An introduction to harmonic analysis

AN INTRODUCTION TO HARMONIC ANALYSIS Yitzhak Katznelson INSTITUTE OF MATHEMATICS THE HEBREW UNIVERSITY OF JERUSALEM Second Corrected Edition DOVER PUBLICATIONS, INC NEW YORK Copyright © 1968, 1976 by Yitlhak KatzncIson All rights rescrved under Pan Amcrican and International Copyright Comentions Publhhed in Canada by General Publishing Compdny, Ltd., 30 Lesmill Road, Don l\1i11s, Toronto, Ontario Published in the United Kingdom by Constable and Company, Ltd., 10 Orange Street, London WC This Do\cr edition, first published in 1976, is an unabridged and corrected republication of the work originally published by John Wiley & Sons, Inc., New York, in 196R Manufactured in the United States of America Dover Publications, Inc 180 Varick Street New York, N.Y_ 10014 Preface Harmonic analysis is the study of objects (functions, measures, etc.), defined on topological groups The group structure enters into the stud)"" by allowing the consideration of the translates of the object under study, that is, by placing the object in a translation-invariant space The study consists of two steps First: finding the "elementary components" of the object, that is, objects of the same or similar class, which exhibit the simplest behavior under translation and which "belong" to the object under study (harmonic or spectral analysis); and second: finding a way in which the object can be construed as a combination of its elementary components (harmonic or spectral synthesis) The vagueness of this description is due not only to the limitation of the author but also to the vastness of its scope 1n trying to make it clearer, one can proceed in various wayst; we have chosen here to sacrifice generality for the sake of concreteness We start with the circle group T and deal with classical Fourier series in the first five chapters, turning then to the real line in Chapter VI and coming to locally compact abelian groups, only for a brief sketch, in Chapter VII The philosophy behind the choice of this approach is that it makes it easier for students to grasp the main ideas and gives them a large class of concrcte examples which are essential for the proper understanding of the theory in the general context of topological groups The presentation of Fourier series and integrals differs from that in [I], [7] [8J, and [28J in being, I believe, more explicitly aimed at the general (locally compact abelian) case The last chapter is an introduction to the theory of commutative t Hence the indefinite article in the title of the book vii viii An Introduction to Harmonic Analysis Banach algebras I t is biased, studying Banach algebras mainly as a tool in harmonic analysis This book is an expanded version of a set of lecture notes written for a course which I taught at Stanford University during the spring and summer quarters of 1965 The course was intended for graduate students who had already had two quarters of the basic "real-variable" course The book is on the same level: the reader is assumed to be familiar with the basic notions and facts of Lebesgue integration, the most elementary facts concerning Borel measures, some basic facts about holomorphic functions of one complex variable, and some elements of functional analysis, namely: the notions of a Banach space, continuous linear functionals, and the three key theorems-"the closed graph;' the Hahn-Banach, and the "uniform boundedness" theorems All the prerequisites can be found in [23] and (except, for the complex variable) in [22] Assuming these prerequisites, the book, or most of it, can be covered in a one-year course A slower moving course or one shorter than a year may exclude some o,f the starred sections (or subsections) Aiming for a one-year course forced the omission not only of the more general setup (non-abelian groups are not even mentioned), but also of many concrete topics such as Fourier analysis on Rn, n > I, and finer problems of harmonic analysis in T or R (some of which can be found in [13]) Also, some important material was cut into exercises, and we urge the reader to as many of them as he can The bibliography consists mainly of books, and it is through the bibliographies included in these books that the reader is to become familiar with the many research papers written on harmonic analysis Only some, more recent, papers are included in our bibliography In general we credit authors only seldom-most often for identification purposes With the growing mobility of mathematicians, and the happy amount of oral communication, many results develop within the mathematical folklore and when they find their way into print it is not always easy to determine who deserves the credit When I was writing Chapter II I of this book, I was very pleased to produce the simple elegant proof of Theorem 1.6 there I could swear I did it myself until I remembered two days later that six months earlier, "over a cup of coffee," Lennart Carleson indicated to me this same proof The book is divided intu chJptcrs, scction~, cll1d subsections he chapter numbers are denoted by roman numerals and the sections and Preface ix subsections, as well as the exercises, by arabic numerals I n cross references within the same chapter, the chapter number is omitted; thus Theorem 111.1.6, which is the theorem in subsection of Section I of Chapter III, is referred to as Theorem 1.6 within Chapter III, and Theorem 111.1.6 elsewhere The exercises are gathered at the end of the sections, and exercise V.I.I is the first exercise at the end of Section I, Chapter V Again, the chapter number is omitted when an exercise is referred to within the same chapter The ends of proofs are marked by a triangle (~) The book was written while I was visiting the University of Paris and Stanford University and it owes its existence to the moral and technical help was so generously given in both places During the writing have benefitted from the advice and criticism of many friends; would like to thank them all here Particular thanks are due to L Carleson, K DeLeeuw, J.