Applied and Numerical Harmonic Analysis

Ahmed I. ZayedGerhard SchmeisserEditors

New Perspectives on Approximation and Sampling TheoryFestschrift in Honor of Paul Butzer's 85th Birthday

Applied and Numerical Harmonic Analysis

Series EditorJohn J. BenedettoUniversity of MarylandCollege Park, MD, USA

Editorial Advisory Board

Akram AldroubiVanderbilt UniversityNashville, TN, USA

Douglas CochranArizona State UniversityPhoenix, AZ, USA

Hans G. FeichtingerUniversity of ViennaVienna, Austria

Christopher HeilGeorgia Institute of TechnologyAtlanta, GA, USA

Stéphane JaffardUniversity of Paris XIIParis, France

Jelena KovacevicCarnegie Mellon UniversityPittsburgh, PA, USA

Gitta KutyniokTechnische Universität BerlinBerlin, Germany

Mauro MaggioniDuke UniversityDurham, NC, USA

Zuowei ShenNational University of SingaporeSingapore, Singapore

Thomas StrohmerUniversity of CaliforniaDavis, CA, USA

Yang WangMichigan State UniversityEast Lansing, MI, USA

More information about this series at http://www.springer.com/series/4968

Ahmed I. Zayed • Gerhard SchmeisserEditors

New Perspectiveson Approximationand Sampling TheoryFestschrift in Honor of Paul Butzer’s85th Birthday

EditorsAhmed I. ZayedDepartment of Mathematical SciencesDePaul UniversityChicago, IL, USA

Gerhard SchmeisserDepartment of MathematicsUniversity of Erlangen-NurembergErlangen, Germany

ISSN 2296-5009 ISSN 2296-5017 (electronic)ISBN 978-3-319-08800-6 ISBN 978-3-319-08801-3 (eBook)DOI 10.1007/978-3-319-08801-3Springer Cham Heidelberg New York Dordrecht London

Library of Congress Control Number: 2014952772

Mathematics Subject Classification (2010): 33C10, 33C45, 33E20, 41A15, 41A35, 41A58, 41A65,42B10, 42C10, 42C15, 42C40, 43A25, 44A15, 46E22, 47B37, 47B38, 47D03, 60G20, 94A08, 94A11,94A12, 94A20

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ANHA Series Preface

The Applied and Numerical Harmonic Analysis (ANHA) book series aims toprovide the engineering, mathematical, and scientific communities with significantdevelopments in harmonic analysis, ranging from abstract harmonic analysis tobasic applications. The title of the series reflects the importance of applicationsand numerical implementation, but richness and relevance of applications andimplementation depend fundamentally on the structure and depth of theoreticalunderpinnings. Thus, from our point of view, the interleaving of theory andapplications and their creative symbiotic evolution is axiomatic.

Harmonic analysis is a wellspring of ideas and applicability that has flourished,developed, and deepened over time within many disciplines and by means ofcreative cross-fertilization with diverse areas. The intricate and fundamentalrelationship between harmonic analysis and fields such as signal processing,partial differential equations (PDEs), and image processing is reflected in ourstate-of-the-art ANHA series.

Our vision of modern harmonic analysis includes mathematical areas such aswavelet theory, Banach algebras, classical Fourier analysis, time-frequency analysis,and fractal geometry, as well as the diverse topics that impinge on them.

For example, wavelet theory can be considered an appropriate tool to deal withsome basic problems in digital signal processing, speech and image processing,geophysics, pattern recognition, biomedical engineering, and turbulence. Theseareas implement the latest technology from sampling methods on surfaces to fastalgorithms and computer vision methods. The underlying mathematics of wavelettheory depends not only on classical Fourier analysis, but also on ideas from abstractharmonic analysis, including von Neumann algebras and the affine group. This leadsto a study of the Heisenberg group and its relationship to Gabor systems and of themetaplectic group for a meaningful interaction of signal decomposition methods.The unifying influence of wavelet theory in the aforementioned topics illustrates thejustification for providing a means for centralizing and disseminating information

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vi ANHA Series Preface

from the broader, but still focused, area of harmonic analysis. This will be a key roleof ANHA. We intend to publish with the scope and interaction that such a host ofissues demands.

Along with our commitment to publish mathematically significant works at thefrontiers of harmonic analysis, we have a comparably strong commitment to publishmajor advances in the following applicable topics in which harmonic analysis playsa substantial role:

Antenna theory Prediction theory

Biomedical signal processing Radar applications

Digital signal processing Sampling theory

Fast algorithms Spectral estimation

Gabor theory and applications Speech processing

Image processing Time-frequency and

Numerical partial differential equations time-scale analysis

Wavelet theory

The above point of view for the ANHA book series is inspired by the history ofFourier analysis itself, whose tentacles reach into so many fields.

In the last two centuries Fourier analysis has had a major impact on thedevelopment of mathematics, on the understanding of many engineering andscientific phenomena, and on the solution of some of the most important problemsin mathematics and the sciences. Historically, Fourier series were developed inthe analysis of some of the classical PDEs of mathematical physics; these serieswere used to solve such equations. In order to understand Fourier series and thekinds of solutions they could represent, some of the most basic notions of analysiswere defined, e.g., the concept of “function.” Since the coefficients of Fourierseries are integrals, it is no surprise that Riemann integrals were conceived to dealwith uniqueness properties of trigonometric series. Cantor’s set theory was alsodeveloped because of such uniqueness questions.

A basic problem in Fourier analysis is to show how complicated phenomena,such as sound waves, can be described in terms of elementary harmonics. There aretwo aspects of this problem: first, to find, or even define properly, the harmonicsor spectrum of a given phenomenon, e.g., the spectroscopy problem in optics; andsecond, to determine which phenomena can be constructed from given classes ofharmonics, as done, for example, by the mechanical synthesizers in tidal analysis.

Fourier analysis is also the natural setting for many other problems in engineer-ing, mathematics, and the sciences. For example, Wiener’s Tauberian theorem inFourier analysis not only characterizes the behavior of the prime numbers, but alsoprovides the proper notion of spectrum for phenomena such as white light; thislatter process leads to the Fourier analysis associated with correlation functions infiltering and prediction problems, and these problems, in turn, deal naturally withHardy spaces in the theory of complex variables.

