA = 1, and Bon/2j =0. •

We shall add a lemma about oscillating weighted polynomials, needed in §2, which gives a uniform estimate for all intervals leBo of length 6 > O.

Lelllllla 1.4. Let a weight w(x) ~ 0 on A, real numbers 0 < 1/ < 1, 6> 0 and an integer m ~ 1 be given. There exist arbitrary large integers n with the following property: Each interval leBo of length 6 contains m interior points t1 < ... < tm so that for some weighted polynomial tPn := wn Pn

(-I)jtPn(tj)~(I-1/)lItPnIlA' j=I, ... ,m (1.3)

(1.4)

Proof. Each maximal interval J C Bo oflength ~ 6 we divide into subintervals of length 6/(m + 1), plus perhaps an additional interval of length < 6/(m+l). If he, k = 1, ... , N are these intervalsoflength 6/(m+l) for all J's, numbered from the left to the right, we have N ~ (m+l)IAI/6. For each he, k = 1, ... , N, by 2 and Proposition 1.1, we select a weighted polynomial ifJ,. := wnkPnk ~ 0, which has the unique maximum ifJ,.(TJ:) = lIifJ,.IIA = 1 at some interior point 'TIc of J,.. Replacing the ifJ,. by their high powers, if necessary, we can assume that

lifJ,.(x)1 < 1//(2m), x E A \ J,. . (1.5)

We shall further assume that all ifJ,. are of the same degree. This can be achieved by replacing them, if necessary, by their powers, ifJ~/nk, where M:= [l,.n,..

Let leBo be an interval of length 6. It contains a sequence of m contingent intervals J,., k = r + 1, ... , r + m, for some r. Then the weighted polynomial

m

tPn(x):= L(-IYifJr+j(X) (1.6) j=l

will have the properties (1.3) and (1.4), where tj := Tr+j. •

434 M. v. Golitschek, G.G. Lorentz, Y. Makovoz

Practically important is the symmetric case.

Proposition 1.5. Let the set A C R be symmetric with respect to zero, let w be even. Then

(i) In the formula (1.2) for the minimal essential set B o, one can restrict oneself to functions wn Pn with even polynomials Pn;

(ii) The set Bo is symmetric;

(iii) If A* := {y ~ 0, .,;y E A} and if Bo is the minimal essential subset of A* for w*(y) := w(.,fY)2, y ~ 0, then

Bo = {x = ±.jY; y E Bo} . (1.7)

Proof. (i) Let Xo E Bo. Then in each neighborhood of Xo there is a point Xl so that for some polynomial Pn E Pn, the function ¢ = wnPn has a unique maximum = 1 of I¢I at' Xl. For a given D > 0, we can assume (by replacing ¢ by ¢m with large m if necessary) that 1¢(x)1 < ! outside of U6(Xl) := {x : Ix - XII < D}. If Ql and Q2 are the even and the odd parts of Pn, and ¢j = wnQj, j = 1,2, then l¢j(x)1 = (1/2)1¢(x) ± ¢(-x)1 < 1/4 outside ofthe set U6(Xl)UU6( -xd. Moreover, for at least one j, I¢j(xdl ~ 1/2. For this j, t/J(x) := ¢J = w2n QJ, where Qj is an even polynomial, t/J(xI) ~ 1/4, and t/J takes all its maxima in U6(±xI).

(ii) follows immediately from (i). (iii) Because of the symmetry of Bo, we consider only points Xo > O.

An even polynomial P2n(x) is = Qn(x2) for some Qn E Pn, and w(x)2n = w*(x2)n. Thus, Xo is a limit point of x> 0 with w(x)2n P2n(x) = IIw2n P2n ll for some even P2n if and only if Xo = Y5, Yo > 0 and Yo is a limit point of y's with w*(y)nQn(Y) = IIw*" Qnll. Thus, Xo E Bo, Xo > 0 is equivalent to Xo =.;Yo for some Yo E Bo· •

3. As a special case, let A = [a, b], a < c ~ b, and w(x) > 0 on [a, c), w(c) = O. Then c rt. Bo. Indeed, the assumptions imply the following. For each c > 0 there is a neighborhood U of c so that for all large n and all Pn E Pn ,

(1.8)

For the proof, we take a < Cl < c, then for some p > 1, A is contained in the interval concentric to [a, cd oflength p(cl-a). Let D > 0, Cl < c-6 < c, U := AU [c - 6,c + 6]. By a simple lemma, if Pn E Pn, IPn(x)1 ~ M on an interval oflength t, then IPn(x)1 ~ M(p+ JP2=1)n ~ M(2p)n, on a concentric interval of length tp, p> 1. If ml is the minimum of w(x) on [a,cl], and m its maximum on U, then for X E U,


< (2P::) n IIwn Pnll[IJ,Cl] ~ (~) n IIwn PnllA ,

if 6 is taken so small that 2pm/ml < 1/2. The conclusion is still true if we assume w(z) > 0 on (a, c).

