Spectral Synthesis (Pure & Applied Mathematics)

Spectral Synthesis Pure and Applied Mathematics A Series of Monographs and Textbooks Editors Samuel Eilenberg and Hym...

Author: John Benedetto

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Spectral Synthesis

Pure and Applied Mathematics A Series of Monographs and Textbooks Editors Samuel Eilenberg and Hyman Bass Columbia University, New York RECENT TITLES

GLENE. BREDON. Introduction to Compact Transformation Groups Connections, Curvature, and WERNERGREUB,STEPHENHALPERIN,AND RAY VANSTONE. Cohomology: Volume I, De Rham Cohomology of Manifolds and Vector Bundles. Volume 11, Lie Groups, Principal Bundles, and Characteristic Classes. In preparation: Volume 111, Cohomology of Principal Bundles and Homogeneous Spaces XIADAO-XING. Measure and Integration Theory of Infinite-Dimensional Spaces: Abstract Harmonic Analysis G. DOUGLAS. Banach Algebra Techniques in Operator Theory RONALD WILLARD MILLER, JR. Symmetry Groups and Their Applications ARTIIUR A. SAGLEA N D RALPHE. WALDE.Introduction to Lie Groups and Algebras T. BENNYRUSHING. Topological Embeddings JAMESW. VICK.Homology Theory: An Introduction to Algebraic Topology E. R. KOLCHIN. Differential Algebra and Algebraic Groups GERALD J. JANUSZ.Algebraic Number Fields Introduction to the Theory of Entire Functions A. S. B. HOLLAND. AND DALEVARBERG, Convex Functions WAYNEROBERTS A.M. OSTROWSKI. Solution of Equations in Euclidean and Banach Spaces, Third Edition of Solution of Equations and Systems of Equations H. M. EDWARDS. Riemann’s Zeta Function SAMUEL EILENBERG. Automata, Languages, and Machines: Volume A. In preparation: Volume B MORRISHIRSCHA N D STEPHENSMALE.Differential Equations, Dynamical Systems, and Linear Algebra WILHELM MAGNUS.Noneuclidean Tesselations and Their Groups J. DIEUDONN~. Treatise on Analysis, Volume IV FRANCON TREVES.Basic Linear Partial Differential Equations WILLIAMM. BOOTHBY.An Introduction to Differentiable Manifolds and Riemannian Geometry BRAYTON GRAY.Homotopy Theory: An Introduction to Algebraic Topology ROBERTA. ADAMS.Sobolev Spaces Spectral Synthesis JOHNJ . BENEDETTO.

In preparation D.V. WIDDER.The Heat Equation IRVINGE. SEGAL.Mathematical Cosmology and Extragalactic Astronomy J. DIEUDONN~. Treatise on Analysis, Volume 11, enlarged and corrected printing

Errata for Benedetto, Spectral Synthesis page

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Spectral Synthesis

-JOHN J.BENEDETT0 Department of Mathematics University of Maryland

ACADEMIC PRESS, INC. New York London San Francisco 1975 A Subsidiary of Harcourt Brace Jovanovich, Publishers

Prof. John J. Benedetto Born 1939 in Boston. Received B.A. from Boston College in 1960, M.A. from Harvard University in 1962, and Ph.D. from University of Toronto in 1964. Assistant professor at New York University from 1964 to 1965; and research associate at University of Liege and the Institute for Fluid Dynamics and Applied Mathematics from 1965 to 1966; employed by RCA and IBM from 1960 to 1965. At University of Maryland, assistant professor from 1966 to 1967, associate professor from 1967 to 1973, and professor beginning in 1973. Visiting positions include the Scuola Normale Superiore at Pisa from 1970 to 1971 and 1974 (spring) and MIT in 1973 (fall); also Senior Fulbright-Hays Scholar from 1973 to 1974.

\(-‘.

B.G.Teubner, Stuttgart 1975

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re-use of illustrations. broadcasting, reproductions by photocopying machine or similar means, and storage in data banks. Under 454 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher. Licensed edition for the Western Hemisphere Academic Press, Inc., New York /London/San Francisco, A Subsidiary of Harcourt Brace Jovanovich, Publishers. Originally published in the series “Mathematische Leitfaden” edited by G. Kothe and G.Trautmann. Library of Congress Cataloging in Publication Data

Benedetto, John. Spectral synthesis. (Pure and applied mathematics, a serics of monographs and textbooks ; 66) Bibliography: p. Includes indexes. I . Spectral synthesis (Mathematics) 2. Locally compact Abelian groups. 3. Tauberian theorems. I . Title. I I . Series. QA3.P8 VOI.66 [QA403] 510’.Rs [515’.78] 75-13765 ISBN 0-12-087050-9 AMS (MOS) 1970 Subject Classifications: ZOHIO, 50C05 Printed in Germany Setting: William Clowes & Sons Ltd.. London Printer: Johannes Illig, Goppingen

Dedicated to John Bertrand and Robert Laurent

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Introduction

The major topic in this book is spectral synthesis. The purpose of the book can be described as follows: A. To trace the development of spectral synthesis from its origins in the study of Tauberian theorems ;

B. To draw attention to other mathematical areas which are related to spectral synthesis; C. To give a thorough (although not encyclopedic) treatment of spectral synthesis for the case of L’(G);

D. To introduce the “integration” and “structure” problems that have emerged because of the study of spectral synthesis. A and B. The first two points are discussed in Chapters 1 and 2, and are the major reasons that such a large bibliography has evolved. By the end of Chapter 2, the significant relationship between Tauberian theorems and spectral synthesis is not only firmly established, but the extent to which this relationship is still undetermined is emphasized by the open “C-set-S-set” problem. Also, Chapters 1 and 2 view spectral synthesis amidst other problems and influences. C. By contrast, Chapter 3 is (or at least is meant to be) more business-like, and synthesis is essentially the only topic under discussion.

It is fashionable and important to exposit spectral synthesis results in the setting of regular Banach algebras (or even more generally). One of the reasons that I have chosen the “special” L’(G) case is because of the two problems mentioned in the fourth point; neither problem has reached the stage where the algebraic structure has been successfully exploited, and a presentation of the analytic techniques available for L’(G) (and not always available more generally) seemed the reasonable thing to do. Of course, L’ is still the right setting for applications and also the proper setting to exposit best the range of topics I have treated.

D. The “integration”prob1em (e.g. Sections 3.2.1, 3.2.2,3.2.4,3.2.8, 3.2.9) is to find the relationship between spectral synthesis and integration theories. The problem was essentially posed by Beurling, and at this time very little is known except that such a relationship exists at a fundamental level.

6

Introduction

The “structure” problem (e.g. Sections 3.2.15-3.2.19) is one with a more secular flavor. Basically, one would like to know the intrinsic properties of a distribution, such as its support, when we have a knowledge of its Fourier transform, and vice-versa. This is precisely the sort of information that one must have at hand in order to solve spectral synthesis problems. Essentially this is the problem of determining the finer structure of Schwartz distributions by using Fourier analysis. Schwartz, of course, studied the structure of distributions and obtained representations (of distributions) in terms of derivatives of ordinary functions. Notation G will always be a locally compact abelian group with dual group r. The problems that we’ve considered are quite classical, and my intent is not altered if one takes G = R, the real numbers, or C = 2, the integers; in these cases f = R, the real numbers again, and r = T = R / 2 d , respectively. For these two examples, “(y,x)” is “eiyx’’. Also, notationally, Q denotes the rational numbers, C the complex numbers, R+ = {r E R : r 2 O},Z+is{O,1,2,. . .}or{l,2,. . .},aXistheboundaryofX,intXistheinterior of X , X “ is the complement of X , and supp cp is the support of the function cp. Depending on the situation, T will be considered as { z E C: IzI = l} or as an interval of length 231. on R.

Bibliography I have already mentioned that the large bibliography is due to the historical notes and the fact that many other topics related to synthesis are mentioned and referenced. I have not provided as large a bibliography for current topics in synthesis, and several issues, in which there is presently a good deal of activity, are not adequately discussed. A sampling of these issues is: complex methods in synthesis; operational calculus problems ; extensions to the non-abelian case; isomorphism problems ; the relation between arithmetic and synthesis; tensor algebra in harmonic analysis; the theory of multipliers and p-spaces; and probabilistic methods in synthesis. Consequently, some of the most active and competent workers in spectral synthesis are not duly listed in the bibliography. On the other hand, we do not feel it is too big a step from this book to their work. Exercises

The exercises contain both problems and remarks. Many of the problems are easy although some of the accompanying remarks, usually referenced, may contain more difficult material. In any case, a reading of the problems will provide added perspective.

Introduction

7

Acknowledgements My first thanks go to C . R . W a r n e r who introduced me to the spectral synthesis problem. There has also been the generous and often ingenious assistance of G. Helzer, R . J o h n s o n , and G . S a l m o n s who were always willing to discussmathematics with me, and whose explicit influence is to be found in some of the exercises. Further, I have benefited mathematically from conversations and correspondence (in ways they may not remember) with S. A n t m a n , A. A t z m o n , L. GBrding, C. G r a h a m , J . - P . K a h a n e , Y . K a t z n e l s o n , R. K a u f m a n , C . M c G e h e e , D . N i e b u r , J . O s b o r n , F. Ricci, D. Sweet, and P. W o l f e . I wrote Chapter2 while I was a guest at MIT and Chapter 3 while I was a guest at the Scuola Normale Superiore; I would like to thank K . H o f f m a n and E . Vesentini of these institutions for the hospitality extended to me. My final thanks go to G . M a l t e s e for being a vital source of encouragement to me and to G . K o t he for asking me to write the book. Boston and Pisa, Summer 1974

J. J. BENEDETTO

Table of contents 1

The spectral synthesis problem 1.1

The Fourier transform of L1(G)........................................ 1.1.1 Prerequisites .................................................. 1.1.2 L1(G)and Lm(G)............................................... 1.1.3 Banach algebras ............................................... 1.1.4 The theory of Fourier transforms for L1(G). Lz(C).and M ( G ) . . . . . . . . 1.1.5 V Q c A(T).supp (p is cr-compact ................................ 1.1.6 The point at infinity is an S-set .................................. 1.1.7 An approximate identity technique ............................... 1.1.8 Wiener's theorem on the inversion of Fourier series ................. 1.1.9 Wiener's Tauberian theorem .................................... Exercises 1.1 : 1.1.1 Non-vanishing Fourier transforms ......................... 1.1.2 A special Banach algebra (cf Exercise 3.1.1). . . . . . . . . . . . . . . . . 1.1.3 A ( f ) = C o ( r ) ........................................... 1.1.4 Riesz products and the Fourier coefficients of CantorLebesgue measures ...................................... 1.1.5 Regular maximal ideals in Banach algebras .................. I J.6 Regular maximal ideals in A ( f ) ............................

.

........................................ 0 is a strong Ditkin set ......................................... The point at infinity is a strong Ditkin set ......................... Points in r a r e strong Ditkin sets ................................

1.2 Approximate identities in A ( T ) 1.2.1 1.2.2 1.2.3 1.2.4 1.2.5 1.2.6 1.2.7

Idempotent measures and strong Ditkin sets ....................... A characterization ofj(E) and k ( E ) ............................... A generalization of Wiener's Tauberian theorem .................... A remark about Drury's theorem ................................

Exercises 1.2 : 1.2.1 The point at infinity is a C-set ............................. 1.2.2 The Tauberian theorem for compact G ..................... 1.2.3 The Tauberian theorem and spectral synthesis properties of points .................................................. 1.2.4 Primary ideals and spectral synthesis properties of points . . . . . . 1.2.5 The Nullstellensatz and spectral synthesis ..................

17 17 17 17 18 20 21 21 22 25

26 27 27 27 29 30 30 30 32 35 37 38 39 40

40 40 40 41 41

1.3

Table of contents

9

1.2.6 Approximate identities and strong Ditkin sets ............... 1.2.7 A characterization of strong Ditkin sets .....................

43 43

Pseudo-measures .................................................... 1.3.1 The space A ' ( f ) of pseudo-measures .............................. 1.3.2 M ( f ) c A'(T) ................................................. 1.3.3 Operations in A ' ( r ) ............................................ 1.3.4 Pseudo-functions .............................................. 1.3.5 Radon measures and A'(T) ...................................... 1.3.6 The support of a pseudo-measure ................................ 1.3.7 The product Tp where TE A ' ( f ) and p E A(T) ..................... 1.3.8 A dual formulation of Wiener's Tauberian theorem . . . . . . . . . . . . . . . . . 1.3.9 Radon measures are synthesizable ................................ 1.3.10 Heuristics for the notion of spectrum ............................. 1.3.1 1 A characterization of the weak closed submodules of X ( f )......... 1.3.12 A basic duality technique ....................................... 1.3.13 Helson sets and S-sets ..........................................

44 44 45 45 46 46 48 49 49 50 51 52 52

Exercises 1.3 : 1.3.1 Wiener's characterization of continuous measures . . . . . . . . . . . . 1.3.2 Unbounded measures in A'@) and Fourier series of Lm(T) . . . . 1.3.3 A property of A & ( f )......................................

53

Ceslro summability on G ................................. Distribution theory ...................................... " { y } ) = M({y}) .........................................

Tp=O .................................................

54 55 56 56 56 57 58

1.4 The spectral synthesis problem .........................................

59

1.3.4 1.3.5 1.3.6 1.3.7

1.4.1 The existence of the spectrum and Wiener's theorem ................ 1.4.2 Heuristics for the problems of spectral analysis and synthesis ........ 1.4.3 Spectra of representations and the spectral analysis problem for

....................................... ................................. Standard characterizations of S-sets .............................. An approximation in G to determine S-sets in r .................... A characterization of S-sets in terms of principal ideals . . . . . . . . . . . . . . Non-synthesis and the principal ideal problem ..................... A characterization of C-sets ..................................... Local synthesis ................................................ The role of uniformly continuous functions in Lm(G). . . . . . . . . . . . . . . . Remarks on non-weak * analysis and synthesis.....................

61 62 63 63 65 65 66 67 68 69

Exercises 1.4 : 1.4.1 Ideals in A ( f ) for which ZJ = I n J ......................... 1.4.2 Estimates with the Fejdr kernel on G ....................... 1.4.3 All subsets of discrete f a r e S-sets (cf . Exercise 1.2.2) . . . . . . . . . 1.4.4 The Fourier transform of a radial function . . . . . . . . . . . . . . . . . . 1.4.5 The classification of closed ideals in algebras of distributions . . .

70 70 71 71 73

Beurling weighted spaces

1.4.4 1.4.5 1.4.6 1.4.7 1.4.8 1.4.9 1.4.10 1.4.1 1 1.4.12

59 61

A technical remark about S-sets

10

Table of contents

1.5 Classical motivation for the spectral synthesis problem .................... 1.5.1 Analysis and synthesis .......................................... 1.5.2 Examples .................................................... 1. The theory of tides .......................................... 2. Cubism .................................................... 3. The theory of filters (in communication theory) .................. 4 . Music ..................................................... 5 . The interferometer .......................................... 6. The Heisenberg uncertainty principle .......................... 1.5.3 Synthesis and the representation problem in harmonic analysis ...... 1.5.4 Integral equations .............................................