ANHA Series Preface vii

Nowadays, some of the theory of PDEs has given way to the study of Fourierintegral operators. Problems in antenna theory are studied in terms of unimodulartrigonometric polynomials. Applications of Fourier analysis abound in signalprocessing, whether with the fast Fourier transform (FFT), or filter design, or theadaptive modeling inherent in time-frequency-scale methods such as wavelet theory.The coherent states of mathematical physics are translated and modulated Fouriertransforms, and these are used, in conjunction with the uncertainty principle, fordealing with signal reconstruction in communications theory. We are back to theraison d’être of the ANHA series!

University of Maryland John J. BenedettoCollege Park Series Editor

Paul Leo Butzer

Paul Leo Butzer was born on April 15, 1928, in Mühlheim, a town on the outskirtsof Germany’s heavy industrial area, known as the Ruhr. His father was a mechanicalengineer and his mother a mathematics teacher.

When Paul was in elementary school, his parents felt increasingly uneasy livingin Germany because of their opposition to Nazism. In March 1937, Paul’s fatherwent to the Netherlands, ostensibly to attend an engineering exhibition, but he didnot return to Germany. From the Netherlands, he went and settled in England, wherethe rest of the family joined him later. Fearing the Nazi’s reprisal, the family sentPaul and his younger brother Karl to Belgium for their safety, and few weeks later,the family reunited in England. Paul’s father joined an engineering company thatwas founded by one of his German friends, Ludwig Loewy, who fled from Germanyto England a few years before and established Loewy Engineering.

The following years were not easy for Paul’s family. Paul went to five schoolsin three countries in 5 years. In January 1941, his father and two other engineers ofMr. Loewy’s firm were commissioned to open a new branch in New York City, butthe family’s plan to go to New York could not be realized because of their Germannationality. They finally settled in Montreal, Canada, where Paul’s father found aposition as an engineer in a company owned by Mr. Loewy’s brother-in-law.

In February 1941, Paul entered Loyola High School in Montreal, and in 1944, atthe age of sixteen, he started undergraduate studies at Loyola College and majored inmathematics, where he was exposed to the Lebesgue integral, Titchmarsh’s theory offunctions, Zygmund’s trigonometric series, as well as Hardy and Wright’s book onnumber theory. He graduated in 1948 with an honors B.Sc. degree in mathematics.

He continued his mathematical education at the University of Toronto andreceived the Sir Joseph Flavelle Prize for his master’s degree. The University ofToronto attracted many top mathematicians from Europe which exposed Paul todifferent teaching styles and research topics. He was introduced to the world ofBritish mathematics, in particular, to the work of Hardy, Littlewood, Titchmarsh,and William J. Webber.

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x Paul Leo Butzer

At the second Canadian Mathematical Congress in Vancouver in August/September 1949, where P.A.M. Dirac, G. Szegö, Laurent Schwartz, andA. Zygmund lectured, Paul was asked to translate and prepare Schwartz’s lectureson Distribution Theory, which were given in French. The lectures were then typedby other students at night and handed out the next morning. That was a toughchallenge for Paul, but as a result, Schwartz suggested to Paul that he should cometo France and write his doctoral dissertation under his supervision. But after somethoughts, Paul decided to do his doctoral work at the University of Toronto underthe supervision of G. G. Lorentz, who came originally from Leningrad. Throughhim, Paul was exposed to Russian mathematics, in particular, to the work of P. L.Chebyshev, S. N. Bernstein, and L. V. Kantorovich.

In 1951, Paul received his Ph.D. with a thesis on Bernstein polynomials. Hewas the youngest student to obtain a Ph.D. in Canada up to that time. He thenreceived a scholarship from the Canadian National Research Council. In 1952, hewas appointed at McGill University in Montreal as a lecturer and a year later as anassistant professor.

In 1955, Paul decided to spend a research year in Europe which he started by athree-week visit to Jean Favard in Paris and from there he went to the Universityof Mainz, Germany, where he stayed for 2 years. He then relinquished his positionat McGill to spend a semester at Freiburg University with Wilhelm Süss, where heearned his Habilitation and became qualified to teach in a regular-track position atGerman universities.

Following the summer semester of 1958 at Würzburg University, he moved toAachen, where his parents had settled, and began teaching as a Dozent at theAachen University of Technology (Rheinisch-Westfälische Technische HochschuleAachen).

In 1962, after receiving an offer from the Rijksuniversiteit Groningen, Nether-lands, the Aachen University of Technology gave him a counteroffer as a chairprofessor, Lehrstuhl A für Mathematik, which he accepted and kept until he retired.

The Aachen University of Technology had a high international reputation for itsengineering faculty, while mathematics rather lived in the shadows, offering primar-ily service courses. Paul very quickly changed that. He established a high-qualitymathematics program making the mathematics department one of the largest andbest in Germany and built up a research group that became known as the AachenSchool of Approximation Theory.

Paul’s contributions to mathematics are overwhelming. Between 1963 and1983, he organized eight conferences at the Mathematical Research Center ofOberwolfach in the Black Forest. The conferences covered a wide spectrum oftopics, from approximation theory, functional analysis, operator theory, harmonicanalysis, integral transforms, interpolation, and orthogonal polynomials and splinesto approximations in abstract spaces. He succeeded in bringing together talentedstudents and renowned mathematicians from both sides of the Iron Curtain todiscuss new ideas and develop new frontiers for the subject. For each conference,he edited a volume containing articles contributed by the conference participants.

Paul Leo Butzer xi

While Paul’s early work emphasized abstract approximation theory, which heeven formed as a subject, he later turned more to applied topics. He establisheda calculus of Walsh functions, also known as dyadic analysis. Together withvarious collaborators, he equipped convergence assertions in functional analysis,numerical analysis, and probability theory with convergence rates suitable forapplications. Furthermore, Paul contributed to combinatorial analysis, fractionalcalculus, analytic number theory, and special functions. One of Paul’s novel ideaswas to reformulate some results in analysis in the context of sampling theory.