4. Similar definitions and facts apply to the sets A = R and A = ~. Let w(z) = e-f(~), where q(z) is an even function on R, increasing to infinity for z -+ 00 and satisfying

"'e-f(~) -+ 0 ... 00 ... , ... -+ . (1.9)

Then, since znw(z)n -+ 0 for z -+ 00, a11 weighted polynomials wn Pn have finite supremum norms IIwn PnIlR. Consequently, for all large {J, [-{J, {J] is an essential subset of R for the weight w, and [0, {J] is an essential subset of ~. Indeed, if b > 0 is fixed and (J > b is so large that (2{J/b)e f (b)-q(P) < 1/2, then as in 3 one proves that

For a compact set A C R and a continuous non-negative weight w on A, there exists a unique monic polynomial Qw,n E 1'n which solves the minimum problem

(1.10)

The polynomial Cw,n := C~,n = Qw,n/ Ew,n is called the weighted Chebyshev polynomial on A. We have:

Theorem 1.6-. (i) wnCw,n has n + 1 distinct extrema eo,n < ... < en,n contained in Bo;

(li) For each Pn E 1'n with IIwn PnllA ~ 1, IPn(z)1 ~ ICw,n(z)1 for z ~ eo,n or z ~ en,n;

( ••• ) CBo cA III wn= wni

(iv) Th~ limit llmn_oo EY,: exists for each wand A.

2 Weierstrass Theorems

We assume in this section that A = [a, b] is a compact interval and that w(z) > 0 on (a, b). Let G c A be a fixed set that is open in Ai we denote by CoCA) the space of all functions l E C(A) which vanish on A \ G. We call G a Weierstrass set for the weight w if aU f E CoCA) are approximable by weighted polynomials tPn = wn Pn: for each e > 0 there should exist a tPn of arbitrary large degree n with III - tPnllA < t.

A useful remark is that a point Zo of a Weierstrass set G belongs to the minimal essential set Bo. For we can take f E CoCA), I ~ 0, with


a unique maximum at xo, I(xo) = II/I1A = 1, then in any neighborhood of Xo, if wn Pn = </> approximates I well, we shall have a point Xl with </>(xI) = II</>IIA. As a corollary, each G is contained in the open kernel 01 Bo, that is the largest set that is open in A and is contained in Bo .

Sometimes this can be inverted: the largest Weierstrass set is the open kernel of Bo. The first example are the incomplete polynomials, where this statement has been conjectured by Lorentz, proved by Saff and Varga [17] and v. Golitschek [2].

We shall call a point Xo E A an approximate zero of the weight w if for each c > 0, there is a neighborhood U of Xo with the property

Plainly, Xo r;. Bo. From 3 of § 1, each zero of w is also an approximate zero of w. By z we denote the number 01 approximate zeros among the endpoints of a,b of A.

In the case that w(x) > 0 on [a, b], the weighted polynomials </> = wn Pn

of degree n form a Haar space c)n on [a, b]. Therefore the polynomial of best approximation to I E C(A) exists and has the following description. If the error of approximation is III - </>11 = c, there are n + 2 points a $ Xl < ... < xn +2 $ b so that for u = 1 or u = -1,

I(Xj)-</>(Xj)=u(-I)jc, j=I, ... ,n+2. (2.2)

This remains valid for each fixed n if IE Co(A) and if w has one or two zeros among the a, b. For example, consider the first possibility with a zero at a. Since I(a) = 0, and since all </> E c)n by 3 are arbitrary small on small neighborhoods of a, the minimal norm of III - </>II A6 will be the same and = c > 0 on A.s := [a + 6, b] for all sufficiently small 6 > O. On A.s, c)n is a Haar space, and we obtain (2.2) with some Xj, a < Xl < ... < Xn +2 $ b.

A point Yo E A is a proper extreme point of a continuous function 9 E C(A) if Yo is not one of the endpoints, and if g(x) $ g(yo) (or ~ g(yo» on some neighborhood (Yo - T/, Yo + T/), T/ > 0 of Yo·

For instance, let a continuous function 9 have at the points Xl < ... < xP'

p ~ 2 differences of alternating sign, beginning with a positive one, of size ~c > 0:

19(XH1)-g(Xj)l~c, j=I, ... ,p-l.

Then 9 has p- 2 alternating proper maxima and minima Y1 < ... < yp-2, with differences of size ~ c. Indeed, there is a maximum of 9 at some yl, Xl < Y1 < X3, and we can apply induction, replacing points Xj by Y1 < X3 < ... < xp.

We set Xo := a or X n +3 := b if a or b, respectively, is an approximate zero of w.

Lemma 2.1. Let I E Co(A) lor some open set G C Bo, and let III -</>IIA = c > 0 lor its weighted polynomial </> E c)n 01 best approximation. If 6 > 0 is


so small that w(f,6) < g/3 , (2.3)

then for all large n there are only two possibilities: (a) ¢ has at least n+ 1 +z sign alternating differences ¢(Zj) - ¢(zHt) of absolute value ~ g/3; (b) at least one of the n + 1 + z intervals [Zj, ZHI] contains an interval of G of length 6.