75 75 75 75 76 76 76 77 77 79 79

2 Tauberian theorems 2.1 The Wiener spectrum and Wiener’s Tauberian theorem ....................

Introductory remark ........................................... Heuristics for generalized harmonic analysis ....................... The Wiener spectrum and generalized harmonic analysis ............ Examples ..................................................... Outline of the following sections ................................. A characterization of the Wiener spectrum in terms of summability kernels ....................................................... 2.1.7 A characterization of the Wiener spectrum in terms of ideals . . . . . . . . . 2.1.8 The primitive of a pseudo-measure on FI .......................... 2.1.9 Generalized harmonic analysis and Wiener’s motivation for his Tauberian theorem ............................................ 2.1.10 Riemann’s summability methods, Tauberian theorems, and generalized harmonic analysis ............................................. 2.1.1 1 Wiener’s original proof of his Tauberian theorem . . . . . . . . . . . . . . . . . . 2.1.12 An algebraic proof of Wiener’s Tauberian theorem . . . . . . . . . . . . . . . . . 2.1.13 Concluding remark ............................................

2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6

Exercises 2.1 : Sets of strict multiplicity .................................. Continuous pseudo-measures .............................. An Abelian theorem ..................................... A comparison of supp T and supp, T ....................... Jexpx(1 iy)sin2e-”dx ................................... The Wiener closure problem for LP(G) ......................

2.1.1 2.1.2 2.1.3 2.1.4 2.1.5 2.1.6

+

2.2 Beurling’s spectrum .................................................. 2.2.1 Fundamental properties of the narrow (i.e., a) and strict (i.e., 8) topologies .................................................... 2.2.2 Remarks on the a and B topologies ............................... 2.2.3 Examples ..................................................... 2.2.4 Beurling’s theorem : supp, T = supp T ............................. 2.2.5 Wiener’s Tauberian theorem and Beurling’s theorem . . . . . . . . . . . . . . . .

81 81 81 82 83 85 85 87 88 89 92 93 95 95 96 97 98 99 99 100 101 101 103 104 105 108

Table of contents

11

Spectral synthesis in the a topology ............................... Unbounded spectral analysis .................................... The spectral analysis and synthesis of almost periodic functions . . . . . . The extension to G of Bohr’s differentiation criterion for almost periodicity .......................................................... 2.2.10 Scattered sets. almost periodicity, and strongly synthesizable pseudomeasures ..................................................... 2.2.11 An historical remark ...........................................

108 109 109

2.2.6 2.2.7 2.2.8 2.2.9

Exercises 2.2: 2.2.1 Properties of the topology ............................... 2.2.2 Properties of the a topology ............................... 2.2.3 Bernstein’s inequality for A‘(FI) ............................ 2.2.4 The harmonic spectrum and suppp T........................ 2.2.5 The uniform closure of B(G) .............................. 2.2.6 Almost periodic functions ................................ 2.3 Classical Tauberian theorems .......................................... 2.3.1 Abel’s and Tauber’s theorems ................................... 2.3.2 A technical lemma ............................................. 2.3.3 Frobenius’ theorem: Cesdro summable implies Abel summable . . . . . . . 2.3.4 Littlewood’s Tauberian theorem ................................. 2.3.5 Slowly oscillating functions and Pitt’s pointwise conclusion to Wiener’s Tauberiantheorem ............................................ 2.3.6 A remark on Tauberian theorems ................................ 2.3.7 l(s) and the fundamental theorem of arithmetic .................... 2.3.8 Number theoretic functions ..................................... 2.3.9 The prime number theorem ..................................... 2.3.10 Fundamental properties of C(s) .................................. 2.3.11 Lambert series ................................................ 2.3.12 Theorems of Chebyshev and Mertens ............................ 2.3.13 Wiener’s Tauberian theorem and the proof of the prime number theorem ...................................................... 2.3.14 Perspective ................................................... 2.3.15 The prime number theorem and the Riemann hypothesis . . . . . . . . . . . . 2.3.16 Tauberian theorems and the Riemann hypothesis . . . . . . . . . . . . . . . . . . . 2.3.17 Salem’s Tauberian characterization of the Riemann hypothesis . . . . . . . 2.3.18 Beurling’s functions and a spectral analysis problem . . . . . . . . . . . . . . . . 2.3.19 Beurling’s Tauberian characterization of the Riemann hypothesis . . . . . Exercises 2.3 : 2.3.1 Tauber’s theorem ........................................ 2.3.2 Hardy’s Tauberian theorem ............................... 2.3.3 ((1 + iy) f O ............................................ 2.3.4 Number theoretic estimates ............................... 2.3.5 A Tauberian remainder theorem ...........................

110 113 114

114 115 116 117 118 118 119 119 119 120 121 124 126 126 127 128 128 130 132 134 136 136 137 137 138 139

141 141 141 141 142

12 Table of contents 2.4

Wiener's inversion of Fourier series .................................... 2.4.1 Remarks on Wiener's inversion of Fourier series .................... 2.4.2 Local membership ............................................. 2.4.3 Wiener's proof to invert Fourier series ............................ 2.4.4 Wiener-Lkvy theorem .......................................... 2.4.5 Other proofs of the Wiener-Uvy theorem ......................... 2.4.6 Generalizations of the Wiener-Uvy theorem ...................... 2.4.7 The maximal ideal space of M(G) and the extension of the Wiener-Lkvy theorem to B(f)............................................... 2.4.8 The Wiener-Pitt theorem ....................................... 2.4.9 The homomorphism problem .................................... 2.4.10 Beurling's criterion for A(T) ..................................... 2.4.1 1 Katznelson's converse to the Wiener-Lkvy theorem . . . . . . . . . . . . . . . . . 2.4.12 Tensorproducts ............................................... 2.4.13 An intrinsic definition of supp T.................................. Exercises 2.4: 2.4.1 Spectral synthesis and local membership .................... 2.4.2 The conjecture of dichotomy .............................. 2.4.3 Norm estimates for I/p ................................... 2.4.4 F o p 4 A(T)for p E A(T) ................................ 2.4.5 Algebra homomorphisms ................................. 2.4.6 Tensor products and support .............................. 2.4.7 Potential theory ......................................... 2.4.8 Inverses in M(G),multipliers, and the Tauberian theorem . . . . .

2.5

The Tauberian theorem in spectral synthesis ............................. 2.5.1 Historical commentary on S-sets and Taubexian theorems ........... 2.5.2 Ditkin's lemma. C.sets. and the major generalization of Wiener's Tauberian theorem ............................................ 2.5.3 Singular support ............................................... 2.5.4 Herzsets ..................................................... 2.5.5 Pisot numbers and spectral synthesis properties of perfect symmetric sets 2.5.6 The finite union of S-sets ....................................... 2.5.7 The intersection of S-sets ....................................... 2.5.8 Sets of spectral resolution and arithmetic conditions . . . . . . . . . . . . . . . . 2.5.9 Totally disconnected sets and measure theoretic properties of pseudomeasures ..................................................... 2.5.10 Measures associated with pseudo-measures ........................ 2.5.11 Varopoulos' lemma ............................................ 2.5.12 E Kronecker implies A'(E) = M ( E ) .............................. 2.5.13 A ' ( E ) = M ( E ) ................................................ Exercises 2.5 : 2.5.1 A norni estimate for A ( r ) ................................ 2.5.2 Lip, (T) E A(T). a > 1/2 .................................. 2.5.3 Properties of C-sets ......................................

142 142 143 144 145 146 146 147 147 149 150 154 154 156

157 158 158 158

160 160 160 161 162

162 163 167 168 171 172 172 173 174 175 178 179 180

181 182 183

2.5.4 2.5.5 2.5.6 2.5.7 2.5.8

Table of contents

13

A classical generalization of Wiener's Tauberian theorem ...... Schwartz's example ...................................... Maximal primary ideals in Banach algebras . . . . . . . . . . . . . . . . . . A classical equivalent form of Wiener's Tauberian theorem .... The union of S-sets ......................................

184 185 187 187 187

3 Results in spectral synthesis

3.1

Non-synthesizable phenomena ......................................... 3.1.1 Introductory remarks on non-synthesis ........................... 3.1.2 Malliavin's idea ............................................... 3.1.3 Malliavin's operational calculus ................................. 3.1.4 Non-spectral functions and Lipschitz conditions . . . . . . . . . . . . . . . . . . . 3.1.5 Principal ideals and non-synthesis ................................ 3.1.6 Malliavin's theorem and a property of A ( f ) ....................... 3.1.7 Helson's theorem on non-S-sets and the structure of ideals . . . . . . . . . . . 3.1.8 Katznelson's proof of Helson's theorem ........................... 3.1.9 The projective tensor product ................................... 3.1.10 Grothendieck's characterization of the projective tensor product . . . . . . 3.1.1 I Further properties of topological tensor products . . . . . . . . . . . . . . . . . . . 3.1.12 Varopoulos' idea .............................................. 3.1.13 Imbeddings of group algebras into tensor algebras . . . . . . . . . . . . . . . . . . 3.1.14 Imbeddings of tensor algebras into restrictions of group algebras . . . . . . 3.1.15 Remarks on Varopoulos' method ................................ 3.1.16 Mappings of restriction algebras ................................. 3.1.17 A characterization of synthesis preserving mappings . . . . . . . . . . . . . . . . 3.1.18 The structure of A(f) and the union of Helson sets . . . . . . . . . . . . . . . . .

188 188 188 189 192 193 193 194 195 197 197 199 201 202 204 208 209 213 213

Exercises 3.1 : 3.1 .1 Non-synthesis of discrete spaces ........................... 3.1.2 A ( T ) and finitely generated closed ideals .................... 3.1.3 Multipliers and spectral synthesis .......................... 3.1.4 A topology for which synthesis always holds ................. 3.1.5 Malliavin's theorem-the completion of Varopoulos' proof . . . . 3.1.6 Conditions for which A ( E , , f ,) = A ( E 2 , f 2 ).................. 3.1.7 A relation between A(R) and A ( T ) , and a result in synthesis ....

215 216 216 217 218 219 219

3.2 Synthesizable phenomena ............................................. 3.2.1 Integral representation and spectral synthesis ...................... 3.2.2 The Beurling integral and a criterion for synthesis .................. 3.2.3 Technical lemmas .............................................. 3.2.4 Applications of the Beurling integral ............................. 3.2.5 The Beurling-Pollard theorem ................................... 3.2.6 Constant values of the primitive of a pseudo-measure . . . . . . . . . . . . . . . 3.2.7 Katznelson's theorem: (p E A ( T ) flB V ( T ) is synthesizable . . . . . . . . . . . . 3.2.8 Spectral synthesis and the Kempisty-Denjoy integral . . . . . . . . . . . . . . . 3.2.9 The general problem of integration and synthesis . . . . . . . . . . . . . . . . . . .

220 220 221 223 225 226 227 230 231 234

14

Table of contents

3.2.10 A characterization of pseudo-measures in terms of LP . . . . . . . . . . . . . . . 3.2.11 Measure theoretic criteria for a pseudo-measure to be a measure . . . . . . 3.2.12 Kronecker's theorem and the condition that A'(E) = M ( E ) .......... 3.2.13 Bounded spectral synthesis ...................................... 3.2.14 A characterization of bounded-S-sets ............................. 3.2.15 The structure problem .......................................... 3.2.16 The balayage problem and spectral synthesis....................... 3.2.17 The structure problem and spectral synthesis ...................... 3.2.18 Refinements of Bochner's theorem and the structure problem ........ 3.2.19 Helson sets and the structure problem ............................

235 238 239 244 244 245 245 246 247 249

Exercises 3.2: 3.2.1 A characterization of A' .................................. 3.2.2 Hadamard measures ..................................... 3.2.3 Arithmetic progressions in the construction of pseudo-measures 3.2.4 Non-pseudo-measures with regular primitives . . . . . . . . . . . . . . . . 3.2.5 A;(T) and derivatives of continuous functions ............... 3.2.6 A(T) =L'(T) * A ( T ) ..................................... 3.2.7 An element of A'(T) \ M ( T ) and its Wiener support . . . . . . . . . . . 3.2.8 Sequence spaces and a strong form of uniqueness ............ 3.2.9 A class of synthesizable pseudo-measures ................... 3.2.10 A definition of integral for spectral synthesis ................. 3.2.1I A property of A(T) ..................................... 3.2.12 Non-synthesizable convergence criteria in A ( T ) . . . . . . . . . . . . . .

250 250 250 252 252 253 253 254 255 255 256 256

References

................................................................

Index ofproper names ...................................................... Index of terms

..............................................................

259 273 277

Index of notation

15

Index of notation The first list of notation includes sets and spaces; the second includes specific elements, operations, and the remaining symbols. Symbols that are used only in the section where they are introduced are not generally included in either list.

I. A ( f ) 1.1.4 Aj(E) 1.1.8 A ( U ) 1.3.6 A:(R) 2.1.3 AP(G) 2.2.8 A' 3.2.1

A , ( f ) 1.1.6 A ' ( f ) 1.3.1 A'@) 1.3.13 AS 2.1.4, E2.1.2 A E ( f ) 2.5.10 A ; 3.2.10

A ( E ) 1.1.8 A A ( f ) 1.3.4 A&!?) 1.3.13 AME) E2.1.1 A ( E , X ) 3.1.16 A"(E) 3.2.14

B ( f ) 1.1.4 g f 3.1.16 B ( X ) 3.2.17

B'"=R'" 2.5.7 BV(T) 3.2.1 B ( X , E ) 3.2.17

bT 3.1.16 BMO 3.2.10

C Introduction C E1.1.4 Cm(fin) E1.3.6 C(R") -E1.3.6 C(E1) @ C(E2) 3.1.9

C o ( f ) 1.1.4 Cm(T) E1.1.4 C?(R") E1.3.6 C"(T) E1.4.5 C ( E , X ) 3.1.11

C,(C) 1.1.5 Co(E) 1.3.13 Cz(fi") E1.3.6 Cb(G) 2.2.1

D o ( f ) 1.3.5 D,(R") E1.3.6 D m 2.5.2

D(fi") E1.3.6 D,O(R") E1.3.6 D(k,j) 2.5.2

D,(R") E1.3.6 D(cp,Z) 2.5.1 D(K,J) 3.1.8

m

E* = lJ Ij,mIj = ~j

3.2.6

EA 3.1.4

1

9 2.5.9

G = f Introduction, 1.1.2

f = C?

Introduction, 1.1.2

Z,, 1.4.7, Zloc(

Gd 3.1.16

I w ( M ) 2.1.5

a) 2.4.2

j ( E ) 1.1.8

k(E) 1.1.8 L'(C) 1.1.2 Lip.(T) E2.5.2

L"O(G) 1.1.2

L(Y, Y)

1.4.3

16

Index of notation

M(G) 1.1.4 M ( I ) 1.3.12 M f ( E ) 1.4.5

MO(f) 1.3.4 M ( E ) 1.3.13

M , ( r ) 1.3.4 MT 1.4.1

O(@) 2.2.8 Q Introduction

R Introduction II Introduction supp Introduction, 1.3.6, 1.4.1 sp@ 1.4.1 SnsIIntl E1.4.4 S U P P ~ 2.1.3,2.1.5 supp. 2.2.4 SUPP, E2.2.1

R+ Introduction s p y 1.4.1 SP,@ 2.1.3 S U P P , ~ 2.2.8

T Introduction 5 a 1.4.1

9 1.4.1 5 ° C 1.4.11

Fsp 1.4.1

V El .4.5 V ( E ) 3.1.9

Va E1.4.5 V(E1,EZ) 3.1.9

V

X” 1.1.3 XI 6 X2 3.1.10

2

Xi @ X2 2.4.12

Z

Z+ Introduction

Z ( X ) 1.1.8

I1 Ilm 1.1.2 11 I(m E2.1.2 I1 IIQ 3.1.9

II IIA, 1.3.1 II IIu 2.5.9

2.5.9

WAP(G) 2.2.8

Zp

Introduction 1.1.8

j.1.3

Xi @I Xz 3.1.11

11.