In addition to his passion for mathematics, Paul always highly appreciatedengineers, seeing them as a source of inspiration for mathematical research. Paul’scollaboration with engineers began in April 1970 when he attended a symposiumat the Naval Research Laboratory, Washington, D.C., where he met the renownedmathematician Joseph Walsh and became aware of Walsh functions and Walshanalysis, which led him to develop a complete theory of “dyadic differentiation.”This collaboration with engineers intensified in 1975 when Paul and some of hiscolleagues at Aachen became interested in sampling theory and its applicationsin signal processing. This theory, which was a thriving topic in communicationengineering since the publication of Shannon’s work in 1949, was studied onlysparingly by mathematicians.

Paul and his group in Aachen worked on many problems raised by electricalengineers, geophysicists, seismologists, and medical doctors. In 1984, he was oneof the main organizers of the “Fifth Aachen Colloquium: Mathematical Methodsin Signal Processing,” which had more than 220 participants. With his Aachengroup, he did pioneering work in signal processing and published many papersin engineering journals, such as the prestigious IEEE journals. They found anappropriate mathematical framework for various engineering problems.

He has served on the editorial boards of 16 journals, including the editorial boardof the reputed Journal of Approximation Theory, on which he served for more than25 years. He has published more than 200 research papers and 28 books, conferencevolumes, and collected works. An example of one of his outstanding publicationsis his 1971 seminal book, Fourier Analysis and Approximation, coauthored withhis former student Rolf J. Nessel. His accomplishments were acknowledged byhonorary doctoral degrees from Liège, York, and Timisuara universities, as well ashonorary memberships in several scientific societies, including the Belgium RoyalAcademy.

With his enthusiasm for mathematics, Paul attracted many talented students. Atleast thirty of them wrote their Ph.D. dissertations under his guidance. He has alwaysbeen very encouraging to students and junior mathematicians, considering them partof a big family and seeing them as the future of the subject. One of his principlesas an editor was “Do not reject somebody’s first paper but rather brush it up ifnecessary. You might cut a growing and later flourishing branch.” Several knownmathematicians had their first papers published on his recommendation.

Since his time as an undergraduate student at Loyola, one of his hobbies hasbeen history, namely, medieval history and history of mathematics. Of particularinterest for him is the medieval history of the Aachen–Liège–Maastricht area.

xii Paul Leo Butzer

He co-conducted two international history conferences in Aachen. His achievementsin this field resulted in more than fifty publications.

The editors of this monograph have known Paul for more than 25 years. GerhardSchmeisser met Paul Butzer for the first time at a conference on approximationtheory in Oberwolfach in March 1979. It was conducted by L. Collatz, G. Meinar-dus, and H. Werner, who represented a group interested in numerical problems.Paul rather took the role of an observer who was invited as a representative ofan alternative group that was interested in approximation in abstract settings. InNovember 1981, Gerhard met Paul again at a conference on approximation theoryin Oberwolfach that was organized by H. Berens and R. DeVore. At that meeting,Gerhard, in collaboration with Q.I. Rahman, presented results on interpolationand approximation by entire functions of exponential type, which he considered acontribution to complex analysis. But Paul immediately recognized their relevanceto signal analysis and invited Gerhard to give a more extensive lecture in Aachen.The two have had close working relationship and a lasting friendship ever since.

After reading R. Higgins’ interesting article “Five Short Stories About theCardinal Series,” Bull. Amer. Math. Soc.,12 (1985), Ahmed Zayed became veryinterested in sampling theory and its applications in signal and image processing.In August 1986, he was about to embark on spending a 1-year sabbatical leave inEurope. Thinking about the best place to visit to do research on sampling theory andintegral transforms, it was evident that this place was Lehrstuhl A für Mathematik,the Aachen University of Technology (RWTH), where Paul Butzer was the headof the group. Paul’s hospitality and encouragement were conducive to a fruitfuland enjoyable leave. During his stay in Aachen, Ahmed, together with Paul andone of his students, G. Hinsen, wrote a joint paper on the relationship betweensampling theorems and Sturm–Liouville boundary-value problems, which appearedin SIAM J. Appl. Math., 50 (1990). This paper opened a new line of research onsampling theorems that stimulated many mathematicians to publish tens of articleson the subject. In 1994, Zayed, together with M. Ismail, Z. Nashed, and A. Ghaleb,organized a conference at Cairo University, Egypt, in celebration of Paul’s 65thbirthday. The location of the conference was so chosen because Paul expressedinterest in visiting Egypt and seeing the Pyramids. Butzer and Zayed maintainedtheir working relationship and friendship over the years.

This monograph is dedicated to Paul Butzer, on the occasion of his 85th birthday,from his friends, colleagues, and students who wish him a long, healthy life andmuch joy with mathematics and history.

Paul Leo Butzer xiii

Paul Butzer’s graduation with an Honors B.Sc. at Loyola College (Montreal) in 1948

Paul Butzer with his parents at their 50th wedding anniversary in 1975

Paul Butzer with his brother, Prof. Karl W. Butzer, in 1994

xiv Paul Leo Butzer

Paul Butzer in fall 2009

Preface

This monograph is a collection of articles contributed by friends, colleagues, andstudents of Paul Butzer, covering different topics to which Professor Butzer hasmade many contributions over the years. The idea of the monograph was conceivedduring the Summer School on “New Trends and Directions in Harmonic Analysis,Fractional Operator Theory, and Image Analysis” that was organized by B. Forsterand P. Massopust and took place on September 17–21, 2012, Inzell, Germany, whereboth editors of this volume, Gerhard Schmeisser and Ahmed Zayed, as well as PaulButzer, were invited speakers. The editors realized that Butzer’s 85th birthday wasfew months away. Thinking about a token of appreciation for Paul’s long-standingcontribution to the field and support for his colleagues and students, it was soonconcluded that a Festschrift in his honor would be an invaluable gift that would lastfor years to come.

The topics covered are sampling theory, compressed sampling and sensing,approximation theory, and various topics in harmonic analysis, to all of whichButzer and his school in Rheinisch-Westfälische Technische Hochschule (RWTH),Aachen, Germany, have contributed significantly. The chapters, which have beencarefully refereed, are grouped together by themes. The first theme is samplingtheory, compressed sensing, and their applications in image processing. Thiscomprises the first ten chapters. The second theme is approximation theory andit consists of three chapters, Chaps. 11–13. The last theme is harmonic analysis andit consists of six chapters, Chaps. 14–19.