Proof. Assume that (b) does not occur. IhHI-Zj < 6, then If(zj+1)f(zj)1 < g/3 and, subtracting two relations (2.2), we obtain

If ZHI - Zj > 6, the Zj and Zj+1 are both at a distance $ 6 from A \ G. As a consequence If(zj)1 < g/3 and If(zHI)1 < g/3, which also implies (2.4). This gives n + 1 changes of sign of the differences (2.4).

Let now a be an approximate zero of w. For all sufficiently large n, by 3 of §1, we have 1¢(a)1 < g/3, and a ft Bo, so that a E A \ G. Hence f(a) = 0, and Zl cannot be a. If Zl - a> 6, then Zl is at a distance::; 6 from A \ G, hence If(zl)1 < g/3. If Zl - a::; 6, then If(zl) - f(a)1 < g/3. In both cases we have by (2.2) with j = 1:

¢(a) - ¢(zt) (¢(a) - f(a» - (¢(zt) - f(zd) - f(zd

We get an additional sign alternation of the differences. If b is an approximate zero, the argument is the same. •

To apply Lemma 2.1, we shall say that the weight w on A has the property (E) if

(E) No weighted polynomial ¢n E c)n, n = 1,2, ... can have n + z proper extrema on [a,b].

In particular, if w is continuously differentiable, then at each proper extremal point Zo we must have !{w(z)nPn(Z)}.~=",o = O. Thus, w will have property (E) if z ~ 1 and if (wn Pn )' is a Haar space on [a, b].

Theorem 2.2. If the weight w has property (E) on [a, b], then the open kernel of the minimal essential set Bo is a Weierstrass set.

Proof. It is sufficient to prove that if f E C(A) vanishes on A \ Bo, then for the ¢n E c)n of best approximation to f we have liminf IIf - ¢n IIA = O. If this is not true, then for all sufficiently large n, IIf - ¢n II > g > o. If we define 6 > 0 to satisfy w(f, 6) < g/3, then the alternative (a) of Lemma 2.1 for the Zj cannot happen for these n.

For all large n there must exist intervals [Zj, ZHI] of type (b). Their number no has the upper bound (b - a)/6. They will divide the n + 2 + z


points Zj into no + 1 groups, with Pi, i = 1, ... , no + 1, Pi ~ 0, points in the i-th group.

To the first interval I of type (b) we apply Lemma 1.4 for one of the large n of this lemma, for ° < TJ < 1/2 and for m = 2(no + 1). The norm IItPn IIA in the lemma can be selected for our convenience. We put

(2.5)

If TJlltPnll < e/9, the i-th group of points Zj will contain Pi - 1 alternating differences of tPn of size ~ c/3 and the same number of alternating differences of tP~ of size ~ c/9. If (I-TJ)lItPnll > IItPnll, the interval I will contain m - 1 alternating differences of tP~ between the points til: of Lemma 1.4. These inequalities for TJ, IItPn II are compatible. With their proper choice, we get on [a, b] at least

no+1

E(Pi-1)-(no+l)+(m-l) = n+2+z-2(no+l)+m-l i=l

n+z+l

alternating differences, or n+z proper extrema of tP~. This contradicts (E) and proves the theorem. •

Here are some of the applications of Theorem 2.2. Weierstrass theorems hold for the following weights:

EX8lllple 2.3. For the incomplete polynomials of type 0, ° < 0 < 1, we have w(z) = zt7, 0' = 0/(1- O}, A = [0,1], z = 1. If

(zt7n Pn(z»' = zt7n-1(O'nPn(z) + zP~(z» = ° in some n + 1 points of (0,1), then (zt7n Pn(z»' = 0, zt7n Pn(z) = const, which is possible only if Pn = 0.

EX8lllple 2.4. Jacobi weights w(z) = (1 + z)6 1 (1 - Z)6 2 , Sl, S2 > 0, A = [-1,1], z = 2. Here (w(z)n Pn(z»' = (1 + z)n61 -1(1 - z)n62 -1Q(z), with Q E Pn+1 cannot have n + 2 zeros in (-1,1), unless Pn = O.

EX8lllple 2.5. Equally simple considemtions apply to the weights w(z) = e-z2 on R, and w(z) = e-z on ~. However, the method seems not to yield Weierstrass theorems for w = e-z "', A = R+ for arbitrary a > O.

The interest of Theorem 2.2 is in the possibility of proving Weierstrass theorems without first determining the minimal essential set. Results similar to Theorem 2.2 are in Borwein and Saff [1] and v. Golitschek [3]. The first paper has been read at the Tampa Meeting, and a manuscript of [3] was circulated there.


3 The Role of the Dirichlet Problem

Let D be a compact subset of R. If q(z) is a continuous real function on D, we consider the following problem: Find a real valued function H(z), z E C* := C U {oo}, which for some real A satisfies

(a) H is continuous on C and harmonic on C \ D

(b) H(z) = log Izl- A + 0(1) for z -+ 00 , (D)

(c) H(z) = q(z) , zED.

In other words, (D) is the Dirichlet problem for a harmonic function H with the logarithmic growth at 00. The set D is called regular if this problem has a unique solution for each function q. We note that A is uniquely determined byH:

A := A(H) := lim (log Izl- H(z» . z-oo

(3.1)

It is known (Tsuji [18]) that D is regular if it is a finite union of disjoint non-degenerate intervals.