II 11 II

1.1.2 3-2.14 Ilv 2.5.9 LCAG 1.1.2 (A) 2.3.3 R2 2.1.10 1.3.1 T * S 1.3.1 L(T)(s) E1.3.6 .s,T 1.3.3 C,p E1.3.5 Ilell(cP,F) 3.2.2 F- T EA’ 3.2.1 K topology 2.2.1 a,( p, 1) 3.2.3 e. 2.5.4 V, E1.2.6 m 1.1.2 A(n) 2.3.8 d(n) 2.3.11 I11

IIA”(E)

(RH) 2.3.15 (C,1) 2.3.3 f 1.1.4 f 1.2.1 T * p 1.3.1 T 2 O 1.3.5 ryf 1.1.9 B ( p , F ) 3.2.2 p, 3.2.2 a topology 2.2.2 H ( E ) 3.1.18 X” Introduction fi E1.2.6 S 1.1.4 ((3) 2.3.7 ~ ( x )2.3.8

Ilg,rll 3.2.2

(PNT) 2.3.9 (R,2) 2.1.10 j 1.1.4 f- 1.3.7 D’T E1.3.6 T = 0 on U 1.3.6 Tp 1.3.7 IBl(v,F) 3.2.2 F- T 3.2.1 /!topology I 2.2.1 ~ ~ ( E2.5.2 6 ) int X Introduction d i E1.2.6 6, 1.3.3 n(x) 2.3.8 y 2.3.11

1 "he spectral synthesis problem 1.1 The Fourier transform of L1(G) 1.1.1 Prerequisites. We suppose as known the basic theory ofharmonic analysis as found in [Rudin, 5, Chapter 11 and the basic theory of commutative Banach algebras as found in [Loo mi s, 1, Chapter 41. In this Paragraph 1.1, we quickly restate some of the necessary results from harmonic analysis in order to establish notation; and proceed far enough along to give a proof of one form of Wiener's Tauberian theorem (Theorem 1.1.3). 1.1.2 L'(G) and Lm(G).Let G be a Hausdorff locally compact abelian group (LCAG), written additively, with Haar measure and dual group r. Haar measure is translation invariant on the Bore1 sets of G and such measures are unique up to multiplicative constants. We write e = r so that f' = G. Haar measure on G or r is denoted by m. L'(G) (resp. L"(G)) is the space of C-valued integrable (resp. measurable and essentially bounded) functions with respect to m, where two functions are identified if they are equal a.e. (resp. equal locally a.e.). (To be more precise about the definition ofL"(G) we make this parenthetical remark. An m-measurable set A is locally null if m(A n K)= 0 for each compact subset K G G. A property holds locally a.e. if it is true outside of a locally null set. Y ' ( G ) is the space of all m-measurable functions CP for which a constant a, > 0 exists such that {x E G : ICP(x)I > a,} is locally null. We write @ Y if {x: @(x) # Y ( x ) } is locally null, and so L"(G) = Ym(G)/.-. Any non-discrete non-6compact LCAG contains a locally null set A with mA = w.) We designate the usual norms on L'(G) and Lm(G)by 11 11' and 11 Ilm, respectively. As is well-known L"(G) is the Banach space dual of L'(G) and we define the duality between L'(G) and L"(G) by

-

Vf

E L'(G)

and V @ E L"(G),

(@, f ) = J @(x)f(x)dx. G

L'(G) is a commutative Banach algebra with convolution

% g E L'(G),

f * Ax) = j f ( y ) g ( x - Y)dY G

as the multiplicative operation. L'(G) has a unit if and only if G is discrete. 1.1.3 Banach algebras. Let X be a commutative Banach algebra. An ideal I G X is regular if X / I has a multiplicative identity. The maximal ideal space, X", of X is the

18

1 The spectral synthesis problem

collection of all the regular maximal ideals in X . y E X' is a multiplicative element if (y, x y ) = ( y , x ) (y, y ) for each x , y E X. The following is a basic theorem in Banach algebras : the set of non-zero multiplicative elements of the dualspace, X', is identijied with X mby the mapping y H (kernel of y) E X", where y E X' \ (0)is multiplicative. As such X" is given the induced weak * topology from X ' ; and the Gelfand transform of x E Xis the continuous function 2:X" --f C defined by f(Y) = (Y, x>.

X is semi-simple if the intersection of all of the regular maximal ideals in Xis (0).If 2 is the set of Gelfand transforms of X then X is semi-simple if and only if X E

x

\{O}

=-

9 E 8 \{O}.

X is a regular Banach algebra if for each y E X" and each closed set E C X" \ { y } , there is 2 E 8 for which f(y) = 1 and 2 ( E ) = 0. L'(G) is a commutative regular semi-simple Banach algebra whose maximal ideal space is r.

1.1.4 The theory of Fourier transforms for L1(G),L2(G),and M(G). The Fourier transform o f f E L1(G)is a function cp :r+ C defined as

AY) = cp(Y) = j f(X)(Y, x ) d x ,

Y E r;

G

and the space of Fourier transforms of L'(G) elements is denoted by A(T).The Gelfand transform of L'(G) is the Fourier transform [Rudin, 5 , Theorem 1.2.21. The Fourier transform map, L'(G) --f A(T) is bijective; A(T) is contained in the space Co(r) of continuous functions which vanish at infinity and it is a Banach algebra with pointwise multiplication and norm

A feature of Fourier transforms (and Laplace transforms) which is the key for their importance in applications is the fact that

v . g EL1(G),

(f*d Y Y ) = .(Y)i(Y).

This fact and some of the above remarks are expressed as Proposition 1.1.1. The Fourier transform bijective map

L'(G) -+ A ( r ) is an isometry and an algebraic isomorphism. M(G), the space of bounded Radon measures, is the Banach space dual of Co(G),taken with the sup norm 11 [la. L'(G) is imbedded in M(G) because of the Radon-Nikodym theorem, and the Fourier transform of p E M(G) is a function cp :r+ C well-defined by

P(Y) = P(Y) =

(Yl

G

4 dlc(x),

y E r.

1.1 The Fourier transform of L1(G')

19

The operation of convolution in L'(G) extends naturally to M(G) and is given by V P ,v

E

M(G) and Vf

E

Co(G),

i = ( A , >.

As such, M(G) is a commutative Banach algebra with unit, 6, defined by (6, f ) =f(0). The space of Fourier transforms of M(G) elements is denoted by B(T); and the Fourier transform map, M(G) +- B(T), is bijective. B(T) is characterized by Bochner's theorem as the space of finite linear combinations of continuous positive definite functions (rp :T -+ C is positive definite if Vyl,

. . ., 7.

E

r

and

vC1,

. . ., C, E c,

CjCk

V(yj

- y k ) 2 0).

J.k

It is easy to check that the elements of B(T) are bounded and uniformly continuous. The inversion theorem is

Theorem 1.1.1. Let f E L'(G) n B(G). T h e n f ~L1(T)and (1.1.1)

f(x)

=

j f(ir)(G)dr.

r

Remark 1. The Haar measure on r in (1.1.1) is uniquely determined by the given Haar measure on G . 2. Note that iff 4 Co(G)then f 4 L'(G) fl B(G).

3. Compare Theorem 1.1.1 with Exercise 1.3.5.

The Purseval-Plancherel theorem is

Theorem 1.1.2. Let L'(G) fl L2(G)and L2(T)be taken with their respective L2-norms. With these norms the Fourier transform is an isometry, (1.1.2)

LI(G)n L ~ ( G ) +

~(r),

onto a dense subspace of L2(T).As such there is a unique extension of( 1.1.2) to a bijective isometry L ~ ( G ) -+ f

H

LZ(r) f

Further, r

We denote the usual L2-norm by

11 [I2.

Corollary 1.1.2.1. (M. Riesz) A(T) =L2(T) * L2(r)and Vrp1

9 E L2(0,

IIP *

911" Q l l P l l z l l ~ l l z .

20

1 The spectral synthesis problem

1.1.5 V cp E A ( f ) , supp rp is a-compact. Let Cc(G)be the space of continuous functions with compact support. Also note that iff, g E L1(G) are equal outside of a locally null set then f = g a.e. We say that U c f is a compact neighborhood if U is compact and int U # 0.

Proposition 1.1.2. a) Iff E L1(G)then there isg =f a.e. (i.e.,f andg really define the same element in L1(G)) such that suppg is a-compact.

b) For each cp E A ( f ) , suppcp is a-compact. Proof. a) Since Cc(G)is dense in L'(G) let lirnllf, -fill = 0, where f, E C,(G), and set Kn= {x:S,(X) # 0) = SUPPA. Therefore, for any measurable set B c A = {xf(x) # 0) \ (U &), we compute that f(x)dx = 0; and so mA = 0. B

By the definition of A and Kn, {x:f(x) # O} c A

u ( U K").

Define for x E U Kn for x $ ( U Kn)'".

f(x)

b.i) For cp E A(T) note that m

m

where En is compact. The compactness follows since cp vanishes at infinity. b.ii) Thus it is sufficient to prove that in a LCAG X the closure of a a-compact set H U H,, H, compact, is a-compact. Let U be a compact symmetric neighborhood of 0 and set

Y=

=

0 (V+ ... +U),

n=l

where the sum on the right-hand side contains n terms. Hence, Y is a a-compact open (and therefore closed) subgroup of X[H e w i t t and R o s s, 1, I, pp. 33-34]. Consequently, we can choose A E X for which

x = Yu Y,, EA is a disjoint union.

Y,=y+ Y,

1.1 The Fourier transform of L1(G) 21

Each H. is covered by a finite number of Yyand so there is a sequence B G A with the property that

H c U Y,. YEB

U Y, is o-compact and closed, the latter fact following since Yyis open and YEB

u yy=(yEuByy)-.

YEB

q.e.d. 1.1.6 The point at infinity is an S-set. Let A,(T) = A(T) fl C,(T). -

Proposition 1.1.3. A,(T) = A(T). Proof. In light of Cor. 1.1.2, the fact that C,(T) is dense in L2(T),and the continuity of the map,

Lyr) x

~ q r -+) A(r) (cp,

@I

H

cp

* *,

we need only observe that supp cp * JI is compact for cp, @ E C,(T). q.e.d. Proposition 1.1.3 is precisely the statement that the point at infinity (of T ) is a spectral synthesis set (cf. Theorem 1.2.1 and Paragraph 1.4). This result is used in the proof of Wiener’s Tauberian theorem. 1.1.7 An approximate identity technique. The following result yields the fact that for any two disjoint closed sets El, E2G r,where El is compact, there is cp E A(T) such that cp = 1 on El and cp = 0 on E,. In particular, A(T) is a regular Banach algebra. We shall have more to say about such matters in Paragraph 1.2 and shall develop the cryptic proof below more fully there. Proposition 1.1.4. Let E c r be compact and let V c T have positive measure and compact closure. There is cp E A(T) such that 0 < cp < 1,

I cpll A d

-

W mW’,

and 1 0

foryEE foryE(E+ V- V)-.

Proof. Define

q.e.d. This result should also be compared with Exercise 2.5.1~.

22

1 The spectral synthesis problem

1.1.8 Wiener's theorem on the inversion of Fourier series. For each cp E A(T), define the zero set of rp to be

zcp = {y € f :cp(y) = 0). Thus, if X 5 A ( f ) then the zero set of X is

z(x)= {? E r :vcp E x,rp(y)

= 0) =

n {zrp:cp E xi.

Further, if E c f is closed then we define k(E) = {cp

E

j ( E )= {rp

E A(T) : E

A ( f ) :E E Zrp}

and fl sUPP ~p = g}.

Zcp is closed, k ( E ) is a closed ideal, j ( E ) is an ideal, and j ( E ) C k(E). Let A(E) = A ( f ) / k ( E )and A,(E) = A ( f ) b v )be taken with their quotient topologies. It is then an elementary fact from the theory of Banach algebras that A(E) and A,(E) are commuta-

tive Banach algebras. Clearly, if Eis not compact then A ( E )and A,(E) do not have units. Also, E is compact if and only if A ( E ) has a unit; and if E is compact, then A,(E) has a unit if and only i f j w ) = k ( E ) (cf. Theorem 1.4.1b). A ( E ) can be considered as the set of restrictions of elements of A(T) to E. Proposition 1.1.5b is Wiener's theorem on the inversion of Fourier series. Proposition 1.1.5. Let E c T be closed. a) The set of maximal ideals of both A(E) and A,(E) is identified with E. VE is compact then E is the maximal ideal space of A(E). b) I f E is compact and Zrp n E = sfor

bJ E E,

$(Y)

E

A(T) then there is $ E A ( f ) such that

= I/rp(Y).

P r o o f . a.i) We shall show that the set of maximal ideals in A,(E) can be identified with E; the same proof works for A(E). We know (e.g. Exercise 1.1.6) that M y= {cp E A ( f ) : rp(y) = 0) is a regular maximal ideal in A(T). Let c:A(T)-+ A,(E) be the canonical map. The canonical image, cMy, of M y in A,(E) is clearly an ideal. If y E E thenjw) G M y .For this y let CJ z cMy,J E A(T),be a proper ideal. (Jis defined as the set of all elements $ E A ( r ) for which $ + j m E cJ.) Thus M y+ j m c J a n d so M y G J since y E E. By the definition of the canonical multiplication in the quotient we see that J is an ideal. Consequently, M y = J since M yis maximal; and thus cMy s A,(E) is a maximal ideal if y E E.

1.1 The Fourier transform of L'(G)

23

Conversely, let cM E A,(E) be a maximal ideal. As before, M is an ideal. If cp EJ'(E) we see that cp E M since ccp is the zero element of cM. Therefore Z M E E. Let J G A(T) be an ideal. Then cM c CJ if M E J. Since cM is maximal we have that CM = cJ. If cp E J then ccp E cM and so cp E M. Hence, J G M , from which we conclude that M is maximal. By Exercise 1.1.6, M = M y ;and y E E since ZM E E. Consequently, ifcM G A,(E) is a maximal ideal then M = M yfor some y

E E.

a.ii) If E is compact then A ( E ) has a unit and so every ideal is regular. b) Using part a) and the basic characterization of invertible elements in Banach algebras with unit, we have that ccp E A(E) is invertible if and only if the Gelfand transform @ never vanishes, i.e.,

zcpn E = 0. Thus, there is t,b E A(T) such that Vdy E E,

*(r) d Y ) = 1

L,: A(r)

-+

*

q.e.d. W i e n e r proved Proposition 1.1.5b for the case r = T and E = T. His proof used a technique of collecting local information to make a global statement; we shall develop this method fully in Paragraph 2.4. In Proposition 1.1.6 we present a different proof to Wiener's original result (trivially generalized to any compact group). Take cp E A(T) and define

*

A(r) cp*

(cf. Exercise 2.4.8 and Exercise 3.1.3). L, is a well-defined continuous linear map. Obviously, L, can be defined for cp E B(T). The resolvent set, p(L,), of L, is the set of z E C such that (L,-~I)-I:

A(r)

-+

A(T)

is a continuous linear map. p(L,) E C is an open set. Proposition 1.1.6. Let

r be a compact group.

a) For each cp E A(T), p(L,)" = { z E C :cp(y) = z, y

E

b) Assume cp E A ( T ) never vanishes. Then (1.1.3)

v+ E A ( r ) 3 e E A(r)such that 1/1=

ecp.

r}= C,.