In what follows, we will give an overview of the content of the monograph sothat the reader can focus on what is interesting for him/her.

In Chap. 1, M. M. Dodson presents an approximate sampling theorem with asampling series inspired by that of Kluvánek. The underlying space of admissiblefunctions is the Fourier algebra A.G/ of a locally compact Abelian group G. Inaddition, a summability condition is needed for the sampling series to exist. SinceA.G/ is a Banach space whose elements need not be square integrable, the author’sresults generalize various other abstractions of Kluvánek’s theorem and include anexact sampling theorem as a special case.

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Closely related to the subject of Chap. 1 is sampling in Hilbert spaces. An accountof sampling in the setting of reproducing kernel Hilbert (RKH) spaces is given byR. Higgins in Chap. 2. One of the main points of the chapter is to show that thesampling theory of Kluvánek, even though it is very general in some respects,is nevertheless a special case of the reproducing kernel theory. A Dictionary isprovided as a handy summary of the essential steps. Another point the authoremphasizes is that the RKH space theory does not always generate a samplingtheorem as found in the Dictionary.

He shows that there are two different types of RKH spaces, one type has noassociated sequence sn such that the set, f�n.t/ D k.t; sn/g ; is complete, let alonebeing a basis or a frame, where k.t; x/ is the reproducing kernel. On the other hand,the other type does have such a sequence, such as the Paley–Wiener space (PW ),for which sn D n, and the set fsin�.t � n/=�.t � n/gn2Z is an orthonormal basisof PW:

Going from sampling in Hilbert spaces to Banach spaces seems to be a naturalstep. In Chap. 3, I. Z. Pesenson discusses sampling in Banach spaces. First, let usrecall that Boas-type formulae represent f .n/.t/ for a bandlimited function f interms of samples taken at uniformly spaced points located relative to t . Theseformulae are more powerful than the differentiated classical sampling series. Theygive easy access to Bernstein inequalities and are of interest in numerical differenti-ation. I. Z. Pesenson generalizes Boas-type formulae within a more abstract setting.Given an operatorD (instead of differentiation) that generates a strongly continuousgroup of isometries etD in a Banach space E , he establishes analogous formulae forDnf when f belongs to a subspace of E that corresponds to a Bernstein space.A sampling formula for etDDnf is also derived. Finally, extensions of the resultsto compact homogeneous manifolds are given.

Sampling series, where the classical sinc function is replaced by a general kernelfunction, have been studied by P. Butzer and his school at RWTH Aachen since1977. To study the Lp-convergence of these series, they introduced a class offunctions ƒp � Lp.R/ and estimated the rate of approximation of these seriesin terms of �-modulus. In Chap. 4, A. Kivinukk and G. Tamberg study series withbandlimited kernel functions and show that in this case, the rate of approximationcan be estimated via ordinary modulus of smoothness. Basics in the proof areJackson- and Bernstein-type inequalities, as well as Zygmund-type sampling series.The main aim of this chapter is to give a short overview of results obtainedpreviously by the authors and to extend the theory of P. Butzer and his school togeneralized sampling operators where the kernel function is defined via the Fouriertransform of a certain even window function.

In Chap. 5, C. Bardaro, I. Mantellini, R. Stens, J. Vautz, and G. Vinti establisha multivariate generalized sampling series and apply it to the approximation ofnot necessarily continuous functions. Error estimates are given in terms of anaveraged modulus of smoothness whose properties are explored in a preliminarysection. A multivariate sampling series built with the help of B-splines is studied

Preface xvii

as an important special case. The power of the method is illustrated by concreteapplications to image processing, including biomedical images as they appear incomputer tomography.

Chapter 6 by H. Boche and U. J. Mönich is motivated by digital signal processing.The authors consider a linear, bounded, time-invariant operator T that maps thePaley–Wiener space PW into itself. Their aim is to study the approximation of.Tf /.t/ by a symmetrically truncated sampling series that involves a sequence ofmeasurements of f . They show that for point evaluations of f; convergence of theapproximation process cannot be guaranteed even if oversampling is used and notnecessarily equidistant sequences of sampling points are admitted. However, forcertain classes of measurement functionals, a stable approximation is possible incase of oversampling, which turns out to be a necessary condition. Without over-sampling, there may exist convergent subsequence of the approximation process.

The next two chapters deal with sparse sampling and compression. In Chap. 7,J. Benedetto and Nava-Tudela discuss sparse solutions of underdetermined systemsof linear equations and show how they can be used to describe images in a compactform. They develop this approach in the context of sampling theory and for problemsin image compression. The idea briefly goes as follows: Suppose that we have afull-rank matrix A 2 Rm�n, where n < m; and we want to find solutions to theequation,

Ax D b; (1)

where b is a given “signal.” Since the matrix A is full rank and there are moreunknowns than equations, there are infinitely many solutions to Eq. (1). The focusis on finding the “sparsest” solution x0 that is the one having the least number ofnonzero entries. If the number of nonzero entries in x0 happens to be less thanthe number of nonzero entries in b, we could store x0 instead of b, achievinga representation x0 of the original signal b in a compressed form. If there is aunique “sparsest” solution to Eq. (1), how does one find such a solution? What arethe practical implications of this approach to image compression? This notion ofcompression and its applications in image processing is the authors’ focus.