In this section we shall see how the solutions H of (D) and A will help us to estimate the weighted polynomials wn Pn , w(z) = exp( -q(z», in particular the weighted Chebyshev polynomials Cw,n and to find essential sets for w.

In our first theorem we shall compare the functions n -1 log IPn (z) I and H(z) since they have similar properties: If Zn := {Zl. ... , zn} are the zeros of a polynomial Pn E Pn , and if an is its leading coefficient, then the function

n

n-1 log IPn(z)1 = n-1 log lanl + n- 1 L log Iz - Zj I j=1

is harmonic on C \ Zn, and = -00 at the Zj. The function n -1 log IPn I -log Izl is harmonic at z = 00 with the value n-1 log lanl. Theorem 3.1. Let D be a compact regular subset of R, let H be the function satisfying (D). Then for each polynomial Pn E Pn the inequality

~ log IPn(z)1 ::; q(z) , zED (3.2)

implies that ~ log IPn(z) I ::; H(z) , z E C. (3.3)

Proof. Let Zn be the zeros of Pn. If jj is the complement of D U Zn in C, the function

v(z) = n-1 log IPn(z)l- H(z)


is harmonic in D. On Zn, v takes the value -00. If Pn has exactly degree n, then v(z) = const +n-1 Ej=llog Iz-zj I-H(z) is harmonic also at Z = 00.

We apply the maximum principle to v in the region C* \ (Zn U D). The function v is :5 0 on its boundary, hence :5 0 elsewhere on C. On the other hand, if Pn is of degree < n, v(z) = -00 at z = 00 and we apply the maximum principle on C* \ (Zn U D U {oo}). •

Remark. This can yield information about the minimal essential set Bo of A, if A :::> D. In the situation of Theorem 3.1, let H(z) < q(z) on an open interval J C R \ D. Then J is disjoint with Bo. Indeed, on each compact subset A C J, H(z) :5 q(z) - 6 for some 6 > O. Then for each weighted polynomial wn Pn with IIwn Pn IIA = 1 we have (3.2), and from (3.3), w(z)nIPn(z)1 :5 e-n6 on A.

We would now like to establish that the upper bound H(z) in (3.3) is asymptotically the best possible. It is natural then to prove this relation for the Chebyshev polynomials Gw,n on D, which, according to Theorem 1.6, have larger values than all comparable polynomials outside of D* (D* is the smallest closed interval, containing D). i.From this theorem, we know also that the sequence n-1log Ew,n, n = 1,2, ... converges, where Ew,n is the deviation of wnQw,n from zero on D. If not indicated otherwise, Ew,n, Qw,n and Gw,n will always refer to the set D.

We fix the continuous weight w(z) ~ 0 on A, w(z) > 0 on DCA, and put

q(z) := -logw(z) . (3.4)

Theorem 3.2. If DC R is regular, and if H is the solution of (D), then

A = A(H) :5 n-llogEw,n, n = 1,2, ... (3.5)

Moreover, one has lim (n-llogEw n) = A

n ...... oo I (3.6)

if and only if, uniformly on compact sets K C C* \ D*

lim (n-1log IGw n(z)l) = H(z) . n-+oo J

(3.7)

Proof. Since IGw,n(z)1 :5 w(z)-n on D, Theorem 3.1 implies that the harmonic functions

vn(z) := n-1log IGw,n(z)l- H(z)

satisfy vn(z) :5 0 everywhere. Consequently, since 1/ Ew,n is the leading coefficient of Gw,n,

A - n-llogEw,n = lim (log 1z1- H(:z» + lim (n-1log IGw n(z)l-log Izl) %-+00 %-+00 '

(3.8) = lim vn(z) :5 0 .

.1-+00


This yields (3.5). Since Vn are harmonic functions bounded from above, they form a normal family, and there exists a subsequence Vn" which converges uniformly on each compact K, to a harmonic function v ~ O. If we have (3.6), then by (3.8), v(oo) = O. By the maximum modulus theorem then v=: 0, and for the sequence Cw •n" we have (3.7) on K. Since this applies to any subsequence of Vn , we have (3.7) for unrestricted n -+ 00. On the other hand, if (3.6) is false, then v(oo) < 0, also v(z) < 0 in some neighborhood of 00, and (3.7) fails. •

The main theorem, Theorem 4.2 of the next section, is that for a regular D and continuous w(x) > 0 on D, the solution of (D) has the form, for some probability measure I-' on D,

H(z) = kloglz-tldl-'(t)-A. (3.9)

The integrals of type (3.9) are called logarithmic potentials (see §4). Since this integral is log Izl + 0(1) for z -+ 00, we have necessarily A = A(H). We need a simple lemma by Mhaskar and Saff:

Lemma 3.3. If the integral (3.9) exists for every z E C and depends continuously on z, then for each c: > 0 and for all sufficiently large n, there exists a monic polynomial Pn of degree n for which

n-1 log IPn(x) I < c: + k log Ix - tl dl-'(t) , xED. (3.10)