-P Kahane, O C McGehee, and W Rudin I would also like to thank the publisher for the friendly cooperation 10 the production of this book YI fLIIAK KAfZNELSON Jerusalem April 1961l Contents chapter I Fourier Series on T I FOURIER COEFF1CIENTS 2 SUMMABILITY IN NORM AND HOMOGENEOUS BANACH SPACES ON T POINTW1SE CONVERGENCE OF u,,(f) 17 THE ORDER OF MAGNITUDE OF FOURIER COEFFICIENTS 22 FOURIER SERIES OF SQUARE SUM MABLE FUNCTIONS 27 ABSOLUTELY CONVERGENT FOURIER SERIES 31 chapter II FOURIER COEFFICIENTS OF LINEAR FUNCTIONALS 34 The Convergence of Fourier Series 46 I CONVERGENCE IN NORM 46 CONVERGENCE AND DIVERGENCE AT A POINT 51 *3 SETS OF DIVERGENCE 55 chapter II I The Conjugate Function and Functions Analytic in the Unit Disc 62 l THE CONJUGATE FUNCTION 62 *2 THE MAXIMAL FUNCTION OF HARDY AND LITTLEWOOD 73 *3 THE HARDY SPACES 81 xi xii An Introduction to Harmonic Analysis chapter IV Interpolation of Linear Operators and the Theorem of Hausdorff-Young I INTER POLA TION OF LINEAR OPERATORS NORMS AND OF 93 THE THEOREM OF HAUSDORFF-YOUNG chapter V 93 98 Lacunary Series and Quasi-analytic Classes 104 I LACUNARY SERIES 104 *2 QUASI-ANALYTIC CLASSES 112 chapter VI Fourier Transforms on the Line I FOURIER TRANSFORMS FOR L '(R) FOURIER-STIELTJES TRANSFORMS FOURIER TRANSFORMS IN I < p ~ 119 120 131 LP(R), 139 TEMPERED DISTRIBUTIONS AND PSEUDOMEASURES 146 ALMOST-PERIODIC FUNCTIONS ON THE LINE 155 THE WEAK-STAR SPECTRUM OF BOUNDED FUNCTIONS 170 THE PALEY-WIENER THEOREMS *8 THE FOURIER-CARLEMAN TRANSFORM KRONECKER'S THEOREM chapter VJ] Fourier Analysis on Locally Compact Abelian Groups 173 179 181 186 LOCALLY COMPACT ABELIAN GROUPS 186 THE HAAR MEASURE 187 CHARACTERS AND THE DUAL GROUP 188 FOURIER TRANSFORMS 190 ALMOST-PERIODIC FUNCTIONS AND THE BOHR COMPACTIFICATION 191 Contents chapter VlIf appendix Commutative Banach Algebras xiii 194 I DEFINITION, EXAMPLES, AND ELEME~TARY PROPERTIES 194 MAXIMAL IDEALS AND MULTIPLICATIVE LINEAR FUNCTIONALS 198 THE MAXIMAL-IDEAL SPACE AND THE GELFAND REPRESENT ATION 205 HOMOMORPHISMS OF BANACH ALGEBRAS 213 REGULAR ALGEBRAS 221 WIENER'S GENERAL TAUBERIAN THEOREM 226 SPECTRAL SYNTHESIS IN REGULAR ALGEBRAS 229 FUNCTIONS THAT OPERATE IN REGULAR BANACH ALGEBRAS 235 THE ALGEBRA M(T) AND FUNCTIONS THAT OPERATE ON FOURIER-STIELTJES COEFFICIENTS 244 10 THE USE OF TENSOR PRODUCTS 249 Vector-Valued Functions 257 Bibliography 261 Index 26] Symbols Page 12 I3 14 15 16 25 31 34 36 37 47 48 50 55 65 73 82 84 101 104 112 116 Symbol Page R,T,Z Ll(T) j(n), sen, S[n !*g Kn, uif), un(j, I) Sn(f), Dn C(T), C(T), U(T) Vn(t), p(r, I) L OO(T), Lip",(T), lip",(T) w(j, h), fl(f, II) A(T) S[P] Sn (Ll(T»" C*{Mn}, C#{Mn} ,H(r) xiv 120 121 123 124 126 127 131 146 147 149 155 156 157 159 160 162 164 187 194 195 205 207 222 223 249 Symbol Ll(R) It A(It) , CoeR) K;., K(,") V;., p; 2, 2c M(R) :FU S(R), S*(R) 1:{v) A{r.,J), AP(R) Wo(f) W(f) a(h) MU) = /0 OF LEO~ARDO DA V[\CI, edited hy J.P Richtcr Extracts from manuscripts reveal great genius; on painting, sculpture, anatomy, scicnces, geography, etc Both Italian and English 186 ms pages reproduced, plus 500 additional drawings, induding studies for Last Supper, Sforza monument, etc 860pp 77/8 x 103,4 USO 22572-0,22573-9 Pa., Two vol set $12.00 AnT NOUVEAU DESIG:-;S I!II COLOR, Alphonse Mucha, Maurice Verneuil, Georges Auriol Full-color reproduction of Combinaison~ ornamentules (c; 1900) by Art Nouveau masters Floral animal, geometric, interlacings, swashes - borders, frames, ~pots - all incredibly beautiful 60 plates, hundreds of designs 3/8 x 1/16 22885-1 Pa $4.00 GRAPIJIC WORKS OF ODILON REDON All great fantastic lithographs, etchings, engra"ings, drawings, 209 in all Monsters, Huysmans, still life work, etc Introduction by Alfred Werner 209pp 1/8 x 1214 21996 Pa.$6.00 EXOTIC FLORAL PA rJ ERl\S IN COLOR, E.-A Seguy Incredibly beautiful full-color pochoir work by great French designer of 20·s Complete Bouquets et frondaisons, Suggestions pour etoffes Richness must he ~een to be believed 40 plates containillg 120 patterns 80pp 9:3/8.1 1214 23041-4 Pa $6.00 SELECTED ETCHI!IIGS OF JA\lES A McN WIII~rL£R, James A McN Whistler 149 outstanding etchings by the great American artist, including selections from the Thames set and two Venice sets, the complete French set, and mallY indi~dual prints Introduction and explanatory note on each print by Maria Naylor 157pp 23194-1 Pa $5.00 93/8 x 1214 VISUAL ILLUSIO'l;S' THEIR CAUS£S, CHARACTERISTICS Al\D ApPLICATIONS, Matthew Luckiesh Thorough description, discussion; 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