24

1 The spectral synthesis problem

c) Assume cp E A ( f ) never vanishes. Then l/cp E A(f). P r o of. c) is immediate from b). To prove b) note that because of part a) we have 0 E p(L,); and SOL;’: A ( f ) exists and is continuous. It is then sufficient to define 8 = L;I($). a) In proving part b) we only used the inclusion

-N

A(f)

c; E P(LJ which we now verify. The opposite inclusion is trivial. Let w E C; and define 6 , = d(w, C,), “d” being Euclidean distance (note that C, is compact). We set Aw = SUP vrr

I d Y ) - WI

and (1.14

$(Y)

=

(cp(y) - ~ ~ ( ( c P W) ( Y) H A : + XJ,

noting that A , 2 6,, $ E A(f), and

ll$llm

=+(A: - &I.

Thus c = +(A? + 43 4 { z :I Z J s IIJlllmI. By B e u r 1i n g ’s formula [Loom i s, 1, p. 751 for the spectral radius of L we conclude that C E p(L,) (cf. the Remark below). Consequently,

L, - cz= ( L , - @Z)(Le- WZ) has a continuous inverse; and so (L,-wZ)-l:

A(f)

+

A(f)

exists and is continuous. Therefore w E p(L,). q.e.d.

Remark. Because of (1.1.3) we mention the Cohenfucrorizurion theorem (1959) : (1.1.5)

A(r)A(r) = A(q.

Actually (1.1.5) is true for quite general algebras and modules (cf. Exercise 1.4.1). Salem proved (1.14 for f = Z and Rudin proved the f = FI case. The reason that the proof of Proposition 1.1.6b does not prove the factorization theorem for A(f), f a-compact (in which case we can trivially define functions cp E A(f)which never vanish), is that, although $ E B(f)in (1.1.4) and L, is well-defined, we can not apply the spectral

1.1 The Fourier transform of t l ( C ) 25

radius formula. Note that the Cohen factorization theorem is trivial for A ( r ) , f compact, since 1 E A ( T ) (cf. (1.1.7)). We refer to [Koosis, 1; PtAk, 11 for elegant proofs of (1.1.5). 1.1.9 Wiener’s Tauberian theorem. With Proposition 1.1.3 and Proposition 1.1.5b we shall now prove the Wiener Tauberian theorem (Theorem 1.1.3). We use the notation TYf(X)

= f ( x -Y).

Theorem 1.1.3. Given r and cp E A(T). cp never vanishes ifand only iff is a-compact and V$ E A(T) and V E > 0,

3 cl, ..., C.E C and 3 xl, ..., x. E G

such that

Proof. i) Given the approximation property (1.1.6), we prove that cp never vanishes. Let p(y) = 0, and choose E > 0 and $ E A(r)such that $(y) > E . This contradicts (1.1.6). ii) The opposite direction provides more of a challenge. If Zcp = 0 then r is a-compact by Proposition 1.1.2. Observe that

This follows from Proposition 1.1.5b by setting E = supp0, choosing 0, which 0, = l/cp on E, and noting that (PO, = 0, where dl = 00,.

E

A ( f ) for

Take E > 0 and $ E A ( f ) . We can choose el E A,(T) such that

II*

- 411” < 4 3

by Proposition 1.1.3, and then can choose 8,

E A,(T)‘such

that

81 = cpez

by (1.1.7). Next, pick 8 = h E A ( f ) with the properties that supph = K is compact and 1102

- ell” < &/(3IIcpIIA).

The left-hand side of (1.1.6) is less than or equal to ll*-~lllA+llcp~~-cp~llA+llcp~-~~,cp~~~~~~~,~llA.

26

1 The spectral synthesis problem

Letting!

= cp

(1.1.8)

it is sufficient to prove that

/lh * f - i 1 c , r x j f / l

1

<~ / 3

for some cl,. . ., c, E C and xl, ..., x, E G. Since the map G +. L1(G), x H rXf, is uniformly continuous there is a symmetric neighborhood U of 0 E G such that vx E u,

I %f -f

111

< e/(311hII1).

Because K is compact there is a finite collection UJ= x, covers K. We set Kl = K rl U,,

(

K,= K \ U K , IT'

) nu,,

+ U,x, E K ,j = 1, . .., n, which

j = 2 ,..., n,

J-s

and c - h(x)dx. Kl

Consequently,

*f(x)-

CJTxjf(x)= J

noting that for y

E K,,

-1

f I hCv)[r~f(x)-rx,f<x>]dy,

J=1

K~

xJ - y E x, - U, = U.

Thus

q.e.d. We do not explicitly use the fact that r is o-compact to prove the sufficient conditions that Zcp = 0.Compare the above proof with Theorem 2.1.4, where the factorization is also explicit but without the use of Proposition 1.1.5b.

Exercises 1.1. 1.I .I Non-vanishing Fourier transforms

a) Let Haar measure on R be chosen so that m(0, 1) = 1 and thus Haar measure on FI is determined by m(0, 1) = 1/2n. Compute

0

f(r) = r(l + ir)

for f ( x ) = e+ex.

1.1 The Fourier transform of L1(C)

27

ii) Thus in both casesf€ A ( n ) never vanishes. b) Let r be a-compact. Construct a non-vanishing element cp E A(r). 1.1.2 A special Banach algebra (cf. Exercise 3.1.1)

Let X be the vector space of Fourier transformsf= cp of functionsf€ L2(T)which are

[

continuous on -,;

41.

Define the norm

a) With this norm and the operation of pointwise multiplication, prove that X is a commutative semi-simple Banach algebra with unit and that its maximal ideal space X" is identified with Z. b) Prove that {cp E X: card supp cp < a} is dense in X (cf. Proposition 1.1.3).

1.1.3 A(T)= co(r) It is trivial to prove that A(T) = Co(T)by the Stone-Weierstrass theorem (cf. Proposition 1.3.2). Obtain the same result by assuming that A(T) is contained properly in Co(T). (Hint: Let p E M ( f ) \ {0} annihilate A T ) . By Fubini's theorem, p E B(G) c L"(G) annihilates L'(G), and thus p is a locally null continuous function. Consequently fl = 0 so that p = 0 (the desired contradiction) since the Fourier transform M ( T ) -+ B(G) is bijective). Note that if card r > No then A(T) is a set of first category in C,(T) even though A ? ) = Co(r).If A(T) = Co(T)then c a r d r < N~ [Segal, 41; the simplest possible proof of this fact is found in [ G r a h a m , 41, 1. I .4 Riesz products and the Fourier coeflcients of Cantor-Lebesgue measures

A Riesz product is an infinite product m

(EI.l.1)

R(r)=n(l+a,cos(r,r+a,)),

O < la,IQ1,a,,rJ,r,al,ER.

j -1

They were introduced by [F. Riesz, 11 to compute the Fourier coefficients of the Cantor-Lebesgue measure supported by the triadic Cantor set C c [0, 27c] (e.g. part c) below). A readable sketch of their properties, particularly those of which we have the least interest in this book, is given in [Keogh, I ] . The partial products, Rk(r),of R(r) are non-negative since 0 < lajl Q 1. A sequence { r , : j = 1, ...} G Z+ is lacunary if inf r I + l / r j> 1. Riesz products are used to prove Sidon's theorem [Zygmund, 2, I, I

28

1 The spectral synthesis problem

pp. 247-2481 : if 2c,e"JYis the Fourier series of cp E L"(T)and {r, :j = 1, ...} is lacunary, 1

then cp E A(T). a) Compute that if a, = a then (El .1.2)

e,-O.fl

' '

b) The perfect symmetric set E E T determined by geometrically as follows: set E = fl E,, and write

{tk:k= 1, , ..} _C (0,1/2) is formed

2k

Ek

u Ej,

J-1

where E: =[O, 212tl] and E: = [2x(1 - tl),2121,

Et = [212t1(1 - 5 2 ) , 2n511,

E: =[O, 2x:t15219

E: = [2n(l - tl),2n(l-

tl) + 2xt1 &I,

and E: = [2x:(l - tl t2),2x:I ,

etc. Thus, Ek is the union of 2, closed intervals Ej, each of length 2xt1 . . .&, and to form E, + we construct Efl+_',,E t : ' c E j ,

j=1,

..., 2,.

Prove that E is a compact totally disconnected set without isolated points, that each y E E E [0,2n] can be written as m

(E1.1.3)

y = 2x:

1 e,r,, 1

where e,=O, 1, r l = l - t l , 1im2'+112t1...t,.

and rk=&...&-1(1-{k)

for k 2 2 , and that m E =

I

Let C"(T) be the space of infinitely differentiable functions cp: [0, 2121 + C such that for each k k 0. The distributional derivative H' of H E L1(T)is defined ~ ( ~ ' (= 0 )(~(~'(212) by 2x

2x

1

where dy = 212. 0

1.1 The Fourier transform of L1(G)

29

The Cantor-Lebesgue function for E is the continuous increasing function F on [0, 2x1 defined as m

F(y) = 2n 2&J/2J on E, I

where y and { E , } are related by (EI.l.3), and extended continuously with line segments. The Cantor-Lebesgue measure p E M(T) corresponding to F with supp p E E is defined by the Riesz representation theorem as p =6

+ F',

where F is the distributional derivative of F. Suppp is defined rigorously in Paragraph 1.3.6. It is easy to check that llplll = 1 and that 2n I

(E1.1.4)

VnE

Z,

P(n) =

*

J 271

einYdF(y).

0

c) Let p be the Cantor-Lebesgue measure corresponding to a given F and E. Compute

n cosxnrj, m

Vn E Z,

p(n) =

J -1

where r, is defined in (E1.1.3). (Hint: An approximation for the Stieltjes integral in (E1.1.4) is

noting that F increases by 1/2kbetween Ej and E j + Calculate that k

n cosnnr, is equal to the expression in (El. IS)). If E is determined by 5,

= 1/3,j = 1, . . ., then

E = C is the (triadic) Cantor set. In this

n cos(2nn/3'). Thus P(3") m

case we have $(n) =

= 1 for each m E Z+and

hence E(n) #

J- 1

o(l/lnl), as In1 -+ a,even though, as is well-known, E(n) = o(l/lnl),as In/ + 03, since F is a function of bounded variation. 1.1 .S Regular maximal ideals in Banach algebras

Let X be a commutative Banach algebra. a) If I

c Xis a (proper) regular ideal prove that Iis contained in a regular maximal ideal.

30

1 The spectral synthesis problem

b) If Z E Xis a regular maximal ideal prove that Zis closed. (Hint: Since l i s regular there is an element u E Xsuch that x - ux E Ifor each x E X. If Z#l then u E I = Xsince Zis maximal. To obtain a contradiction first note that m

- y)'x - u Z ( u - y ) J x

m

0

for llu - yll < 1 . Thus X is the ideal generated by 1 and y , and so y $ Z if llu - y 1 < 1. Hence u $2). c) Prove that every maximal ideal in X is regular if and only if X X = X. Thus, by the Cohen factorization theorem, every maximal ideal in A ( f ) is regular (and therefore closed). Closely related to this matter, and at one point using a lemma by C o h e n, [V a r o p o u 1o s, I ] investigates the continuity of positive linear forms on Banach algebras X with an involution. 1.1.6 Regular maximal ideals in A ( f )

For each y E r define M , = ($9

E

A ( f ) : cp(y) = O } .

Prove that Z E A(r)is a closed regular maximal ideal if and only if Z = M yfor some r. It is interesting that Theorem 1.2.5, which is a form of the Tauberian theorem, provides an immediate proof of the fact that closed maximal ideals in A ( f ) are regular (cf. Exercise 1.1.5).

yE

1.2 Approximate identities in A(T) 1.2.1 0 is a strong Ditkin set. The construction of approximate identities, which are norm bounded and contained in specific subsets of A ( f ) , constitutes an essential technique in spectral synthesis.

The first result involves a standard technique used to approximate 6, as a measure on G,in various distribution spaces and for various topologies. It is our simplest approximation procedure for A(T), and the convolution trick that is employed is an essential tool for much of this section.

fo.

Notationally, set f ( x ) = Recall that G is metric if and only if f is a-compact [Hewitt and Ross, 1, I, p. 3971.

1.2 Approximate identities in A ( f )

31

and for all compact subsets E 5 r (1 2.2)

lim cpu = 1, u

uniformly on E.

r f G is metric, or, equivalently, ifr is a-compact, {cpu} can be chosen as a sequence.

Proof. i) For each compact neighborhood U of 0 E G let V be a compact symmetric neighborhood of 0 E G satisfying V + V E U ; and choose a non-negative element g, E L2(G)such that suppg, E V and g,(x)dx = 1.

1 G

Define (1.2.3)

fu = g,

* g',

B0

and

fu = cpu.

ii) Clearly, fu is a continuous positive element of L'(G). By the Plancherel or inversion theorem, cpu E L'(T); and so by the inversion result fu E A(G). The fact that fu E A,(G) is then clear from (1.2.3); in fact, supp fu G U. iii) Now

iv) Since Vis symmetric we can takeg,(x) g, is real, we have

= g,(-x)

(e.g. g,

= (l/mV),y,) so that because

G

G

thus g, is real. Consequently, in this case, cpo 2 0 by (1.2.4). Also from (1.2.4), we compute

v) Now, given E > 0 andf E L1(G),choose U as above, so that by the uniform continuity of the function y H 7, f (cf. the proof of Theorem 1.1.3), we have that

v Y E u,

Il5f -fll1 < E .

32

1 The spectral synthesis problem

I

Consequently, sincefU 2 0 and G f&) dx = 1, j f " ( Y ) Il%f-fll1dY < 8;

Ilf*fU-flll

G

and so (1.2.1) is true. For (1.2.2) let rp E A(T) equal 1 on E and apply (1.2.1). vi) To define precisely the directed system, set U compact basis elements at 0.

> W if U c W, where U and Ware q.e.d.

A subset X E A(T) contains an approximate identity {cp,} if {q,} E X is a directed system and, for each cp E X , 1imI)cp- 'p'p,l,, = 0; an approximate identity {cp,} contained (I

in X is bounded if {qa}is 1 IIA-normbounded.

1.2.2 The point at infinity is a strong Ditkin set. Our next two results were first stated generally by [G o d em e n t, 2, p. 1261. The major difference between them and Proposition 1.2.1 is that the functions are chosen in Ac(r). Proposition 1.2.2. There is a directed system (cp,} 2 O , f , E A(Gh II~,lla = 1, and

G A,(T),

where 3, = cp,

such that

fa, cpa

limcp,(y)= 1.