It is very easy to sketch the proof of the lemma. It follows from the hypothesis that for some points tl < ... < tv+!, tj E D, the Stieltjes integral (3.9) can be approximated by its Stieltjes sums 2:j=II-'(Ij) log Ixtjl, where Ij = (tj, tHd. In turn, the numbers I-'(Ij) can be approximated by rational numbers kj In, kl + ... + kj = n. The Stieltjes sum will be then close to

This allows us to prove

Theorem 3.4. Let the solution of the problem (D) be of the form (3.9). Then (i) we have

A(H) = lim (n- l log Ew n) n-+oo I

and consequently also the weak asymptotics (3.7). (ii) If in addition

H (x) ~ q( x), x E A ,

(3.11)

(3.12)


then D is an essential set for w on A and one has IIwn PnllA = IIwn PnIlD, Pn E Pn .

Proof. (i) Let J.l be the measure in (3.9) and let Pn be the monic polynomial of Lemma 3.3 with this J.l and a given e > O.

We know from Theorem 1.6 that wnQw,n attains its maximum and minimum at n + 1 points of D with alternating values ±Ew,n. At least one of the extreme points e of wnQw,n satisfies IQw,n(e)1 :$ IPn(e)1, for otherwise the polynomial Qw,n - Pn of degree :$ n - 1 would have n zeros. For this point e,

n-1 log IQw,n(e)1 < e + iv iog Ie - tl dJ.l(t) .

Since e is an extremal point of wnQw,n we have w(e)nIQw,n(e)1 = Ew,n and thus

log w(e) + n-1 log IQw,n(e)1 = n-1 log Ew,n . (3.13)

Since e E D one has H(e) = -log w(e). It follows from (3.9) and (3.13) that

H(e) + n-1 log Ew,n = n-1 log IQw,n(e)1 < e + iviog Ie - tl dJ.l(t)

e+H(e)+,x .

Hence we get n -1 log Ew,n < A + e for all sufficiently large integers n. Since e> 0 is arbitrary, A ~ lim n-1 log Ew,n. Combining this with (3.5) we get (3.11).

(ii) It is sufficient to consider only Pn E Pn with IIwn PnllD = 1. Then we have (3.2), hence

~ log IPn(z)1 :$ H(z) :$ q(z) , z E A ,

that is, IIwn PnllA :$ 1. •

4 The Mhaskar-Saff Theory

Or, better, this is the kernel of their investigation: an application of the extended potential theory ([12], [13]). All results of this section are due to them. In what follows, A will be a compact subset of R, w( z) ~ 0 a continuous weight on A, q(z) := -logw(z). We always assume that A contains an interval on which w(z) > O. Mhaskar and Saff study logarithmic potentials with weight

u(z) := - L log Iz - tlw(t) dJ.l(t) , z E C, (4.1)


where I' is a probability measure on A. (In the classical potential theory (Tsuji [18]), the weight w(x) = 1 on A.) The energy integrals Iw(l') L u(x)w(x) dl'(x) as functions of 1', have a finite infimum

(4.2)

The key fact is that this infimum is attained by a unique probability measure I'w called the equilibrium measure. This measure has a compact support B* := Aw = supp I'w; B* does not contain atoms, and w( x) > 0 for x E B*.

The ambitious goal of Mhaskar and Saff is to prove, under certain assumptions on w, that B* is the minimal essential set Bo. Other problems could be: to find the extremal measure I'w; to prove that B* is an interval and find it; to prove the corresponding Weierstrass theorems.

Applications of potential theory begin with the following theorem about the function

flog Iz - tl dl'w(t) - Aw, z E C lB-

Theorem 4.1* (of Frostman type). The /unction Hw(z) satisfies

Hw(x) ~ q(x) quasi-everywhere on A

Hw(x) ~ q(x) everywhere on B* = Aw.

(4.3)

(4.4)

(4.5)

(4.6)

Thus, Hw "almost" satisfies the conditions of Theorem 3.4. However, we have only that Hw is upper semi-continuous on A, not continuous. Further, from (4.5) and (4.6) it follows that Hw(x) = q(x) is only quasi-everywhere on B*, that is, with a possible exception of a set of capacity zero, likewise (3.12) for Hw is valid not necessarily everywhere on A.

Nevertheless, using classical potential theory, Mhaskar and Saff [12], [13] derive something like Theorem 3.4. Their key assumptions are that A is a finite union of compact intervals, and that B* = Aw is regular for the Dirichlet problem (D) of §3. This last assumption is not a pleasant one, although there seems to be no reason why, for smooth w, B* should not be regular. But this will follow only later. In Theorem 4.5, we shall show that for some special w, B* is an interval.

Theorem 4.2*. If A is a finite union of compact intervals, and if B* is regular, then

Hw(x) = q(x), x E B*, Hw(x) ~ q(x), x E A \ B* . (4.7)


As a consequence of Theorem 3.4, B* in this case is an essential set for w.