Conventional sampling techniques are based on the Shannon–Nyquist theoremwhich states that the required sampling rate for perfect recovery of a bandlimitedsignal is at least twice its bandwidth. Although the class of bandlimited signalsis a very important one, many signals in practice are not necessarily bandlimited.A low-pass filter is applied to the signal prior to its sampling for the purposeof anti-aliasing. Many of these signals are sparse, which means that they have asmall number of nonzero coefficients in some domain such as time, discrete cosinetransform, discrete wavelet transform, or discrete Fourier transform. This propertyof such a signal is the foundation of a new sampling theory called compressedsampling (CS). In Chap. 8, M. Azghani and F. Marvasti give an overview ofcompressed sensing and sampling, as well as popular recovery techniques. TheCS recovery techniques and some notions of random sampling are investigated.Furthermore, the block sparse recovery problem is discussed and illustrated by anexample.

xviii Preface

In Chap. 9, M. Pawlak investigates the joint nonparametric signal sampling anddetection problem when noisy samples of a signal are observed. Two distinctdetection schemes are examined. In the first scheme, which deals with what is calledthe off-line testing problem, the complete data set is given in advance and one isinterested in testing the null hypothesis that a signal f takes a certain parametricform. That is, given a class of signals F and a data set, we wish to test the nullhypothesisH0 : f 2 F against an arbitrary alternativeHa W f … F:

In the second scheme, which deals with what is called the on line detection, thedata are collected in a sequential fashion and one would like to detect a possibledeparture from a reference signal as quickly as possible. In such a scheme, onemakes a decision at every new observed data point and stops the procedure whena detector finds that the null hypothesis is false. In practice, only discrete andoften noisy samples are available, which makes it difficult to verify whether atransmitted signal is bandlimited, time limited, and parametric or belongs to somegeneral nonparametric function space. The author suggests a joint nonparametricdetection/reconstruction scheme to verify the type of the signal and simultaneouslyrecover it.

P.J.S.G. Ferreira in Chap. 10 gives a brief account of a phenomenon called“superoscillation.” Briefly, a superoscillating function is a bandlimited functionthat oscillates faster than its maximal Fourier component, and it may do so overarbitrarily long intervals. Superoscillating functions provide direct refutation of thestatement that a bandlimited function contains no frequencies above a limit, say�=2; and so it cannot change to substantially new values in a time less than one-halfcycle of its highest frequency, that is, 1=�. Although the theoretical interest insuperoscillating functions is relatively recent, a number of applications are alreadyknown in quantum physics, super-resolution, sub-wavelength imaging, and antennatheory.

In the second group of chapters, which is dedicated to topics in approximationtheory, we have Chap. 11 by K. Runovski and H. J. Schmeisser in which theyconsider Lp spaces of 2�-periodic functions. They introduce a general modulusof smoothness, called the �-modulus, which is generated by an arbitrary periodicmultiplier � , and then explore its properties. They also establish a Jackson-typeestimate and a Bernstein-type estimate for this modulus. The authors also introducea constructive method of trigonometric approximation and describe its quality interms of the �-modulus. Moduli related to Weyl derivatives, Riesz derivatives, andfractional Riesz derivatives are covered by this approach.

Chapter 12 by L. Angeloni and G. Vinti deals with approximation by a familyof integral operators T! of Mellin type in a multivariate setting on R

NC, where! 2 RC. The authors introduce the notion of multidimensional variation V in thesense of Tonelli and study convergence and order of approximation in the variationfor certain classes of functions f , that is, they investigate the convergence to zeroof V ŒT!f � f � as ! ! 1.

The chapter 13 by F. Stenger, Hany A. M. El-Sharkawy, and G. Baumann is acontribution to error estimates for Sinc approximation, that is, for approximationby a truncated classical sampling series also known as cardinal series. The authors

Preface xix

determine the exact asymptotic representation up to an O.n�2/ term for theLebesgue constant, where n is the number of consecutive terms in the truncatedcardinal series. In numerical experiments, they compute the Lebesgue function andthe Lebesgue constant when the cardinal series is transformed to the interval .�1; 1/by a conformal mapping and compare these quantities with the corresponding onesfor three other approximation procedures that have been proposed in the literature.

The third group of chapters is a collection of articles on different topics inharmonic analysis. In Chap. 14, O. Christensen describes some open problemsin frame theory and presents partial results. First, he introduces the necessarybackground for frame and operator theory, as well as wavelet and Gabor analysis.The problems under consideration deal with extension of dual Bessel sequences todual frames, generalization of the duality principle in Gabor analysis, constructionof wave packet frames, generation of Gabor frames byB-splines, and determinationof good frame bounds. He also includes a conjecture by Heil–Ramanathan–Topiwala and a conjecture by Feichtinger. The latter has been answered affirmativelywhile the author was preparing his contribution.

Closely related to Chap. 14 is Chap. 15 by B. Forster in which she provides anoverview of the importance of complex-valued transforms in image processing.Interpretations and applications of the complex-valued mathematical image analysisare given for Fourier, Gabor, Hilbert, and wavelet transforms. Relationships tophysical imaging techniques such as holography are also described. The role ofamplitude and phase of the Fourier coefficients in images is illustrated by severalinstructive examples.

Chapter 16 by D. Mugler and S. Clary introduces a new method to compute thefrequencies in a digital signal that is based on the discrete Hermite transform thatwas developed by the authors in previous papers. Particularly for an input signal thatis a linear combination of general sinusoids, this method provides more accurateestimations of both frequencies and amplitudes than the usual DFT. The method isbased primarily on the property of the discrete Hermite functions being eigenvectorsof the centered Fourier matrix, analogous to the classical result that the continuousHermite functions are eigenfunctions of the Fourier transform.

Using this method for frequency determination, a new time-frequency repre-sentation based on the Hermite transform is developed and shown to provideclearer interpretations of frequency and amplitude content of a signal than thecorresponding spectrograms or scalograms.

In Chap. 17, P. Massopust considers three quite distinct mathematical areas,namely, (1) fractional differential and integral operators, (2) Dirichlet averages,and (3) splines. He shows that if these areas are generalized to complex settingsand infinite-dimensional spaces, then an interesting and unexpected relationshipbetween them can be established.

In Chap. 18, H. G. Feichtinger and W. Hörmann propose a concept of generalizedstochastic processes within a functional analytic setting. It is based on the unpub-lished thesis of the second-named author. Although the thesis was written severalyears ago, it has gained new interest because of recent developments. Considering

xx Preface

the Segal algebra S0.G/ of a locally compact Abelian group G, the authorsdefine a generalized stochastic process � as a bounded linear mapping of S0.G/into a Hilbert space and obtain various results that generalize those on classicalstochastic processes. This concept has some advantages over other generalizationsof stochastic processes.

