Applications of Hypergroups and Related Measure Algebras

A Joint Summer Research Conference on Applications of Hypergroups and Related Measure Algebras

July 31-August 6, 1993 Seattle, Washington

William C. Connett Marc-Olivier Gebuhrer

Alan L. Schwartz Editors

Recent Titles in This Series

183 William C. Connett, Marc-Olivier Gebuhrer, and Alan L. Schwartz, Editors, Applications of hypergroups and related measure algebras, 1995

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(Continued in the back of this publication)

http://dx.doi.org/10.1090/conm/183

43 32 46 15 42 5

31 33 34 29 20 4

51 25 30 17

Participants of the AMS-SIAM-IMS Summer Research Conference on Applications of Hypergroups and Related Measure Algebras

Seattle, Washington, July 31-August 6, 1993

Akkouchi,Mohammed, Morocco 36 Kunze, Ray, U.S. Askey, Richard A., U.S. 35 Lang, Christopher, U.S. Bakali, AlIal, Morocco 24 Lazhari, Nejib, Tunisia Berezansky, Yu, Ukraine 13 Levitan, B. M., Russia (now U.S.) Bloom, Walter, Australia 3 Litvinov, Gregory L., Russia Chebli, Houcine, Tunisia 47 Mabrouki, Mohammed, Morocco Connett, William C., U.S. 23 Mahmoud, Nour el Houda, Tunisia Dunkl, Charles F., U.S. 28 Noumi, Ma.satoshi, Japan Effros, Edward G., U.S. 44 Osilenker, Boris P., Russia Ehring, Martin, Germany 38 Podkolzin, Gleb, Ukraine Esterle, Jean, France 22 ROsIer, Margit, Germany Eymard, Pierre Henri, France 7 Ross, Kenneth A., U.S. Fitouhi, Ahmed, Tunisia 9 Schwartz, Alan L., U.S. Folland, Gerald B., U.S. 8 Shapiro, Rosa, Russia Fournier, John, Canada 6 Singh, Ajit Igbal, India Gallardo, Leonard, France 11 Sunder, V. S., India

(not shown) Gangolli, Ramesh, U.S. 19 Szwarc, Ryszard, Poland 39 Gasper, George, Jr., U.S. 12 Trimeche, Khelifa, Tunisia 40 Gebuhrer, Olivier, France 18 Vainerman, Leonid, Ukraine 50 Hamza, Moncef, Tunisia 45 Voit, Michael, Germany 1 Hanes, Lorna, U.S. 49 Vrem, Richard, U.S.

48 Heyer, Herbert, Germany 2 Weiss, Y sette, Germany 26 Jewett, Robert, U.S. 10 Wildberger, N.J., Australia 14 Jorgensen, Palle E. T., U.S. 21 Xu, Darning, U.S. 16 Kalyuzhnyi, Alex, Ukraine 41 Xu, Zengfu, PRC 37 Koornwinder, Tom H., The Netherlands 27 Zeuner, Hansmartin, Germany

Applications of Hypergroups and Related Measure Algebras

CoNTEMPORARY MATHEMATICS

183

Applications of Hypergroups and Related Measure Algebras

A Joint Summer Research Conference on Applications of Hypergroups and Related Measure Algebras

July 31-August 6, 1993 Seattle, Washington

William C. Connett Marc-Olivier Gebuhrer

Alan L. Schwartz Editors

American Mathematical Society Providence, Rhode Island

Editorial Board Craig Huneke, managing editor

Clark Robinson J. T. Stafford Linda Preiss Rothschild Peter M. Winkler

The Joint Summer Research Conference on Applications of Hypergroups and Related Measure Algebras was held at the University of Washington, Seattle, Wash-ington, from July 31 to August 6, 1993, with support from the Division of Inter-national Programs of the National Science Foundation, Grant No. INT-93122080j the Division of Mathematical Sciences of the National Science Foundation, Grant Nos. DMS-9208407 and DMS-9221892j the International Science Foundation, Grant Nos. 0136/1 and 0136/2, and the University of Missouri Research Board.

1991 Mathematics Subject Classification. Primary 43A62; Secondary 34B24, 43A10, 16W30, 60B99.

Library of Congress Cataloging-in-Publication Data Joint Summer Research Conference on Applications of Hypergroups and Related Measure Algebras (1993: Seattle, Wash.)