The final general step is to obtain the equilibrium measure JJw as the weak*-limit of certain discrete measures lin, n -+ 00. Let Z1 n < ... < Zn n

be the zeros of the weighted Chebyshev polynomial C: n. (They interla~e the extreme points eo,n < ... < en,n of this polynomial.) For any Borel set DCA, we put IIn(D) = (l/n)Nn(D), where Nn(D) is the number of the Zk,n ED.

Theorem 4.3*. If A is a finite union of compact intervals and if B* = Aw is regular, one has the weak*-convergence lin -+ JJw. In particular, for any interval I = [c, d], (l/n)Nn(I) -+ JJw(I). Moreover, B* is actually the minimal essential set for w.

To get rid of the assumption of the regularity of B*, Mhaskar and Saff use

Lemma 4.4*. If I := (ell e2) is an open interval of A \ B*, whose endpoints belong to B* = Aw, then Hw E C[6,6], and Hw(ei) = q(ei), i = 1,2.

Our best concrete theorem is as follows:

Theorem 4.5. Let A be a union A = Uj=1 Jj of finitely many compact intervals Jj, whose interiors JJ are pairwise disjoint. Set 0:0 := min {z : Z E A} and Po := max{z : Z E A}. Let w be continuous and non-negative on A, let q(z) = -log w(z) satisfy on each of the Jj at least one of the following three conditions:

(i) q(z) is convex; (ii) q(z) is continuously differentiable and (z - o:o)q'(z) is increasing; (iii) q(z) is continuously differentiable and (z - Po)q'(z) is decreasing.

Then the support B* := Aw of the equilibrium measure JJw consists of finitely many compact disjoint intervals, Ij C Jj, j = 1, ... , r; some of the Ij may be empty. Consequently, Aw is in this case regular.

(Results (ii) and (iii) have been communicated to us by Mhaskar.)

Proof. If B* nJj is not empty, let [Zj, Yj] be the smallest compact interval containing this set. We have to prove that [Zj, Yj] C B* . If this is not true, there exists a compact interval [6,6] C [Zj,Yj] for which 6,e1 E B*, and I := (e1,e2) is disjoint with B*. By Lemma 4.4, Hw E C[elle2] and q(ei) = Hw(ei), i = 1,2.

(i) Since w is continuous, q is convex on [e1,e2], and w(ed > 0, W(e2) > 0, hence q is continuous also on [e1,e2]. For Z E I, the second derivative


is negative. It follows that Hw, hence also v := Hw - q are strictly concave on [6,6]. This contradicts the properties V(ei) = Hw(ei) - q(ei) = 0, i = 1,2 from Lemma 4.4 and v(x) = Hw(x) - q(x) ~ 0, x E I, by (4.5). It follows that B" n Jj is the interval [Xj, Yj].

To derive the same conclusion in case (ii), we begin with v := Hw - q E C[6,6] and v(6) = v(6) = O. Moreover, for all x E I,

«x - O!o)H~(x»' = H~(x) + (x - O!o)H::'(x)

1 ( Ix - 010 ) 1 t - 010 = - - ( )2 dpw(t) = - ( )2 dpw(t) < 0 . B. X - t x - t B. X - t

It follows that the functions (x - O!o)H~(x) and (x - O!o)v'(x) are strictly decreasing on I. Combining v(6) = 0 with v(x) ~ 0, x E I of (4.5) we deduce that v'(6) ~ O. The function (x - O!o)v'(x) has value ~ 0 for x = 6 and is strictly decreasing, so that v'(x) < 0 for 6 < x ~ 6. This contradicts the fact that v(ed = v(6) = O.

Similarly, in case (iii) we obtain that (x - .Bo)v'(x) is strictly increasing on I, that v'(x) > 0 for el < x ~ 6, and obtain a contradiction. •

5 Determination of a Minimal Essential Interval

Let A be an interval. We assume that Aw = I = [a, b] is also known to be an interval. In addition, we assume that q(x) := -logw(x) is absolutely continuous on I.

TheoreUl 5.1. Under the above assumptions, if b is not an endpoint of A, and if q'(x) exists and is continuous in some left neighborhood of b, then

11b rz.::a ;: a q' (x) V ~ dx = 1 . (5.1)

Similarly, if a is not an endpoint of A and if q' exists and is continuous in a right neighborhood of a, then

11b ~-.x - q'(x) -- dx = -1 . 1r a x-a

(5.2)

Proof. We shall discuss only the case of (5.1), because the second case follows from the first by replacing [a,b] by [-b,-a], and w(x) by w(-x).

We can apply Theorems 4.2 and 3.4. In particular, we know that the Chebyshev polynomials C:,n = Cw,n satisfy

lim .!.log ICw n(x)1 = H(x) , x E A \ I , n~oo n '

(5.3)


where H(z) = Hw(z) is the solution of the Dirichlet problem (D) with D = I and with the boundary values q(z) = -logw(z) on I. i,From the properties of the Poisson integral it follows that this H is given by the formula

H(z) = h(u) -log lui, z = '1'(u) , (5.4)

where '1'( u) is defined in the circle lui :5 1 of the complex plane by

a+h b-a 1 '1'(u) = -2- + -4-(u + u- ) . (5.5)