VyyE,

U

I f G is metric, or, equivalently, ifr is a-compact, {cp,}

can be chosen as a sequence.

P r o o f . i) For each n and each compact neighborhood V of 0 E r we can find (e.g. [Hewitt and Ross, 1, I, pp. 254-2551) a compact neighborhood U G f such that V c Uand (1.2.5)

m(U- f(l U + V)) mU

1 < -. n

This condition is obviously satisfied for r = R. Generally, (1.2.5) is a reasonable statement; its proof for arbitrary r, although not requiring any structure theory or other powerful results, is however quite technical. We shall assume (1.2.5) as known. ii) Consider the triplet a = (n, V , U ) , with the notation from (1.2.5), and set 1

* xu B 0.

cp. = - x u

mu

1

1

nl

n2

Define al> a2 if - < -, V1 E V2,and U1c U,.Clearly, if we are given a1 and a2 we can choose a3 such that n3 > max(n,, n2) and V , = U 1U U2.Thus, {pa}is a directed system.

1.2 Approximate identities in A ( r )

33

iii) cp, E A , ( f ) by Corollary 1.1.2and& E A(G) by the inversion theorem. Also, by the computation in Proposition 1.2.1,

iv) Given y E f.It remains to verify that lim cp,(y) = 1. If = (n, V, U)and y

E

V then,

0

using (1.2.5) and the fact that m(y + U)= mu, we compute 12 Va(y)=

m(U n ( y

mu

+ U))b m(y + U)- m(U- n ( V + U)) 2 1 - - 1. mU

n

This yields the result. q.e.d. Another approach to deal with (1.2.5) is given in [Hewitt and Ross, 1, 11, pp. 2993011. We are now in a position to make a refinement of Proposition 1.1.3. Let E E f be we define the "multiplier norm" closed. For cp )-E

IllcpIII = suP{llcpII/llAlllII/IIA:II/ E W )\ (0)). Clearly, IIIcp((I< I((P((~. E c f is a strong Ditkin set if there is a directed system {cp,)

sj(E)

such that

v cp E k ( E ) ,

1ipIIcp -

(PcpUIIA

=0

and SUP Ill P a 111 < a) U

(cf. [Saeki, 7, Definition 21). If r is a-compact and metrizable then E G f is a strong Ditkin set if and only if there is a sequence {cp, :n = 1, ...} s j ( E ) such that

v CP E k ( E ) ,

lim

n-tm

1 1 -~ V V n / I A = 0.

It turns out that intervals [a, b] 5 FI are strong Ditkin sets (e.g. Exercise 1.2.6c), and so, by drawing the appropriate picture, we see that {cpn:n= 1, . ..} can not generally be 11 IIA-normbounded.

34

1 The spectral synthesis problem

Using Proposition 2.5.3 and the fact that closed sets E G T for which mE = 0 are disjoint from some translates of roots of unity, the following is easy to check: given a closed set E c T-forwhich mE = 0 ; then 9 Ej(E),

llPll.4 = lIlplll

(cf. the remark on sets of measure 0 in Section 1.2.4). The following result implies that the point at infinity (of r)is a strong Ditkin set.

Theorem 1.2.1. There is a directed system {q,} E A@), with each IIq,IIA= 1, such that for all rp E A(T), lim"rp - (p(pollA = 0. In particular, if E c r is compact then (I

lim cp, = 1,

uniformly on E.

0

ZfG is metric, or, equivalently, if

Proof. i) Take {cp,}

is a-compact, then {cp,}

can be chosen as a sequence.

as in Proposition 1.2.2.

We shall prove that if Y is a compact neighborhood of 0 E G then

(1.2.6)

(I

where!,

1

lim f,(x)dx= 1, V

= (pa.

The idea of proof for (1.2.6) is straightforward. Note that f,E LZ(G)sincef:
andf, E A(G). Thus, by Parseval's formula,

so that if

iv E A(r)n L1(T)(instead of just being in A(T) fl Lz(T))then

by Proposition 1.2.2 and the inversion theorem. We omit the technical detail to handle the case that E L2(r)\ L1(r);it is simply a matter of constructing functions g and h for which g < xv < h, g(0) = h(0) = 1, and 8, h E A(r)n L1(f), by the convolution technique we've been using. ii) Take?= cp E A(T). For any e > 0 choose af, and a compact neighborhood U of 0 E G such that

v Y E u,

IlTYf

-fll,

< d3

1.2 Approximateidentities in A(f)

+ J”

1

AX)

+~”jJ

jfa(Y)dy - j fa(y)f(x-y)dy fa(y)f(x-y)dy

35

dx

dx

1.2.3 Points in f are strong Ditkin sets. Strong Ditkin sets were first defined (as such) and systematically studied by [W i k, 11, and a complete classification of strong Ditkin sets for quite general groups has been made in [Meyer, 2; M e y e r and R o s e n t h a l , 1; R o s e n t h a l , 1; G i l b e r t , 2; B. S c h r e i b e r , 1; Saeki, 7] (Saeki'spaper wassubmitted for publication in the spring of 1969). Theorem 1.2.2b, which implies that each one-point set in f is a strong Ditkin set, goes back much further and we shall discuss its history in Chapter 2. The following result specializes Proposition 1.1.4 in a useful way; the additional conclusion, (1.2.7), is necessary to proveTheorem 1.2.2. The relevant parts of Proposition 1.2.3 and Theorem 1.2.2 should be compared with Exercise 2.5.1~.

Proposition 1.2.3. Given r > 1, E > 0, a compact subset K E G, I E r,and a neighborhood W E r of I . There is cp E A J f ) , such that cp = 1 on a neighborhood of 1,cp = 0 on W " , 0 < cp < 1, IIcplla < r, and

Proof. Without loss of generality take A = 0. By the definition of the topology on r, the subset U 5 r, for which (1.2.8)

Vy

E

U and V x E K,

11 - (y, x)l < ~ / 2 r ,

is an open neighborhood of 0 E I'. Let Y be any compact symmetric neighborhood of 0 E r such that V - V C U fl W.

36

1 The spectral synthesisproblem

By the regularity of Haar measure, there is a compact neighborhood Eof 0 E r satisfying m ( E - V )< r 2 m V ; and Ecan bechosen so that E + V - V c U n W. Setting cp = (1/mV)xv * xE- (as in Proposition 1.1.4) we obtain all the conclusions except (1.2.7); in particular, cp = 1 on E and cp = 0 off of E V - V. We have

+

(mV)(cp(.)(.,x)-cp(.))= xv (1.2.9)

*((.,X)XE-Y-XE-Y)

+((.,x)XB-Y)*((.,x)XY-xY).

Using Cor. 1.1.2, (1.2.8), and (1.2.9), we compute V

XE

K,

E

(mV)llcp(*)(*, x) - c p ( * ) I I A

< -r [ m V m ( E -

V ) ] ' / 2 < E(mV).

q.e.d.

Theorem 1.2.2. Given A E r and r > 1. a) There is a directed system {JI.} 0 Q $ a G 1, ll$allA < r, and

b) There is a directed system {d,}

v a,

118allA

c A,(T) such that JIa = 1 on a neighborhood of I ,

G A,(T)

r l j({A}),norm bounded by

< 1 +r,

.such that (1.2.11)

V cp E k({A}),

lim/Icp- cpOallA = 0. a

c) If'G is a-compact, or, equivalently, vr is metric, the directed system in a) can be chosen as a sequence. I f G and r are both a-compact the directed system in b) can be chosen as a sequence. Proof. Without loss of generality, let A = 0. a) The triplets tl = (n, K, W ) ,where K c G is compact and Wis a compact neighborhood of 0 E r, form a directed set with an order defined by a > @: na B ne, Ka 2 KO, W , E W p

For each tl we choose 8. = $a, corresponding to cp in Proposition 1.2.3, such that

v x E Ka,

][$a(*)

-

('9

x)

$~(*)IIA

< l/naa

1.2 Approximate identities in A(f)

cp E k({O})and

Now, for!=

E

37

> 0, take a compact set K c G for which

j If(x)l dx < 4(4r). K-

I

= p(0) = O,

Since f(x)dx G

we have

Thus,

b > a,

II(P)(IBIIA

< E.

b) Choose {ps}as in Theorem 1.2.1 and {$,} as in part a). Set 8, = cpB - cps $, and define a, > a2 if b1>/I2and ql>tf2. Then, if cp E k({O}),we observe that - cpeuliA

I/p- VPBI/A + tIV$rtlAIt(PrltA.

The result follows from Theorem 1.2.1 and part a). q.e.d. 1.2.4 Idempotent measures and strong Ditkin sets. One of W i k ’ s major tools in studying strong Ditkin sets, and one used and refined by various of the subsequent investigators mentioned earlier, centered about P. J. C o hen’s classification of idempotent measures p (i.e., those for which p * p = p). We refer to [Rudin, 5, Chapter 31 and [Am e m i y a and I t 6,1] for a full discussion and elegant proof, respectively, of C o h e n ’s result. In order to state Cohen’s theorem (in the context developed by Wik and R o s e n t h a l ) we define W c 9(r)to be the smallest algebra of subsets of r (i.e., closed under finite unions and complements) which contains all of the cosets of arbitrary subgroups of r. The result is Assume X c r is not in 9 and that {cp, :n = 1, . . .} c B(T) satisfies thefollowing property : (1.2.12)

limcpn(y)=

n +m

0

for y

1

foryg X ;

E

X,

then limllp,,/l,= 03, where p,, = p,,. n

38

1 The spectral synthesis problem

Using this fact it has been shown, for example, that the only strong Ditkin sets in I‘= T with Haar (i.e., Lebesgue) measure 0 are finite sets and that those with positive Haar measure contain an interval. If p E M(G) is idempotent and llplll > 1 then llplll 2 (1 + 4‘)/2 and this constant is attained by the norm of some such p [Saeki, 2; 31. 1.2.5 A characterization of j v ) and k(E). The point for us at this moment is that although A(T) is normal (cf. Proposition 1.1.4 and the remark preceding it) (1 -2.12) tells us that it has definite possibilities for being rather “abnormal”. In order to determine just how regular A ( T ) can be, we shall investigate the regularity properties of the ideals, i.e., we shall investigate its “local regularity” properties.

Proposition 1.2.4. Let I E A(T) be an ideal. a) A , ( f ) n I is norm dense in I. b) If E s I‘ is compact and ZI fl E = 0 then there is cp E I such that cp = 1 on E. Proof.a)Consider{q,} E A,(f)asinTheorem 1.2.1. ForeachqEI,limllq - qqJA=O a

and cpq. E I. b) By hypothesis, for each y E E, there is qyE Z such that cp,(y) = 1. Since I is an ideal, $, = cp, q, E I. Thus $, 2 0, and there is a neighborhood V, of y such that $, >, 1/2. By the compactness we can choose a finite set, { V,,, . . ., V,”},of such neighborhoods which covers E. Define

We apply Proposition 1.1.5b (noting that ZJ/fl E = 0) and have the existence of 8 E A(T) for which cp = 8$ = 1 on E. q.e.d. Proposition 1.2.5. Let ZE A(r)be a closed ideal and set E = ZI. a) Ifq E AJT) andZZ c intZcp (Le.,ZZ n supp cp = 0) then cp E I. b ) j T ) c I c k(E),and so k(E) is the largest closed ideal with zero set E a n d j v ) is the smallest closed ideal with zero set E.

Proof. a) By Proposition 1.2.4b there is $ E Z such that 1(1 = 1 on supp cp. Therefore cp = fp1(1 E I. b) The statement about k(E) is obvious. From the Proposition 1.2.4a) and the fact that I is closed we need only to prove that A,(r)n j ( E ) c I.

1.2 Approximate identities in A ( f )

39

Let cp E A , ( f ) fl j ( E ) have the property that ZI fl supp cp = 0 (since ZI = E). Apply part a). q.e.d. 1.2.6 A generalization of Wiener's Tauberian theorem. We have the following generalization of Wiener's Tauberian theorem. Theorem 1.2.3. Let I E A ( f ) be a closed ideal and assume that ZI s intZcp,

where cp E A(f). Then cp E I.

Proof. By Theorem 1.2.1 or Exercise 1.2.1 let lipllcp - cpcpuJIA =0, where cp, E A , ( f ) . Then supp cpcp, E supp cp and supp cp fl ZI = 0, so that, from Proposition 1.2.5a), cpcp, E I; and hence cp E I since I is closed. q.e.d. Note that both Theorem 1.1.3 and Theorem 1.2.3 depend essentially on Proposition 1.1.3 and Proposition 1.1.5b. We do not use the full strength of Theorem 1.2.1 in Theorem 1.2.3 but only employ Exercise 1.2.1 which depends heavily on Proposition 1.1.3. To see that Theorem 1.2.3 actually generalizes Theorem 1.1.3 we first need the following result recorded by [G o d e me n t, 2, p. 1271, Proposition 1.2.6. a) I f I E L'(G) is a closed ideal then it is a closed translation invariant subpsace. b) Zf I c L'(G) is a closed translation invuriant subspace then it is a closed ideal.

Proof.a)ForanyfEL'(G)choose(g,:n= (cf. the Hint of Exercise 1.2.1). Also,

1, ...} ~L'(G)suchthatlimI(f-f *g,II,=O

f * g,)

n

= (7%gn) *f, and

so

lip II(rxgn) */ - T x f ! ( l = 0. Thus, if I is a closed ideal and f E I we have 7xf E I since (rxgn)* f E I. b) The proof of part b) is the proof of (1.1.8) which in fact did not involve the hypothesis that f never vanished. q.e.d. An Hahn-Banach argument is also often used to verify Proposition 1.2.6 [Rudin, 5, pp. 157-1581, Theorem 1.2.4. Let I E A(T) be a closed ideal. r f for all y E f there is an element cp E I for which cp(y) # 0 then I = A(T).

Proof. Take cp E A ( f ) and note that ZI= 0 by hypothesis. Thus, we apply Theorem 1.2.3 directly. q.e.d.

40 1 The spectral synthesis problem As a corollary we obviously obtain Theorem 1.1.3 because of Proposition 1.2.6. Further, in the context of Banach algebras, Wiener’s Tauberian theorem has the form

Theorem 1.2.5. Let I E A(r)be a proper closed ideal. Then I is contained in a closed regular maximal ideal (cf. Exercise 1.1.5a). Proof. E = Z I # 0, for otherwise we contradict Theorem 1.2.4. Thus, IE k(E) E My,where y E E and M y is the maximal ideal corresponding to y (cf. Exercise 1.1.6). q.e.d. 1.2.7 A remark about Drury’s theorem. The discussion prior to Proposition 1.2.4 about the separation properties of A(f) can be rounded out for the time being by mention of some recent theorems due to D r u r y et al. Essentially these results determine a property of A(T) which allows us to separate Helson sets (e.g. Section 1.3.13) from closed sets while maintaining a norm boundedness (cf. (1.2.12)). We are necessarily vague at this point but will be more explicit in Section 3.1.18.