Applications of hypergroups and related measure algebras : a Joint Summer Research Con-ference on Applications of Hypergroups and Related Measure Algebras, July 31-August 6, 1993, Seattle, Washington / William C. Connett, Marc-Olivier Gebuhrer, Alan L. Schwartz, editors.

p. cm. - (Contemporary mathematics, ISSN 0271-4132; v. 183) Includes bibliographical references. ISBN 0-8218-0297-6 (acid-free) 1. Hypergroups-Congresses .. 2. Measure algebras-Congresses. I. Connett, William C.,

(William Carroll), 1939-. II. Gebuhrer, Marc-Olivier, 1945-. III. Schwartz, Alan L. (Alan Lee), 1941-. IV. Title. V. Series: Contemporary mathematics (American Mathematical Society); v. 183. QA174.2.J65 1993 515'.785-dc20 94-42671

CIP

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Contents

Preface

Limit theorems for some hypergroup structures on ]Rn x [O,oo[ N. BEN SALEM AND M. N. LAZHARI

Nuclear spaces of test functions connected with hypercomplex systems and representations of such systems

xi

1

Yu. M. BEREZANSKY 15

Hypercomplex systems and hypergroups: Connections and distinctions Yu. M. BEREZANSKY AND A. A. KALYUZHNYI 21

The Hardy-Littlewood maximal function for Chebli-Trimeche hypergroups WALTER R. BLOOM AND ZENGFU Xu 45

Sturm-Liouville hypergroups H. CHEBL! 71

Continuous 2-variable polynomial hypergroups WILLIAM C. CONNETT AND ALAN L. SCHWARTZ 89

A survey of Fourier algebras PIERRE EYMARD

Expansion in series of Laguerre functions for solution of perturbed Laguerre equations

111

A. FITOUHI AND M. M. HAMZA 129

Asymptotic behaviour of the paths of random walks on some commutative hypergroups

LEONARD GALLARDO 135

Bounded measures algebras: A fixed point approach MARC-OLIVIER GEBUHRER 171

Progress in the theory of probability on hypergroups HERBERT HEYER

ix

191

x CONTENTS

Discrete hypergroups associated with compact quantum Gelfand pairs TOM H. KOORNWINDER213

Transmutation operators and the inverse spectral problem B. M. LEVITAN 237

Limit theorems for the spin process MOHAMMED MABROUKI 245

Differential operators with matrix coefficients and transmutations N. H. MAHMOUD 261

Generalized product formula for orthogonal polynomials B. P. OSILENKER 269

An infinitesimal algebra of the hypergroup generated by double cosets and nonlinear differential equations

G. B. PODKOLZIN 287

Convolution algebras which are not necessarily positivity-preserving MARGIT ROSLER 299

Signed hypergroups-a survey KENNETH A. Ross 319

On the relation between subfactors and hypergroups V. S. SUNDER 331

Connection coefficients of orthogonal polynomials with applications to classical orthogonal polynomials

RYSZARD SZWARC 341

Generalized transmutation and translation operators associated with partial differential operators

K. TRIMECHE 347

Gel/fand pairs of quantum groups, hypergroups and q-special functions LEONID VAINERMAN 373

Central limit theorems for Jacobi hypergroups MICHAEL VOIT 395

Finite commutative hyper groups and applications from group theory to conformal field theory

N. J. WILDBERGER 413

Kolmogorov's three series theorem on one-dimensional hypergroups HANSMARTIN ZEUNER 435

Preface

The Joint Summer Research Conference held July 31 to August 6, 1993, at the University of Washington in Seattle and entitled, "Applications of Hypergroups and Related Measure Algebras," was a major event for researchers in this field. There were fifty-two registered participants from thirteen countries. Fourteen plenary talks were given by invited speakers, and there were thirty-seven more technical talks ranging in length from fifteen to ninety minutes.

The most important single thing about this conference was that it brought to-gether for the first time representatives of all the major groups of users of hyper-groups. Workers from schools in France, Germany, Russia, Poland, Tunisia, the Ukraine, etc. talked to each other about how they were using hypergroups in fields as diverse as special functions, probability theory, representation theory, measure algebras, Hopf algebras, and Hecke algebras.

This led to fireworks. Hypergroups occur so often and in so many different and important contexts, that mathematicians all over the world have been discovering the same mathematical structure hidden in very different applications, and pub-lishing theorems about these structures, in many cases without even knowing that they were talking about hypergroups. It was time for this conference to happen.

One positive consequence of this clash of different points of view was the real-ization of the need for a common perspective, if not a common theory. This need has been a stimulus for further research and has led to a number of efforts to bring some coherence to the varieties of notation and language in common usage. This effort has stimulated a lively exchange on the internet as the various issues are being thrashed out.