Finally, in Chap. 19, E. E. Berdysheva and H. Berens study a multivariategeneralization of an extremal problem for bandlimited functions due to Paul Turán,known as the Turán problem. A simple equivalent formulation of this problem is asfollows: Let F be a nonnegative function on R

d withRRdF .x/dx D 1 such that its

Fourier transform is `1-radial and supported in the `1-ball of radius � . Determinethe supremum of F.0/ over all such F . The authors consider a discrete multivariateTurán problem and state a conjecture for any positive integer dimension d: Theproof of the conjecture for d D 2 was given in one of their earlier papers. Here theysettle the case for d D 3; 5; 7:

Erlangen, Germany Gerhard SchmeisserChicago, IL, USA Ahmed I. ZayedMay 2014

Contents

1 Abstract Exact and Approximate Sampling Theorems . . . . . . . . . . . . . . . . 1M.M. Dodson

2 Sampling in Reproducing Kernel Hilbert Space . . . . . . . . . . . . . . . . . . . . . . . . 23J.R. Higgins

3 Boas-Type Formulas and Sampling in Banach Spaceswith Applications to Analysis on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39Isaac Z. Pesenson

4 On Window Methods in Generalized Shannon SamplingOperators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63Andi Kivinukk and Gert Tamberg

5 Generalized Sampling Approximation for MultivariateDiscontinuous Signals and Applications to Image Processing . . . . . . . . . 87Carlo Bardaro, Ilaria Mantellini, Rudolf Stens, Jörg Vautz,and Gianluca Vinti

6 Signal and System Approximation from General Measurements . . . . . 115Holger Boche and Ullrich J. Mönich

7 Sampling in Image Representation and Compression . . . . . . . . . . . . . . . . . . 149John J. Benedetto and Alfredo Nava-Tudela

8 Sparse Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189Masoumeh Azghani and Farokh Marvasti

9 Signal Sampling and Testing Under Noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215Mirosław Pawlak

10 Superoscillations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247Paulo J.S.G. Ferreira

xxi

xxii Contents

11 General Moduli of Smoothness and Approximationby Families of Linear Polynomial Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269K. Runovski and H.-J. Schmeisser

12 Variation and Approximation in Multidimensional Settingfor Mellin Integral Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299Laura Angeloni and Gianluca Vinti

13 The Lebesgue Constant for Sinc Approximations . . . . . . . . . . . . . . . . . . . . . . 319Frank Stenger, Hany A.M. El-Sharkawy, and Gerd Baumann

14 Six (Seven) Problems in Frame Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337Ole Christensen

15 Five Good Reasons for Complex-Valued Transformsin Image Processing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359Brigitte Forster

16 Frequency Determination Using the Discrete Hermite Transform . . . 383Dale H. Mugler and Stuart Clary

17 Fractional Operators, Dirichlet Averages, and Splines . . . . . . . . . . . . . . . . . 399Peter Massopust

18 A Distributional Approach to Generalized StochasticProcesses on Locally Compact Abelian Groups . . . . . . . . . . . . . . . . . . . . . . . . . 423H.G. Feichtinger and W. Hörmann

19 On a Discrete Turán Problem for `-1 Radial Functions . . . . . . . . . . . . . . . 447Elena E. Berdysheva and Hubert Berens

Chapter 1Abstract Exact and Approximate SamplingTheorems

M.M. Dodson

Abstract The Fourier algebra A.G/ of an lca group G is shown to be a naturalframework for abstract sampling theory. As a Banach space, the Fourier algebraprovides a broader setting than the Hilbert space of square-integrable functions ongroups, although corresponding norm (or energy) information is lost. An approx-imate sampling theorem is proved for those functions in the Fourier algebra of alocally compact abelian group which are summable on the sampling subgroup; anexact sampling theorem is a special case.

Keywords Kluvánek’s theorem • Approximate sampling • Locally compactabelian groups • Fourier analysis

1991 Mathematics Subject Classification. 94A05, 42A99

1.1 Introduction

Sampling theory can be described briefly as the study of reconstruction of continu-ous functions from values on a discrete sampling set by means of a sampling seriesinvolving the sample values. The sampling set can be either regular (equally spaced),for example, when it is a subgroup, or irregular, when it is a union of subgroups andtheir translates. The modern theory began with the Whittaker–Kotelnikov–Shannon(WKS) theorem [34], a cornerstone of communication theory and analogue/digitalsignal processing. The theorem establishes a theoretically exact reconstructionof continuous, finite energy, bandlimited signals (i.e., using more mathematicallanguage, of continuous square-integrable functions f WR ! C with boundedspectrum) from values on a sampling set that consists of a discrete subgroupof R. Discovered many times [24], it has been studied and extended by numerous

M.M. Dodson (�)Department of Mathematics, University of York, York YO10 5DD, UKe-mail: mmd1@york.ac.uk

© Springer International Publishing Switzerland 2014A.I. Zayed, G. Schmeisser (eds.), New Perspectives on Approximationand Sampling Theory, Applied and Numerical Harmonic Analysis,DOI 10.1007/978-3-319-08801-3__1

1

2 M.M. Dodson

mathematicians and engineers (see for example [10, 25–27] and references therein)and has a surprising number of mathematical connections [9, 16, 24]. Subsequently,the WKS theorem was generalised in different ways in order to address someof its practical limitations; these included important approximate results for non-bandlimited functions (i.e., with unbounded spectrum) [6, 7, 11]. Further details arein Sect. 1.3.3.

In his remarkable paper of 1965 [28], Kluvánek extended the WKS theoremto abstract harmonic analysis. He replaced the real line R by a locally compactabelian Hausdorff1 (lca) group G and L2.R/ by L2.G/. Being Hilbert spaces offunctions on lca groups, they share many pleasing properties, such as the notion oforthogonality and a Fourier–Plancherel theory. Kluvánek used these to show that anyfunction in L2.G/ with compactly supported Fourier–Plancherel transform can bereconstructed from values in a discrete subgroup (sampling set) of G (a convenientform of Kluvánek’s theorem is given in Theorem 1 in Sect. 1.3.2 below). This settingoffers an elegant and unifying framework for a wide variety of seemingly disparatesampling results [17,18,24]. On the other hand, some of the finer detail of classicalanalysis is lost. In particular the Fourier series of functions of bounded variation arenot uniformly convergent and instead summability conditions are required.