Exercises 1.2

1.2.1 Thepoint at infinity is a C-set Prove that for each cp E A(f) there is a sequence {cp,:n = 1, . . .} G A , ( f ) such that limllcp - cpcpnllA = 0. This is equivalent to the statement that the point at infinity of r is a C-set (cf. (E1.2.3) and Section 1.4.9). The proof is much simpler than the proof of Theorem 1.2.1. (Hint: Given f = cp E A(f) and E > 0. Using the uniform continuity of the function G + L’(G), x H~,f,find $ E A(T) such that IIcp < 4 2 (cf. Proposition 1.2.1); from Proposition 1.1.3 there is 8 E A,(T) such that /I$ - ellA< E/(2IIcpIIA). Then Ilcp - cp% < 4.

1.2.2 The Tauberian theoremfor compact G A statement of Theorem 1.2.3 for the case of G compact (and f discrete) is: if I E A(f) is a closed ideal and cp E A(r)satisfies the condition, (E1.2.1)

ZIcZcp,

then cp E I. Prove this directly and briefly from first definitions and Exercise 1.2.1. (Hint:$ E A , ( f ) has the form 2 ayxy,where card F < co, and cp$ = 2 byxyE I ; apply YE

F

YlZ1

Exercise 1.2.1).

1.2.3 The Tauberian theorem and spectral synthesis properties of points a) For arbitrary G and T we would like to find conditions which, when coupled with (El.2.1), imply that cp E I. Prove that if a closed ideal IE A(r)and cp E A(f) satisfy (El.2.1) and if ZI= {y}, a set consisting of a single point, then cp E I. (Hint: From

1.2 Approximate identities in A(f)

41

Theorem 1.2.2, lim IIcp - cpcpUIIA = 0, and note that ZI c int Z(cpcp,);apply Proposition (I

1.2.5). Let us remark that this theorem uses the Tauberian result, Proposition 1.2.5, and Theorem 1.2.2. Actually the full strength of Theorem 1.2.2 is not necessary but only the fact that {y} c r is an S-set, i.e., (E1.2.2)

V

E

> 0 and V cp E k({y}),

3 $ E~({Y)) such that IIcp - $11"

< E.

3 $ ~ j ( { y } ) such that IIcp - ~

$ 1 1<~ E ,

(E1.2.2) or the stronger statement, (E1.2.3)

V

> 0 and V cp E k({y}),

E

can be proven directly in an easier way than Theorem 1.2.2 (the analogy is the same as that between Exercise 1.2.1 and Theorem 1.2.1). b) Prove (El .2.3) without resorting to the complicationsof Theorem 1.2.2. (El .2.3) is the statement that {y} is a C-set (cf. Exercise 1.2.1 and Section 1.4.9). 1.2.4 Primary ideals and spectral synthesis properties of points We continue to discuss the subject of Exercise 1.2.3 but from an algebraic point of view. A primary ideal I E A(T) is an ideal that is contained in precisely one closed regular maximal ideal (cf. Exercise 1.1.5b, c). Prove that the following statements are equivalent (and true by Exercise 1.2.3): a) Every closedprimary ideal in A(r)is a (closed regular) maximal ideal. b) Every one point set {y} E I' is an S-set. (Hint: To prove that a) implies b) note thatj({y}) is primary taking into account that each closed regular maximal ideal in A(T) is of the form k({y})(e.g. Exercise 1.1.6). Conversely, if I is primary we havej({y}) c I E k({~})by Proposition 1.2.5, and so a) follows immediately from b). 1.2.5 The Nullstellensatz and spectral synthesis

Let X be a commutative ring. An ideal 1 c X is prime if (E1.2.4)

X

~

I E and x $ I

=. ~ E I ;

and an ideal I G X is algebraically primary if (E1.2.5)

xy E I and x 4 I

3

3 n E Z' such that y"

a) Prove that each closed primary ideal in primary.

E

I.

A(r)is prime and

hence algebraically

b) Prove that each closed algebraically primary ideal I, and hence each closed prime

42

1 The spectral synthesis problem

ideal, in A(f) is primary (and thus maximal). (Hint: I is contained in a maximal ideal by Theorem 1.2.5. Let I E My, fl My*, where M,is defined in Exercise 1.1.6, and take ‘pi E~({Y,})such that cp,(y,) # 0 for i # j and $ = cpl ‘pz= 0. Now ‘pi 4 M,,for i # j and hence ‘pi # I does not imply cp; E I). Thus the closed primary, closed prime, closed algebraically primary, and maximal ideals are equivalent in A(T).

Remarkl. The fundamental problems of both the number theoretic and algebraic geometric aspects of commutative ring theory have analogues in the algebraic view to spectral synthesis that we have hinted at in Exercise 1.2.4, and which we shall develop. For this reason we recall that a commutative ring X is Noetherian (named after E. N o e t h e r because of her fundamental paper in 1921) if every ideal in Xis finitely generated. An ideal in a Noetherian ring is the finite intersection of algebraic primary ideals. We first make a remark on the number theoretic situation. The Noetherian rings which are integrally closed and in which proper prime ideals are maximal are called Dedekind domains, and these are characterized by the property that each ideal is a unique finite product of prime ideals. This result follows from the above general intersection property of ideals in Noetherian rings. The ring of integers in an algebraic number field is a Dedekind domain; and, consequently, we have K u m m e r ’ s very important unique factorization property. (In his attempt to prove Fermat’s conjecture, K u m m e r introduced the notion of an ideal and investigated the unique factorization of integers in certain algebraic number fields.) A basic example from classical algebraic geometry of a Noetherian ring is the polynomial ring A’= C [ z , w ] , z, w E C . The fact that the maximal ideals in C [ z , w ] are all generated by two elements, P&) = z - u and P,(w) = w - b, where u, b E C , is the first version of the NullsfeNensarz [Kaplansky, 2, p. 191. The idealJgenerated by the polynomial P(z) = z is prime but not maximal. It turns out that if S E C x C and J G C [ z ,w ] is the largest possible ideal such that VPEJ,

P=OonS,

then J is the intersection of a finite number of prime ideals. Thus the zero set of the ideals J and I generated by P(z) = z and P(z) = zz, respectively, is the w-“axis” ; clearly I c J properly and I is an algebraic primary ideal. If I c C [ z , w ] is an ideal we define fi= {P:3 n > 0 such that P“ E I }Then . the full Hilbert Nirllstellensatz [Kaplansky, 2, p. 191 depends essentially on the theorem that flis the intersection of the maximal ideals containing it.

A(r).In light of Exercise 1.2.5 and our remark on Noetherian rings wc ask if each closed ideal in A(r)is the intersection of maximal ideals. This problem of “unique factorization in A ( V ’ is the problem of spectral synthesis (cf. Paragraph 1.4).

2. We return to the situation of

1.2 Approximate identities in A ( f )

43

1.2.6 Approximate identities and strong Ditkin sets a) Recalling the convention from Exercise 1.1.1 for Haar measure on R and FI, we define

Prove that A, =

21[

A

x , ,* ~xAI2where xA,2 is meant to be the characteristic function of

[ - I / 2 , 4 2 ] E FI for this exercise. Computef,

E L'(R), where],

= A,,

as

for x = 0,

Thus llAAlla= I,fA 2 0,fA(x) = I f l ( I x ) ,and A,(y) = A ,

(3

and it is an approximate identity for L1(R) when we let A

. {!,:A

+=

> 0}is the Fejkr kernel co.

b) The de la Vullke-Poussin kernel, { V A : I> 0 } , is defined on FI as

v 1. > 0,

vA(y)= 2 A 2 1 ( y ) - A,(?),

and its graph is obviously a trapezoid. From a), V , 2 0 and IIV,IIA Q 3. Prove directly that for each cp E k({O}), Il;m - 0 II VA vIIA= 0

(e.g. [ K a h a n e and Salem, 4, p. 170; Wik, 1, p. 571). Thus {0} c FI is a strong Ditkin set (cf. Theorem 1.2.2); in fact if cp EA,(FI) nk({O})and 0, € A , @ ) is 1 on [-n,n] then for large enough n,

IIv

- (0,

- V 1 / n ) ( P l l A = IIVl/n(PIIA.

c) Prove that each closed interval [a, b] G FI is a strong Ditkin set. (Hint: Define q n = ?-a

Vl/n

+ T-b

Vlln

+ $n,

where $" E Ae(FI) is 0 on [a, b]" and cpm = 1 on [a, b]. Choose 0"as in b) and check that if cp E k([a,b ] ) f l A,(R) then lim!Icp- (0, - Vn)IIa

= 0).

1.2.7 A characterization of strong Ditkin sets

Let G be a-compact and metric. Prove that E G r is a strong Ditkin set if and only if there is a sequence {p, :n = 1, . ..} G M(G)such that p,, = I on a neighborhood of E and

44

1 The spectral synthesis problem

cp E k(E),

such that for each!= 1imIIp.

n-m

*fll,

=O.

(Hint: If E is strong Ditkin let p,, = 6 - A wherejn = (P, and cp,, is the “strong Ditkin approximate identity” for E; this does it. For the converse let& = 0, be an approximate identity for A(f) with llO,,ll., 6 1. Set cpn = On (1 - fin) E K E ) . For each cp E H E ) ,

1.3 Pseudo-measures 1.3.1 The space A ’ ( 0 of pseudo-measures. The Banach space dual of A(f) is A ’ ( f ) , the space of pseudo-measures. The transpose of the Fourier transform map, L1(G)+ A ( f ) , is the function F : A ’ ( f ) -+

L“(G)

defined by (1.3.1)

V T E A ’ ( T ) and V f e L ’ ( G ) ,

(FT,f)=(T,f).

We write FT= f’, and call the Fourier transform of T. The canonical norm on A ’ ( f ) is given by IITIIA.= 11f‘11,,,for each T EA ’ ( f ) . We can well-define the convolution map

”f) x A ‘ ( f )

+

A‘(f)

(T,S)

H

T*S

(1.3.2) as Vf

E L’(G),

( T * S,f) = (%,f ).

Thus V T, S EA ’ ( f ) ,

n

T* S=

f3.

The following is immediate. Proposition 1.3.1. A ’ ( f ) is a commutative Banach algebra with multiplication defined by (1.3.2) and with unit 6 . Further F : A ’ ( f ) -+ L“(G) is a bijective isometry and algebraic isomorphism.

Note that (1.3.3) V T, S E A ’ ( f ) and Vf= cp E A(f),

= (T7, (SA,cp(A

+Y)))

since S f E L’(G). We shall give an intrinsic(in A ’ ( f ) )definition of T * S i n Section 2.4.13.

1.3 Pseudo-measures

45

1.3.2 M ( f ) c A ' ( f ) . Proposition 1.3.2. a) M ( f ) is a subalgebra o f A ' ( r ) . b) The duality, (1.3.4)

V T E M ( f ) and

V cp E A(f),

(T, cp) = JcpdT, r

is consistently defined so that (1.3.5)

V T EM ( f ) ,

T(X)= I c y , x)dT,, r

where f' in (1.3.5) is defined by (1.3.1) and T is considered as an element of A ' ( f ) in the pairing of( 1.3.4).

Proof. a) If T E M ( f ) , Tis defined and linear on A ( f ) since A(f) E C,,(f), Tis uniquely determined on A(f);in fact the algebra A(f) is separating and self-adjoint so that A m = C,(f) (in the sup norm) by the Stone-Weierstrass theorem (cf. Exercise 1.1.3).

If limIIqnlla= 0, where cpn E A(f), then limIIcpnllm= 0. Thus, lim(T, cp,) n

n

= 0, and we

n

conclude that T E A ' ( f ) . b) For (1.3.5) note that (T,, (y, x)) exists when T E M ( f ) . Also, by Fubini's theorem (which we can use since T E M ( f ) ) ,

v f E L'(G),

(T,P) = J N ( T y 7(y, 4 )dx. G

Consequently, since f'€Em(G)and ( f ' , : f )

= (T,f),

we have f'(x)= (T,, (7, x)). q.e.d.

1.3.3 Operations in A ' ( f ) . By the definition of Q and the integral representation in Proposition 1.3.2 we set '0' T E A ' ( f )

and V cp E A(f),

( R cp)

= ( T , 9).

We define the translate T , T of T E A(r)by y as lfcp E A ( 0 ,

(7,

T, cp) = (T,, 447 + 4) = (T,

For the case of 6 we also write ~ , = 6 6,. Note that we have the imbedding

f y

+. A ' ( f ) H

T,6.

T-,

cp).

46

1 The spectral synthesis problem

In this context r is the maximal ideal space of the Banach algebra A(T) and the induced weak * topology on r from A'(r)is precisely the given locally compact topology on r taken as the dual group of G. If p

E

M ( r ) and cp E A(T) then p

* d Y ) = (PA, cp(Y -

A(r).

If T E A'(r)we still have (T*, cp(Y

in fact, withf (1.3.6)

- 4)E 40;

= cp E A(r),

(7'1,

&y-l)) =(TA, ( f ( * ) ( ~ *))A(-A)) ,

= ~ T ( x ) ( ~ ( - x )-(xv),) ~ x * G

Thus we define the convolution

and we write

* d Y ) = ( T A , d?- A))' When we use the notation, T * rp, we shall mean (1.3.7) just in case cp E A(r)fl A'W).

and not the convolution (1.3.2)

1.3.4 Pseud*functions. We shall be interested in various subfamilies of A ' ( f ) . Besides M(T) c A'(T) we now define the space AA(Z")of pseudo-functions as Ah(T) = { T E A'(T) : ? vanishes at infinity}.

A&(r) is quite

important for the study of Riemann's sets of uniqueness (e.g. [Benedetto, 6, Chapter 31 and Exercise 2.1.1). Notationally we set

~ , , ( =r )M ( r ) n ~ x r ) and note that M,(T) is a subspace of Mc(r),the space of continuous bounded Radon measures on r (e.g. Exercise 1.3.1 and Exercise 2. I .2). 1.3.5 Radon measures and A Q . Now write

ccm = u c m , where K E r is compact and C K ( I ' ) = { ( P E C ~S ( ~U) P: P ~ ~ E K } .

1.3 Pseudo-measures

47

cK(f)is a Banach space with the sup-norm; and a linear functional defined on C c ( f ) which is continuous on each c K ( f ) is a Radon measure. The topological vector space properties of the spaceDo(f) of Radon measures are found in [Bourbaki, 1,Chapitre 3; Schwartz, 5, Chapitres 1 and 31; for our purposes we note that M ( f ) E D o ( f )(cf. Exercise 1.3.2), and that a directed system {cp,} c C , ( f ) converges to 0 if and only if 3 K c f , compact, such that V a, supp cp. (1.3.8)

cK

and li? cp. -- 0, uniformly on f .

The notation “ D O ( f ) ”is used because of its relation to distribution theory, e.g. Exercise 1.3.6.

Proposition 1.3.3. Given T E A ‘ ( f ) and assume that ( T , cp) 2 0 for each non-negative function cp E A ( f ) . Then T E M(f). Proof. i) We first prove that T E D o ( f ) . Clearly A,(T) = C J f ) (with the topology described above). Let {cp,} c A c ( f ) satisfy (1.3.8) and choose a non-negative function cp E A , ( f ) equal to 1 on K. From the uniform convergence there is (6,) E R tending to 0 such that y

I

9

cpU(Y)(

‘ $(Y)* &Q

A straightforward calculation yields the fact that the real and imaginary parts of each cp, are in A , ( f ) , and so we assume that cp, is real-valued. Thus -&Q$

VU

&U$9

so that, by the positivity assumption, lim (T, cp,)

= 0, and,

hence, T E D o ( f ) .