We feel strongly that these different perspectives and approaches should be avail-able in a single volume. We hope that these proceedings of the conference will be a valuable reference tool and starting point for the wide audience of users of hy-pergroups and will make it easier for an even larger audience to use these natural structures in their work.

The conference, and these proceedings, would not have been possible without the help of many people and organizations. Beginning with the very nascence of the idea for a conference, support, encouragement, and advice was supplied by various members of our scientific committee:

xi

xii PREFACE

Richard A. Askey, University of Wisconsin Walter Bloom, Murdoch University, Australia Houcine Chebli, Faculte des Sciences de Tunis Charles F. Dunkl, University of Virginia Jean Esterele, Universite de Bordeaux Pierre Henri Eymard, Universite de Nancy, France Gerald B. Folland, University of Washington Leonard Gallardo, Universite de Bretagne Occidentale, France Ramesh Gangolli, University of Washington George Gasper, Jr., Northwestern University Herbert Heyer, Universitat Tubingen Rodger Howe, Yale University Andrzej Hulanicki, Wroclaw University Joe Wiley Jenkins, SUNY at Buffalo and the NSF Tom H. Koornwinder, University of Amsterdam Ray Kunze, University of Georgia Boris Levitan, University of Minnesota Jean Ludwig, Universite de Metz Kenneth A. Ross, University of Oregon Khelifa Trimeche, Faculte des Sciences de Tunis We want to thank the American Mathematical Society, the Society for Industrial

and Applied Mathematics, and the Institute of Mathematical Statistics for select-ing our suggested topic as a Joint Summer Research Conference. Specific help, advice, and encouragement preceding the conference was supplied by the Associate Executive Director of the AMS, Jim Maxwell, who is a real problem solver; Chris Harkness from the Conferences Department of the AMS was the on-site coordi-nator of the JSRC, and in addition to his usual duties he assumed diverse other responsibilities ranging from the careful selection of refreshments for the reception to taking the photograph that appears as the frontispiece of this volume; finally, we thank Donna Harmon of the Publications Department of the AMS for constant as-sistance, advice, and direction in the preparation of these proceedings. Her helping hand was there from the time of the conference until the day of production.

Of course, events such as these are only possible if funding can be found. We thank the Division of Mathematical Sciences of the National Science Foundation for its support via the AMS, SIAM, and IMS for funding the Joint Summer Research Conference series. In addition, the following are to be thanked for having the vi-sion to supply the funds that made possible the participation by our colleagues from Morocco, Poland, Russia, Tunis, and the Ukraine: The International Science Foun-dation, The International Program Division of the National Science Foundation, and the University of Missouri Research Board. Some of the conference planning and editorial work was performed incidental to activities taking place under a grant for international collaborative research from NATO.

PREFACE xiii

Finally, we wish to thank the referees for their efforts. In addition to the usual work of reviewing, some spent a tremendous amount of time and effort to help several non-English speaking authors prepare what was in several cases their first publication in English.

The organizers: William C. Connett, Olivier Gebuhrer, and Alan Schwartz, Co-Chairs.

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theory and algebraic geometry, 1992 132 Mark Gotay, Jerrold Marsden, and Vincent Moncrief, Editors, Mathematical aspects of

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noncommutative rings, 1992 129 John R. Graef and Jack K. Hale, Editors, Oscillation and dynamics in delay equations,

1992 128 Ridgley Lange and Shengwang Wang, New approaches in spectral decomposition, 1992 127 Vladimir Oliker and Andrejs Treibergs, Editors, Geometry and nonlinear partial

differential equations, 1992 126 R. Keith Dennis, Claudio Pedrini, and Michael R. Stein, Editors, Algebraic K-theory,

commutative algebra, and algebraic geometry, 1992

(See the AMS catalog for earlier titles)

Applications of Hypergroups and Related Measure Algebras William C. Connett, Marc-Olivier Gebuhrer,

and Alan L. Schwartz, Editors "The most important single thing about this conference was that it brought

together for the first time representatives of all major groups of users of hyper-groups. [They] talked to each other about how they were using hypergroups in fields as diverse as special functions, probability theory, representation theory, measure algebras, Hopf algebras, and Hecke algebras. This led to fireworks."-from the Introduction

Hypergroups occur in a wide variety of contexts, and mathematicians the world over have been discovering this same mathematical structure hidden in very different applications. The diverse viewpoints on the subject have led to the need for a common perspective, if not a common theory. Presenting the proceedings of a Joint Summer Research Conference held in Seattle in the summer of 1993, this book will serve as a valuable starting point and reference tool for the wide range of users of hypergroups and make it easier for an even larger audience to use these structures in their work.

ISBN 0-8218-0297-6

9 780821 802977