The function '1'-1 maps I onto the circle T, the points a, b,oo onto -1,1,0 respectively, the set R \ I onto the open interval -1 )) d¢ hu ---()- 211" t 1-2rcos(¢-t)+r2'

·f u = rei , r:5 1 . (5.6)

It is not too difficult to prove that

h'(l) = lim h(u) - h(l) = .!.1" q'(z)Vz - adz. (5.7) u_1- u - 1 11" II b - z

Equivalent to (5.3) is the relation

lim .!.log Iw(ztCw n(z)1 = G(z) , z E A \ I , (5.8) n_oo n '

where G(z) = h('1'-l(Z» -log 1'iI-1(z)l- q(z). We also have

G(z) = g(u) := h(u) -log lul- q('iI(u» , z = 'iI(u) . (5.9)

The function g(u) is defined on -1 :5 u :5 1. If b is not an endpoint of A, we prove that

g'(l):= lim g(u) - g(l) = 0 . u-1- u-1

(5.10)

This would imply (5.1), because W'(l) = 0 and therefore h'(l) = g'(l) + 1. Since h(u) = q('1'(u» for lui =:: 1, for u = 1 in particular we get g(l) = O. Thus g'(l) < 0 would imply that for some Uo < 1, close to 1, we would have g(uo) > 0 and Zo := '1'-l(uo) E A. Then (5.8) would yield w(zo)nCw,n(ZO) -+ +00, contrary to the fact that IIwnCw,nllA = 1.

On the other hand, let g'(l) > O. From (5.7) we see that g'(l) is continuous with respect to b. Let glo hlo Glo W1 correspond to the interval 11 : [a, b1], a < b1 < b, and the old function q on 11. For h1 sufficiently close to h, gH1) > 0, and therefore gl(U) < 0 on an interval (uo, 1). By '1'1, this interval is mapped on some J := (b1' h1 + 6), 6 > O. Hence G1(z) = H1(Z)-q(z) < 0 on J. By the Remark to Theorem 3.1, (hl. b1 +6) contains no points of the minimal essential set, a contradiction. •


Theorem 5.2. Let w(z) = e- q(.,), where q(z) is an even continuousfunction, increasing on R+, which satisfies (1.9). Let also zq'(z) be increasing on R+. If b is the unique solution of the equation

~ 11 bzq'(bz) dx = 1 , 7r 0 ~

then [-b, b] is the minimal essential set of w on R.

(5.11)

Proof. We take f3 > ° so that [0, f3] is an essential set for w* (y) = w( VY)2 , Y ~ 0. By Theorem 4.5(ii) we know that the minimal essential subset of [0, f3] that is, of R+ is also an interval of Bo. Since w* (y) = e-2q(,fi) has its maximum at 0, ° E Bo, hence Bo = [0, b*]. By Proposition 1.5, the minimal essential set of R for tV is [-b, b], with b = ../b* given by (5.1):

1 = .!.lbo q'(VY) J Y dy = ~ 1b zq'(z) dz = ~ 11 bzq'(bz) dz . 7r 0 ..;y b* - Y 7r 0 vb2 - Z2 7r 0 ~ •

We shall find the minimal essential sets for some popular weights.

Example 5.3. Exponential weights. (See Mhaskar and Saff [12] and Rahmanov [15])

The exponential weights

(5.12)

a> 0, have the properties of Theorem 5.2 where q(z) := IzIO'. The endpoint b = ba of the minimal essential set [-ba , ba ] is the unique solution of

2ab~ 11 za -- ~dz=l.

7r 0 v 1- Z2 (5.13)

For the evaluation of the integral (5.13) we use the formulas

11 za 11</2 . -jir{(a + 1)/2} . ~dz = sma zdz = r( /2) o vl-x- 0 a a

Since r (a + 1) _ -jir(a)

2 - 2a - 1ar(a/2) , we get

_ (2a - 2r(a/2)2) 1/0'

bO' - r(a) . (5.14)

Theorem 5.4. For a> ° and Pn E Pn , n = 1,2, ... ,

(5.15)

448 M. v. Golitschek, G.G. Lorentz, Y. Ma.kovoz

where In(a) := [-ban1/a,ban1/a]. This is not true ifba is replaced by any smaller number.

Proof. This follows from Theorem 5.3 by the substitution y = n1/ a z .•

Example 5.5. Jacobi weights. This is the case when A = [-1,1] and, with 81 > 0, 82 > 0,

w(z) = (1- z)-1(1 + Z)-2, -1:S z :S 1 . (5.16)

The function q(z) := -log w(z) = -81Iog(1 - z) - 821og(1 + z) is strictly convex in A, and w( -1) = w(l) = 0. By Theorems 4.3 and 4.5, the support Aw of the equilibrium distribution measure I'w in a subinterval [a, b] of A, with -1 < a < b < 1, and Aw is also the minimal essential set for w.

Theorem 5.6. (Saff, Ullman and Varga [16]) For the Jacobi weight8 (5.16).

a = O~ - O~ - v'X, b = O~ - O~ + v'X (5.17)

where 01 := 8d(1 + 81 + 82). O2 := 82/(1 + 81 + 82) and ~ := {I - (01 + (2)2}{1- (01 - (2)2}.