At the 2005 Sampling Theory and Applications meeting held in Samsun, Turkey,Paul Butzer posed the question whether Kluvánek’s exact theorem implied anabstract approximate version, on the lines of [11]. Although the implication couldnot be proved, an approximate form of Kluvánek’s theorem (Theorem 2 below)for functions in L2.G/ was established [3]. The functions lie in a class F 2.G/ �L2.G/, which consists of continuous L2.G/ functions with integrable Fourier–Plancherel transforms and which is the abstract analogue of the class F 2 of functionsin the approximate WKS theorem [11]. By contrast with the classical case, thefunctions are also required to be square-summable on the sampling setH , a discretesubgroup of G. Earlier a similar approximation result [18] had been established forfunctions in the class F 1.G/ � L1.G/ of continuous integrable functions withintegrable Fourier transform s, again under a summability condition not needed inthe classical case. Thus approximation formulae hold for functions f 2 F 1.G/

satisfying a summability condition and for those f 2 F 2.G/ satisfying the square-summability condition f 2 `2.H/ (see Sect. 1.3.3).

In this paper, abstract sampling theorems are discussed. The principal result is anapproximate theorem for functions in the Fourier algebraA.G/ [defined in (1.15)] ofan lca groupG (Theorem 3). The Fourier algebraA.G/ is a Banach space containingboth F 1.G/ and F 2.G/ (Proposition 1) and is thus a broad framework for samplingtheory. The form of Theorem 3 is similar to those of the approximation theoremsfor functions in F 1.G/ (Corollary 3.7 in [18]) and F 2.G/ (Theorem 2), but nointegrability conditions with respect to G are needed and the error term is different.Just as the approximation theorems in the L2.G/ and L1.G/ settings require

1The Hausdorff separation condition can be weakened to T1 and is usually omitted.

1 Abstract Sampling Theorems 3

summability conditions, an additional summability condition, namely f 2 `1.H/,is needed for functions in A.G/. An abstract Fourier algebra version of Kluvánek’stheorem follows immediately (Theorem 4).

Familiarity with abstract harmonic analysis and lca groups, as given in [12,22, 23, 31, 32], will be assumed. Some background to sampling theory in abstractharmonic analysis and suited to this paper can be found in [1, 3, 13, 14, 18]. The lcagroup framework covers a wide variety of seemingly disparate results in samplingtheory, including some in electronics and signal processing [13, 17, 18, 24]. Firstsome communication engineering terminology background is explained and somedefinitions are given to fix notation.

1.2 Terminology and Notation

In engineering, a signal is modelled by a continuous function of time f WR ! C

(suitable integrability is usually assumed). The real line R represents time t and thecomplex number f .t/ is the output. Recall that when the frequencies are bounded,the signal is bandlimited. In the abstract harmonic analysis considered here, the timedomain R of signal theory is replaced by an lca group G and Lebesgue measure byHaar measuremG . This measure is pretty well behaved and has regularity propertiesnot very different from those of Lebesgue measure. In particular, it is translationinvariant. As with Lebesgue measure, a “null set” has zero Haar measure andexpressions such as “almost everywhere” or “almost all” refer to sets of full Haarmeasure.

1.2.1 The Groups and Their Duals

ThroughoutG will be an lca group, with translation invariant Haar measuremG andHaar integral

Z

G

f .x/dmG.x/ DZ

G

f:

The function spaces Lp.G/, 1 6 p < 1, consist of those functions f WG ! C forwhich the integral

RG

jf jp < 1 (functions differing on a null set are identified).The Haar measure of a compact set is finite.

When G is discrete and countable, each point x 2 G has a non-zero measure or“mass”, written mG.fxg/ D mG.f0g/ by translation invariance. The uniqueness ofHaar measure implies that the integral over G of a function f in L1.G/ reduces toa constant multiple of a sum, so that for each f 2 L1.G/,

4 M.M. Dodson

Z

G

f .x/ dmG.x/ D mG.f0g/X

x2Gf .x/: (1.1)

Thus whenH is discrete and countable, the sum

X

h2Hf .x C h/ D 1

mH.f0g/Z

H

f .x C h/dmH.h/

and is constant on cosetsH C x.Each lca group G has a dual G^, which will be written , defined as the set

of continuous homomorphisms (or characters) WG ! S1, where S

1 is the unitcircle. The dual is also an lca group, with Haar measure m , and its dual ^ isisomorphic to G. The dual of the real line R is also R, that of S1 is Z and that of Zis S

1. The value at x 2 G of a character of is a unimodular complex numberwritten .x; /, reflecting this duality. When G D R, the character can be taken tobe the exponential e2�ix . Multiplication is componentwise. Note that .0; / D 1 D.x; 0/, so that .x; / D .x;�/ D .�x; / and that .x; / .x; / D 1.

The frequency domain of engineering is also represented by the real line R, sinceR is self-dual and thus the Fourier transform f ^ of a signal f (assumed integrable)is also a function from R ! C. Thus in the lca group setting, the frequency domaincorresponds to the dual of G. The spectrum of a signal is defined to be the set ofthe non-zero frequencies, so that, in the abstract setting, the spectrum is the supportsupp f ^ of the Fourier transform f ^ and is a (closed) subset of the dual group .

1.2.2 Subgroups and Quotient Groups

A closed subgroup H of G is also a lca group, as is the quotient group G=H .The complete set of coset representatives of a quotient group will be called atransversal [19].

Given a closed (lca) subgroup ƒ in the dual group , the transversal � say ofthe (lca) quotient group =ƒ plays an important part in sampling. By definition,a transversal � consists of one and just one point from each distinct coset Œ� Dƒ C , i.e., � \ Œ� consists of a single point in �. Thus translates of � by non-zero elements in ƒ are disjoint. Transversals are not unique and there is always ameasurable one [19]. Note that when the quotient group is compact, the transversalis relatively compact and not necessarily compact.