0

ii) From Proposition 1.3.2 and the fact that T E D o ( f ) ,we have (1.3.9)

V cp E A , ( f ) ,

(T, cp) = IcpdT. r

Since T E A ’ ( f ) , sup{l(T, cp)l: cp E A , ( f ) and IcpII, Q l} = IITII,,. < co; consequently, by Proposition 1.2.2, (1.3.9), the positivity of T E D o ( f ) ,and the monotone convergence theorem, dTQIITIIAr

03.

r

ldT

Thus, by standard integration theory, e.g. [Bourbaki, 1, pp. 154-1551, IITllI = and therefore T E M ( f ) (as well as 11 TI1 = 11 TI/,,,).

q.e.d.

48

1 The spectral synthesis problem

If T E A’(r) and (T, cp) b Ofor all non-negative cp E A(T)then we write T b 0. Obviously, D o ( f )= M ( r ) when r is compact. 1.3.6 Thesupportofapseudo-measure. Our next project is to define the support of T E A’(f); this is a crucial notion from the point of view of synthesis. For an open set

U c r we define

Given T EA’(r); T = 0 on U if

Proposition 1.3.4. Let U c r be open and take T E A ‘ ( f ) . If for each y E U there is an open neighborhood V, E U of y such that T = 0 on V, then T = 0 on U. Proof. Let cp E A ( U ) and choose {cpn:n= 1, ...} E A , ( r ) such that limIIcp - cpcpnl(A = 0, n

e.g. Exercise 1.2.1. We shall prove that (T, cpcp,)

= 0.

For each V, we use Proposition 1.1.4 to choose $, E A,(r)and an open set N , E V, for which $, = 1 on N, and $, = 0 off of V,. { N y:y E U}is an open cover for the compact set supp qcpnand therefore we can choose Nn,,. . ., NYnE {N,: y E U}as a finite subcover.

Clearly, PcPn=CPVn[l

-$,,)(I

-$yJ.-.(l

-$TJI-

When we expand 1 - (1 - $yl) .. .(1 - $ym) we obtain a sum of products of the $, so that each of the terms has support in one of the V,,. By hypothesis, then, (T, cpcp,) = 0. q.e.d. (suppT) to be the Because of Proposition 1.3.4 we well-define the support of’T E A’(r) complement of the union of all open sets U E r such that T = 0 on U.Thus, equivalis the intersection of all closed sets K for which T = 0 on ently, the support of T E A’(r) K”.Clearly, Proposition 1.3.5. Assume that the sequence {T. : n = 1, ...} c A’(T) conuerges to T E A‘(T) in the weak * topology. If E c r is a closed set such that Vn, then supp T C E.

suppT,~E

1.3 Pseudo-measures 49

1.3.7 The product Tt$ where T E A‘Q and E A ( f ) . In order to give further basic properties of the support (e.g. Theorem 1.3.1) we must define the following “multiplication” :

Tq is well-defined by the formula

v $ EW),

(Tcp, k) = (T, cp$).

Observe that V T E A‘W) and V cp E A(r), suppTcp E suppTrl suppcp.

Setf-(x) =f(-x) and note thatf- = cp- iff= cp. Clearly (1.3.1 1)

n

Tcp(x)= f * f-(x),

where f= cp ;

and if Zs A(T) is a closed ideal, (1.3.12)

(T,I)= 0

0

TZ=O.

We shall verify (1.3.12). Since the point at infinity is a C-set, the implication from right to left is true for any subset Z c A(T). The other direction is clear using the fact that I is an ideal. The following is also immediate (e.g. Exercise 1.3.4). Proposition 1.3.6. Given T E A’(T). T = 0 on an open set U E r i f and only i f for each q E A(U), Tcp = 0. 1.3.8 A dual formulation of Wiener’s Tauberian theorem. Because of the Hahn-Banach theorem, Theorem 1.3.1c is equivalent to Theorem 1.1.3 when we assume that cp never vanishes (cf. the discussion in Section 2.1.13). Once again, the proof depends essentially on and follows easily from Proposition 1.1.5b. Theorem 1.3.1. Given T E A’(T) and cp E A(T). a) Zfq = 0 on an open set U c r then Tq = 0 on U. b) ZfT = 0 on an open set U E r then Tcp = 0 on U. c) ZfTq = 0 then cp = 0 on suppT. Proof. a) and b) are clear. c.i.) Let U c r be a relatively compact open set such that cp never vanishes on 0. Letting $ E A ( U ) we shall first prove that (T, $) = 0. From Proposition 1.1.5b choose 8 E A(T) for which 8 = l/cp on 0 so that $* = $8 E A(U).

50

1 The spectral synthesis problem

Therefore, by hypothesis on Tcp, (1.3.13)

(T, $) = ( T , cp$,>

= (TV, $,>

=O.

ii) Let V = U { UaE r : Uais relatively compact and open, and cp never vanishes on Ua}. From Proposition 1.3.4 and (1.3.13), T = 0 on V; hence (1.3.14)

suppTs V".

iii) If y E suppT we shall assume that cp(y) # 0 and obtain a contradiction. There is an open relatively compact neighborhood W of y such that cp is not 0 on so, as above, T = 0 on W. On the other hand, W fl V" 20, and this contradicts (1.3.14).

and

q.e.d. 1.3.9 Radon measures are synthesizable. The following is a standard measure-theoretic fact [Bourbaki, 1, pp. 68-71].

Theorem 1.3.2. a) Given T E Do(T) and assume cp E C,(T) oanishes on suppT. Then (T, cp> =o. b) For each p E M(T) there is a directed system {pa}E M(I') such that cardsuppp, < OD, supp pa c supp p, lim pa = p in the weak * topology a(M(T), COW)),and 1 pall = 11 pII a

,.

Proof. We shall prove a). i) Set K=suppcp and E=suppT. By (1.3.8) there is M K< 0 such that for each 8 E C,(T) with supp 8 c K we have I(T, @I < M K l l e l l m . Given E > 0, we shall prove that [(T, cp)l < e. ii) Let V = { y E T :1 cp(y)l < e/(2MK)}; then V is open since cp is continuous, and E E V. Clearly, E" is an open neighborhood of the compact set V". There is a standard procedure to adapt Urysohn's lemma to locally compact spaces, i.e. there is a continuous function $ : r + [0, I ] such that $ = 1 on V" and supp $ c E" (we of course have much stronger results from Paragraphs 1.1 and I .2). iii) Note that E n suppcp$ = 0 and hence (T, cp$) = 0. Further, cp = cp$ on K f l V" and Q IcpI on I'. Consequently, since cp = 0 on K -,

1.3 Pseudo-measures

51

Therefore, noting that supp(cp - cp$) E K , we compute I(T, CP>I= I(T* co - CP$>I < MK

IIv

-

( ~ $ 1 1 m < 8. q.e.d.

In terms of harmonic analysis, Theorem 1.3.2 says that each Radon measure in A’(r)is synthesizable (cf. Paragraph 1.4. and Section 3.2.13). Using this fact and Proposition 1.1.3 we have the following result (cf. Theorem 1.3.la) which provides a converse to Theorem 1 . 3 . 1 in ~ case T E Do(T). Proposition 1.3.7. Given T E A’(r)fl Do(r) and assume cp E A(T) vanishes on suppT. Then Tcp = 0. 1.3.10 Heuristics for the notion of spectrum, Now is a convenient time to begin discussion on the notion of spectrum. We shall give various examples in Paragraph 1.5 and develop the idea carefully in Chapter 2.

During the plague years of 1665-1666, I s a a c Newt o n made an important contribution to spectral analysis and synthesis. He discovered that sunlight, as a special case of white light, is actually composed of the continuous spectrum of colors from red to violet; he did this by letting beams of sunlight pass through a prism. Different colors of light correspond to different wavelengths of the sinusoidal light waves emitted by a source [Rossi, 1, Chapter 31; and each color has its own characteristic index of refraction (with regard to a prism) which is dependent on the frequency of the wave. In the case of a finite spectrum (of frequencies) this means that a complicated wave @ is synthesized in terms of its spectrum {A, : k = 1, .. ., n} c R as n

(1.3.15)

@(x)=

2 ake”k*. 1

By a standard computation (which we shall develop in Chapter 2 ) the means

2R

\

@(XI e-lYxdx

-R

tend to 0 if y # I , and to a, if y = I , (for the case of (1.3.15)). The point is that if quite general “waves” @ E Lm(R)have “spectra” contained in R,and if @ has a formal Fourier expansion similar to (1.3.15) (but infinite), then it is possible to utilize various means on @ and specify the “spectrum” of @. Abel [Be u r 1in g, 51 and Riemann [Po 11a r d, 21 summability have been used for this purpose; and, in this context, Proposition 1.3.6 and Theorem 1.3.la, b are proved using the Riemann localization principle, e.g. [Benedetto,6,pp. 55ff.;Bochner,pp. l≪KahaneandSalem,4,pp. 166-1671. Since (1.3.15) is the Fourier transform of a measure with support { I l , ..., An}, we would like to associate the concept of support with the intuitive notion of a spectrum for even more general phenomena than (1.3.15). [Beurling, 2; 61 and [Godement, 21 defined the spectrum of f = @ E Lm(G)equivalent to supp T ; Be u r 1in g’ s formulation was in

52

1 The spectral synthesis problem

terms of the narrow topology on L"(R) (cf. Paragraph 2.2) and G o d e m e n t ' s (viz. (1.4. I)) was in terms of the weak * topology on L"(G). 1.3.11 A characterization of the weak * closed submodules of A'(r).A'(r)as a group under addition is a module over the ring A(f). Proposition 1.3.8. The Fourier transform, A(r) -+ L"(G), is a bijectionfrom the space of weak * closed submodules of A'(T) onto the space of weak * closed translation invariant subspaces of L"(G). Proof. i) If M E A'(r)we write .T = &EL"(G). F is weak * closed (resp. a vector space) if and only if M is weak * closed (resp. a vector space). ii) Given a weak * closed translation invariant subspace Y C L"(G), 1- = cp- E A(T), and f E J ;from (1.3.1 I) M is a submodule if T * f E 9. Choose a directed system {pu}5 M(G) as in Theorem 1.3.2b which converges tofin the weak * topology a(M(G),C,(G)). T * pu E .T by the translation invariance. Ifg E L'(G) then (f, (m,g(Jc+ v)>> = < i.*L s ) , i.* P u , g ) = (Pu, ( i.(Jc), g(x + Y ) ) ) since (T(x),g(x + y ) ) is bounded and continuous. -+

(

(The fact, p. 1681.)

?'* f E Y,can be proved by the Hahn-Banach theorem [Benedetto, 2,

iii) Let M E A ' ( f ) be a weak * closed submodule. It is sufficient to prove that (1.3.16)

V T E M and

where T,(y, x,)

E A'(r) is

(T,(Y, xo),

VX,EG,

TY(y,xo)€M,

obviously well-defined hy

cpm

= (TY, cp(7)

( 7 9

xo));

in fact, if = @ E F then ((Tl(y, xo))*,g ) = ( @ ( x + x,), g(x)), and so T-+, @ E X Let {$.} z A(T) be an approximate identity for A(r)as in Proposition 1.2.1. Thus, since M i s a module and $.(.) (-,x,) E A(r),T,$,(y)(y, x,) E M ; and we have (1.3.16) from (1.2.1). q.e.d. 1.3.12 A basic duality technique. For Ic A(f) and M c A'(T), we define (1.3.17)

M ( I ) = { T EA ' ( r ) : V cp E I , Tp =O},

and (1.3.18)

I ( M ) = {p E A ( r ) : V T E M , Tcp = 0},

1.3 Pseudomeasures

53

respectively. It is trivial to check that:

i f I C A(I? is a closed ideal then M ( I ) E A‘(T) is a weak * closed submodule, and

if M c A‘(T) is a weak * closed submodule then I ( M ) c A(T) is a closed ideal. The following is a fundamental fact in functional analysis:

If X is a normed space and X’, taken with the weak * topology, is denoted by Xi,then (1.3.19)

(Xi)’= X.

Because of (1.3.19) we can formulate the dual form of the Hahn-Banach theorem (for the locally convex space X i and its dual X) which we now use. Proposition 1.3.9. Let I, I, C A ( r ) denote closed ideals and let M, M, C A’(r)denote weak * closed submodules. Then a) II = Z2 o M ( I l ) = M(Zz). b) Mi = M2 ’ c) M

0

I(M1) = Z(M2).

= M(Z(M)).

d) I = I( M( I)). Proof. a) If cp E I2 \ Il then apply the Hahn-Banach theorem to obtain a contradiction. b) If T E Mz \ M 1then apply the dual Hahn-Banach theorem, which we have from the setting of (1.3.19), to obtain a contradiction. c) Clearly, M c M(I(M))and J c I(M(J)),for J = I(M). On the other hand, if we apply “I”to M c M ( I ( M ) )we see that

from which we can conclude that

c) then follows from a). A similar argument works ford). q.e.d. 1.3.13 Helson sets and S-sets. After proving some theoretical results about A ’ ( r ) , examples of elements T E A’(r) \ M ( T ) are in order. Bochner’s theorem and the fact m = L“(G)obviouslyprovide a means-although a bit too hygienic-to compute that A examples. We promise hair-raising calculations, subsequently, for the “mathochist”.

54

1 The spectral synthesis problem

For the time being we introduce a certain subclass of A ’ ( f ) \ M ( r ) in a non-constructive way. In order to do this let E c r be closed, and write A’@) ={nA’(r):suppTc E } = J ~ ) ’ - , A i ( E ) = { T EA ‘ ( E ) : V cp E k ( E ) , ( T , cp)

=

0 } = k(E>’ = A ( E ) ’ ,

and M ( E ) = { p E M ( r ):S U ~ pP E E ) = c,(E)’,

where C,(E) is the space of continuous functions defined on E which vanish at infinity. From Proposition 1.3.2 and Theorem 1.3.2a, M ( E ) E A i ( E ) and, of course, A ( E ) E C,(E). By a standard Banach space argument we have Proposition 1.3.10. Let E c r be a closed set. A ( E ) = Co(E)ifand only if&(E)

= M(E).