Proof. By Theorem 5.1, a and b are a solution of the two equations

Ii" ( 81 82 ) 18-a d - --+-- -- z = -1 11' 4 z-1 z+l b-z

(5.18)

Ii" ( 81 82 ) ~-z d - --+-- -- z = 1. 11' 4 z-l z+l z-a

We shall show that the numbers a, b in (5.17) are the unique solution of this system.

For "I = ±1, elementary calculations yield

1"_1 Jz-a dz = 4 Z+"I 6-z

ll1_1_J6~z dz - 1I'J6+"I -11'.

4 Z+"I z-a a+"I

Hence, the system (5.18) is equivalent to

81 1- -- +82 1- --( [8-1) (v'ffi)+ 1 6-1 6+1

= -1

= 1.


With the numbers (h, 92 , the system for a and b becomes

(5.19)

(5.20)

The solution of (5.19)-(5.20) is not difficult: we multiply (5.19) with the factor (1 - b)V(l + a)(l + b), and subtract from it (5.20) multiplied with (1 - a)V(l + a)(l + b). We obtain

292 = V(l + a)(l + b) . (5.21)

Similarly, we multiply (5.19) and (5.20) with (1 + b)V(l - a)(l- b) and (1 + a)v(l - a)(l - b), respectively, and subtract the results from each other. This yields

291 = V(l - a)(l - b) . (5.22)

It follows that

ab = 29~ + 29~ - 1, a + b = 29~ - 29~ ,

and further that a and b are the two solutions of the quadratic equation

(z - a)(z - b) = z2 - 2z(9~ - 9n + 29~ + 29~ - 1 = 0 .

This establishes (5.17). • The work of G.G. Lorentz has been supported by an ARP grant of the

State of Texas.

References

[1] Borwein, P. and Saff, E.B., On the denseness of weighted incomplete approximation; these Proceedings.

[2] M. v. Golitschek, Approximation by incomplete polynomials, J. Approx. Theory, 28 (1980), 155-160.

[3] M. v. Golitschek, Weierstrass theorem with weights, manuscript, available at the Tampa 1990 Conference.

[4] A.A. Gonchar and E.A.Rahmanov, Equilibrium measure and the distribution of zeros of extremal polynomials, Math. USSR Sbornik, 53 (1986), 119-130.


[5] A.A. Gonchar and E.A.Rahmanov, On the simultaneous convergence ofPade approximants for systems of functions of Markov type, Proc. Steklov Inst. Math. 157 (1981), 31-48.

[6] J .B.B. Kemperman and G.G. Lorentz, Bounds for polynomials with applications, Indagationes Math., 88 (1979), 13-26.

[7] G.G. Lorentz, Approximation by incomplete polynomials (problems and results), in: "Pade and Rational Approximations," E.B. Saff and R.S. Varga, eds., Academic Press, New York, 1977, pp.289-302.

[8] G.G. Lorentz, M. v. Golitschek, Y. Makovoz, "Constructive Approximation, Advanced Problems," book in preparation.

[9] D.S. Lubinsky and E.B. Saff, Uniform and mean approximation by certain weighted polynomials, with applications, Constr. Approx. 4 (1988), 21-64.

[10] D.S. Lubinsky and E.B. Saff, Strong asymptotics for extremal polynomials associated with weights on R, Lecture Notes in Math. 1305, Springer, Berlin, 1988.

[11] D.S. Lubinsky, B.N. Mhaskar and E.B. Saff, A proof of Freud's conjecture for exponential weights, Constr. Approx. 4 (1988), 65-84.

[12] B.N. Mhaskarand E.B. Saff, Extremal problems for polynomials with exponential weights, Trans. Amer. Math. Soc., 285 (1984), 203-234.

[13] B.N. Mhaskar and E.B. Saff, Where does the sup norm of a weighted polynomial live?, Constr. Approx. 1 (1985), 71-91.

[14] B.N. Mhaskar and E.B. Saff, Where does the Lp norm of a weighted polynomial live?, Trans. Amer. Math. Soc. 303 (1987), 109-124.

[15] E.A.Rahmanov, On asymptotic properties of polynomials orthogonal on the real axis, Math. USSR.-Sb., 47 (1984), 155-193.

[16] E.B. Saff, J .L. Ullman and R.S. Varga, Incomplete polynomials: an electrostatics approach, in: "Approximation Theory, III," E.W. Cheney, ed., Academic Press, New York, 1980, pp.769-782.

[17] E.B. Saff and R.S. Varga, Uniform approximation by incomplete polynomials, Internat. J. Math. and Math. Sci. 1 (1978),407--420.


[18] M. Tsuji, "Potential Theory in Modern Function Theory," 2nd edition, Chelsea, New York, 1958.

M. v. Golitschek Inst. fUr Angewandte Mathematik Am Hiebland 8700 Wiirzburg GERMANY

Y. Makovoz Deptartment of Mathematics University of Mass. at Lowell Lowell, MA 01854-2882 U.S.A.

G.G. Lorentz Deptartment of Mathematics RLM 8-100 University of Texas Austin, TX 78712 U.S.A.

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