The annihilator of ƒ � in G, denoted byƒ?, is given by

ƒ? D fx 2 GW .x; �/ D 1 for all � 2 ƒg

1 Abstract Sampling Theorems 5

and is a closed subgroup of G. For convenience, ƒ? will be denoted by H . It canbe shown that H? D ƒ?? D ƒ [32, Lemma 2.1.3] and that H is isomorphicto the dual of =ƒ, i.e., H Š .=ƒ/^ (algebraically and topologically) [32,Theorem 2.1.2];H will be identified with the dual .=ƒ/^. The sampling set Z=2win the classical sampling theorem as stated in [3, Theorem 1] is an example of thesubgroupH .

1.2.3 The Weil Decomposition Formula

Letƒ be a subgroup of , withmƒ normalised so that the Weil coset decompositionformula

Z

'./ dm./ DZ

=ƒ

Z

ƒ

'. C �/ dmƒ.�/ dm=ƒ.Œ�/ (1.2)

holds for ' 2 L1./ (see [23, Chap. 7, Sect. 28.54 (iii)] or [31, Chap. 3, Sect. 4.5]).Using coset decomposition (1.2), it is straightforward to verify [13, Lemma 1] thatthe Haar measure of the transversal� of the compact group =ƒ satisfies

m.�/ D mƒ.f0g/m=ƒ.=ƒ/ < 1: (1.3)

1.2.4 Lattices

From now on it will be assumed that ƒ is a discrete countable subgroup of , sothat the quotient group=ƒ is compact with finite Haar measurem=ƒ.=ƒ/ < 1.Further assume that from now on� is a measurable transversal in of the compactgroup =ƒ, so that by (1.3), m.�/ < 1. It will also be assumed that thediscrete annihilatorƒ? D H ofƒ is also countable. For convenience, such discretecountable subgroupsH and ƒ will be called lattices.

Since the lattice ƒ is discrete and countable, the sum

X

�2ƒ'. C �/ D 1

mƒ.f0g/Z

ƒ

'. C �/dmƒ.�/

is invariant under translates by elements in ƒ, is constant on cosets Œ� D ƒ C

and converges almost everywhere when ' 2 L1./ (see Sect. 1.3.5 below). As theannihilatorH ofƒ is also discrete and countable, the same observations hold forH .

6 M.M. Dodson

1.2.5 Integral Transforms

Much of classical Fourier theory holds for lca groups [32, Sect. 1.2]. If the functionf WG ! C is integrable, i.e., if f 2 L1.G/, the Fourier transform f ^W ! C off is defined by

f ^./ DZ

G

f .x/.x;�/dmG.x/ (1.4)

and is in C0./, i.e., f ^ is continuous and vanishes at 1 [32, Theorem 1.2.4].When G D R, the Fourier transform will be taken to be

f ^.u/ DZ

R

f .t/ e�2�itudt: (1.5)

This normalisation follows [3] and differs from that in [11]. When f; g 2 L1.G/,the convolution f � g is given by

f � g.x/ DZ

G

f .x � y/g.y/dy 2 L1.G/; (1.6)

being finite almost everywhere. More details are in [32, Sect. 1.1.6.].The inverse Fourier transform f _W ! C for f 2 L1.G/ is defined by

f _./ DZ

f .x/.x; /dmG.x/ D Qf ./; (1.7)

where Qf .x/ D f .�x/, and is in C0./. Because of the connection with signalprocessing, the inverse Fourier transform '_WG ! C of ' 2 L1./, defined by

'_.x/ DZ

'./.x; /dm./ (1.8)

and in C0.G/, can be thought of as representing a signal, with its domain G beingan abstraction of the real line model for time.

The Fourier inversion formula for lca groups states that for f 2 L1.G/ withFourier transform f ^ 2 L1./, there is a unique Haar measurem on such that

f .x/ DZ

f ^./.x; /dm./ D f ^_.x/ (1.9)

for x in G, i.e., f ^_ � f [31, Sect. 4.1]. This measure will always be chosen for .The repeated Fourier transform of f satisfies f ^^ � Qf .

The L2.G/ theory for lca groups is also similar to the classical case [23, Chap. 8,Sect. 31], [31, Chap. 4, Sect. 4], [32, Sect. 1.6]. Plancherel’s theorem holds forfunctions f in the subspace L1.G/ \L2.G/, i.e.,

1 Abstract Sampling Theorems 7

Z

G

jf j2 DZ

jf ^j2; (1.10)

and its image .L1.G/ \ L2.G//^ under the Fourier transform ^ is dense in L2./.The restricted Fourier transform ^jL1.G/\L2.G/ is thus an isometry in the L2 normwhich can be extended by continuity to the isometric Fourier–Plancherel operatorFof L2.G/ onto L2./, with inverse F�1WL2./ ! L2.G/. The Fourier–Planchereltransform Ff is a limit of a sequence of approximating integrals of functions inL1.G/ \ L2.G/ and is unique modulo null sets. For each f 2 L2.G/, kf k2 DkFf k2, i.e., Plancherel’s theorem holds and

Z

G

jf � F�1Ff j2 D 0 or F�1Ff � f: (1.11)

Similarly FFf � Qf . If f 2 L1.G/ \ L2.G/, the Fourier transform f ^ and theFourier–Plancherel transform Ff coincide.

When G is a compact lca group, the characters .�; �/ on G, where the � runthrough the discrete dual group ƒ, are an orthonormal basis of L2.G/ [30, Sect.38C].

1.2.6 Function Spaces Associated with Sampling

Recall that � is a Haar measurable transversal of the compact quotient group =ƒ,so that m.�/ < 1.

Paley-Wiener Space

The classical Paley-Wiener space PWw is defined by

PWw WD ff 2 L2.R/\ C.R/W suppFf � Œ�w;w�g;

using the normalisation for the Fourier transform (1.5). The functions in PWw aresquare integrable and continuous, with Fourier–Plancherel transform Ff squareintegrable and integrable, i.e., bandlimited signals in engineering terminology.They are also examples of complex functions of exponential type (for moredetails see [10, 33, Chap. 19]). The content of the WKS theorem is that, in thenormalisation (1.5), such functions can be reconstructed exactly from samplesf .k=2w/ from the sampling set Z=2w.

The abstract analogue of the Paley-Wiener space is

PW�.G/ D ff 2 L2.G/\ C.G/W suppFf � �g: (1.12)