A closed set E 5 r is an Helson set if A ( E ) = C,(E) (cf. the remark at the end of Exercise 1.1.3). Consequently if E is not Helson we always have non-measures in A’(E). To satisfy your curiosity for the moment we note that a closed interval in F1 is not an Helson set. [Korner, 11, and afterwards [Kaufman, 21, have provided examples of a different kind and on a much deeper level by proving that Helson sets exist for which A’(E) \ A;@) # 0. As we’ll see, there are uncountable Helson sets E such that (1.3.20)

A ’ ( E )= M ( E ) ;

and diverse conditions on an Helson set to establish (1.3.20) have been given in [Benedetto, 81. The above approach to find elements in A‘(r)\ M ( T ) is justified because of its relationship to spectral synthesis. In fact, A#) is the space of synthesizable pseudo-measures supported by E ; and a closed set E E f is a set of spectral synthesis, or, briefly, an S-set, if A’(E)= A;@). Exercises 1.3 1.3.1 Wiener’s characterization of continuous measures

a) Recall that p E D o ( f ) is continuous if p({y}) = 0 for each y theorem [Wiener, 1; Lozinski, 11: (E1.3.1)

V p E M(T),

1 2 Ip({y})12 = lim -

YET

and so p E M(T) is in MJT) (e.g. Section 1.3.4) if and only if

E

r. Prove W i e n e r ’ s

55

1.3 Pseudo-measures

Thus, p $ M,(T) if p

E

M(T)and

1/21 = 1. It is not difficult to generalize (E1.3.1) to any " 1 not only 2 N + l -N

r and for a general class of means

where 3" = 'pu E A(T), U goes through a compact neighborhood basis at 0 E r, supp 'pu c U, and 0 < q,(y) < ~ ~ (=01 )(cf. Proposition 1.2.1 and (1.2.3) where the roles of G and r are reversed). (Hint: Set v = p * fi so that 0 = 1/21'; hence, from (1.3.1) and (1.3.4),

~fU(x)l/2(x)l' dx.

v({O)) =

G

Then note that v({O})

=

2 lp({y})['). Y

Compare this proof with the proof in [Wiener, 7, pp. 146-1491 and the methods introduced in Paragraph 2.1.

+

b) Using a) prove that M0(T)c M,(T). (Hinr : Write the integral in (E1.3.2) as K

I

K-

where K is compact and 1/21 < E on K-). In light of a) M,(T) can be viewed as the largest class of complex measures p such that the arithmetic means of tend to 0 at infinity. This is the approach used in [Ben e d e t t 0, 101 to generalize AA(T), e.g. Exercise 2.1.2. 1.3.2 Unbounded measures in A'@) and Fourier series of L"(T) a) Prove that T=

2'

6,E

n Do(fi)) \ ~ ( f i ) .

"1 (Hint: T(x) -2 - sinnx and, as is well-known, e.g. [Edwards, 5, I, pp. 112-1131, this I n sereis converges pointwise and has uniformly bounded partial sums (i.e., the series is boundedly convergent)).Note that f is not continuous. b) Prove that there is no element @ E L"(T)for which (E1.3.3)

&(n)

20

and

2 &(n) = m.

(Hint: Let @ be defined on T = R/2nZ and note that T= 2&(n)6, E Do@). Use Proposition 1.3.3 and (E1.3.3) to prove that T $ A'@). Compute T).(Compare this statement and proof with [Edwards, 5, I, Section 9.21, which is Bochner's theorem for T and which also yields b)).

56

1 The spectral synthesis problem

On the other hand there are functions @ E (nLP(T))\ L"(T)which satisfy (E1.3.3), JJ

e.g. m

1

exp ixn[log n].

Incidentally, the first distributional derivative of dJ in (E1.3.4) can be considered as an element of A'(T). 1.3.3 A property ofAA(f)

Prove that if T EA;(r)and cp E A ( f ) then Tcp E AA(T). (Hint: Use (1.3.11) and the dominated convergence theorem). 1.3.4 Tcp = 0

a) Prove Proposition 1.3.6. (Hint: For $ E A ( U ) choose a sequence {cp,: n = 1, A(T) such that limn$ - $cpn(lA = 0, and write ( T , $) = (T, $cpn) ( T , $ - $cp.>).

+

n

...} E

b) Prove Proposition 1.3.7. 1.3.5 Cestiro summability on G

a) Take {cp,}

(E1.3.5)

c A C ( f ) , 3 #= cp, as in Theorem 1.2.1 and define Vf=

cp E A ( 0 ,

C, d x ) = /cp,(y) ~ ( y(G) ) dy. r

Prove that iff'is continuous at x then lim C, ~ ( x=)f ( x ) ; further, prove that the con(I

vergence is uniform on compact sets K C G wherefis continuous, and that iffis uniformly continuous (e.g. iff€ Co(G))on G then limC,cp =f,uniformly on G. (Hint: If U

V is a compact symmetric neighborhood of 0 E G then

(E1.3.6) The definition off, from Proposition 1.2.2 is then used to show that the right-hand side of (El.3.6) tends to 0). C a p is the C e s ~ r omean and cp, can be compared with A, in the Exercise 1.2.6. If

f E Co(G) fL'(G) l and 3= cp E L1(r), then (E1.3.5) and the dominated convergence theorem imply that

v x E G,

f ( x )= r

(cf. Theorem 1.1.1).

cpwmdY

57

1.3 Pseudo-measures

b) Given cp E L"(T). Prove that cp = /IE B(T) if and only if {C, cp} E M(G) converges to p in the weak * topology a(M(G), C,(G)). (Hint: First show that cp = 0 E B(T) if and only if {I[ C, cpII ,} is bounded). The intrinsic characterization of A(f) or B(f) is a very difficult problem even for A(T). The prognosis in [LCvy, 21 for a satisfactory characterization is negative and LCvy's view of such mathematical problems is quite interesting. For the sake of clarification, an "intrinsic characterization" should read something like : cp is the Fourier series of an element in L'(2) if and only if cp E Lip,(T), a > 1/2 (unfortunately, only the sufficient condition is true in this case, cf. Exercise 2.5.2). Because of the difficulty in verifying the conditions, Bochner's theorem, that cp E A(T) if and only if cp is a linear combination of positive definite functions, can not be considered a viable "intrinsic characterization". A survey of known results for the A(T) case is given in [Kahane, 7; 131; we mention in particular the work of S. Bernstein, Salem, SteEkin, Szasz, and Zygmund (cf. Section 2.4.10 and [Wik, 31). Related results for A(FI),B(R)andA(f),B(T)aregivenin [Berry, l ; C r a m e r , l ; D y s o n , l ; E b e r l e i n , 2 ; R y a n , 1 ; Schoenberg, 1; Simon, 11.

1.3.6 Distribution theory Since A'(T) is a special class of distributions we quickly recall some of the distribution spaces and refer to the classics, [Schwartz, 5 ; G e l f a n d , 11, for anything else. C"(FI") is the space of infinitely differentiable functions on R" and C;(FI") is the subset

a

of Cm(FIn)whose elements have compact support. We write Ds = ays,

a

-,

aYSt

(rd = rapidly decreasing). The following convergence criteria are used: {cp,} -C C,"(R")converges to 0 E C:(R") if there is a compact set K 2 R" such that supp cp, E K and Vs,

lip D'cp. = 0, uniformly on K

(cf. (1.3.8)); a sequence {cp, : m = 1, . . .} E Cm(Rn)converges to 0 E Cm(FIn)if V K , compact, and Vs,

l i p D " q , = 0, uniformly on K ;

and a sequence {cp, : m = 1, ...} c C,md(R")converges to 0 E C,XFI") if V r and Vs,

li,m y' Dscpm(y) = 0, uniformly.

C"(R") and C,md(R")are FrCchet spaces and C:(FI") is complete but not metric.

58

1 The spectral synthesis problem

Prove that the Fourier transform (of L1(Rn))defines a homeomorphic bijection, C,md(R")--f C,md(FI")(cf. Exercise 2.5.2). The duals of C;(FI"), C,md(FI"),and Cm(FIn) are, respectively, D(R") the space of distributions, D,(FI")the space of tempered distributions, and D,(R") the space of distributions with compact support. Also, A'@") E D,(FI")c D(F1")and D,(FI") c D,(R"). If C(W) is the space of complex-valued continuous functions on FIn taken with the topology of uniform convergence on compact sets (e.g. Paragraph 2.2), then its dual, D;(FI"), is the space of Radon measures with compact support. Clearly, D:(FI") c M(FI"). With the above notation, the distributional derivative DsT of T E D(R") is defined as V

~p E

(D'T,

C,"(FI"),

~ p )=(-l)"'

( T , DJq)

(cf. Exercise 1.1.4, Section 2.1.8 and Section 3.2). The Fourier transform F: D,(R") -+ D,(R")is defined as the transpose of the ordinary Fourier transform C,",(R")-+C,",(FI").Clearly, F is an extension to D,(R") of the map defined in (1.3.1). We can formulate the Fourier transform in terms of the Laplace transform in the following way. Let E be the space of entire functions of exponential type which are tempered on vertical lines and assume that it is taken with the compact-open topology (cf. Exercise 2.2.3b). The Laplace transform

L ( T ) ( s )= (Tx,esx), where T E D,(R) and s = (r iy E C, defines a bijection

+

L:D,(R)

-+

E

whose restriction (E1.3.7)

L,:D,(R)

-+

T

Cm(FI)

UT)(iY)

is continuous. The restriction L , of L , to C,"(R), L,:C,"(R)

+

Crmd(J0,

is continuous and extends to a bijective homeomorphism Le:C,md(R)

+

CXfl)

whose transpose is the bijective homeomorphism

F :a ( R )

+ Dt(R).

1-3-7 A'({yH = W{YN In Paragraph 1.2 we saw that a one-point set is an S-set (Theorem 1.2.2b and Exercise 1.2.3). In light of Exercise 1.3.6 and our formulation of spectral synthesis in terms of

1.4 The spectral synthesis problem

59

pseudo-measures, prove the result distributionally in FI" ; and, in fact, show that A'({y}) = M({y}).(Hint: If 0 E FI" is the support of T E D(R") then it is an elementary fact from distribution theory (e.g. [Horvhth, 1 , pp. 343-344; Schwartz, 5, p. 1001) that T = 2 a, Ds6, a finite sum). In 1949, [Riss, I ] proved the theorem for any LCAG using a theory of distributions for groups (cf. [Katznelson, 5, pp. 152-1531).

1.4 The spectral synthesis problem 1.4.1 The existence of the spectrum and Wiener's theorem. F E L"(G) (with or without subscripts) will always denote a weak * closed translation invariant subspace. For 9- E L"(G) we define

(1.4.1)

s p Y = { y ~ r : ( y *)EF}, ,

and let .TSp be generated by s p y . Clearly (1.4.2)

F pc F

and (1.4.3)

SP YSp = SP 9.

It is conceivable that s p y = 8 if F is non-zero; we shall use Wiener's theorem to prove that such can not be the case. If M E A ' ( f ) we write SUPP

M = U {SUPP T :T E M } .

For the following recall the notation of (1.3.17) and (1.3.18), and note that (1.4.4)

( Z ~ ~ ) " ( X=) (

Proposition 1.4.1. Let M Proposition 1.3.8).

y,~).

sA'(r)be a weak * closed submodule and set & = F (cf.

a) y E s p y o V cp E I ( M ) , ( q 6 ) cp = 0. b) SPY= Z(I(M)). c) s p y = suppM. Proof. a) If y (1.4.5)

E

s p 9 then z,6

V cp E I ( M ) ,

E

M by (1.4.3) and so, by the definition of I(M),

( T , ~ ) ( P= 0.

Conversely, given (1.4.5) we have ZY6 E

M(I(M));

60

1 The spectral synthesis problem

but

M(I(M))= M from Proposition 1.3.9~. Consequently, y E s p y by (1.4.4). b) If y E Z(Z(M)),(1.4.5) is immediate; and so y E s p y from part a) (the direction that uses Proposition 1.3.9). Conversely, if y E s p y and cp E I(M) we see that

v k+ E A m ,

(cpk+) (7) = 0,

by the trivial direction in part a). Thus cp(y) = 0 and hence y E Z(Z(M)). c) We have (1.4.6)

Z(I(M))G sp F c supp M

from b) and the fact that if y E F then r,6 E M.Let y E supp M so that y E supp S for some S E M.If y 4 Z(Z(M))let q ( y ) # 0 for some cp E I(M). Note that Sq = 0 since S E M and cp E Z(M). Thus cp = 0 on supp S by Theorem 1.3.1 c, a contradiction. Therefore suppM c Z(I(M)) which, when combined with (1.4.6), yields c). q.e.d. Consequently,

Proposition 1.4.2. a) I f 9 c L"(G) is non-zero then (1.4.7)

s p y # 0.

b) V E c r is closed there is a weak * closed submodule M E A'(T) such that,for F = k,

E = SP F = SUPP M = Z(Z(M)). Proof. a) is clear from Proposition 1.3.8, Proposition 1.4.1 and the hypothesis that 9# (0). b) Let Z = k ( E ) so that Z I = E and I = Z(M(Z))by Proposition 1.3.9d. Setting F = (M(Z))^we have s p y = Zk(E)(=E) because of Proposition 1.4.lb. q.e.d. Clearly, M = A&!?) in the proof of Proposition 1.4.2b. Note that if a weak * closed submodule MTE A'(r)is generated by T E A'(r)and if &T = Fa, where f'= @, then Proposition 1 . 4 . 1tells ~ us that SP Fa = SUPP

T.

1.4 The spectral synthesis problem

61

In this case we write sp@= sp .T@. Thus the spectrum of a given element @ E L“(G) is suppT, where T = @ (cf. Section 1.3.10). A crucial feature of this definition of spectrum is (1.4.7) which implies that (1.4.8)

V @ E L Y G ) \ {0},

sp Qi # 0.

It is important to h t e that Theorem 1 . 3 . 1 ~is the key step to prove (1.4.7). 1.4.2 Heuristics for the problems of spectral analysis and synthesis. The spectral analysis and synthesis problems have a diverse array of ancestors. One that significantly influenced the way we have defined s p a is the theory of convolution integral equations (cf. Section 1.5.4); these equations have the form

where we are givenf’EL’(G) and Y and wish to determine a solution @ satisfying certain conditions. Iff never vanishes, Y = 0, and = @ E L“(G), we have Z(I(A4,)) = sp @ = 8 by Proposition 1.4.1b; hence, by Proposition 1.4.2a, @ = 0 is the only L“(G) solution t o J ( @ )= 0 whenfnever vanishes (cf. [Rudin, 6, p. 2181). The chief developers of this approach to the notion of spectrum were [Carleman, 1, pp. 7 4 7 8 and pp. 111-1161 and [Beurling, 1, p. 346; 2; 51 (cf. the motivation in [Herz, 5, pp. 185-1861, noting that the above integral equation with L’(G) kernelf could just as well have a measure kernel); although [Wiener, 5; 6; 71 was certainly a key figure in the area. The fact that (1.4.8) is valid for the notion of spectrum defined by(1.4.1) tells us that the “spectral analysis” of the (non-zero) phenomenon @ yields a non-empty set of elementary “waves”, viz.

{(Y? .) Y E SP @I. The converse problem is the problem of spectral synthesis: when and how can @ be reconstructed in terms of ((7, :y E sp @}? a )

1.4.3 Spectra of representationsand the spectral analysis problem for Beurling weighted spaces. Except when specifically mentioned to the contrary our definition of spectrum shall be (1.4. I). Consequently, because of (1.4.8), we shall not generally worry about spectral analysis problems.

Remark. [Domar, I] (cf. [Vretblad, 11) solved the spectral analysis problem for Beurling weighted spaces [Beurling, 11 A ( p ) ={f=cp E A(T1:J IJ’lP < ml, G

62

1 The spectral synthesis problem

+

where p satisfies p ( x ) 2 p(0) = 1, p ( x y ) < p ( x ) p ( y ) , and the non-quasi-analyticity condition VXxEG,

2

< 00.

1

In the process he introduced the following general notion of spectrum which includes (1.4.1). Let X be a complex commutative Banach algebra with maximal ideal space

X”’, let Y be a complex topological vector space, and let R:X

-+

L(Y, Y )

be a continuous representation; then the R-spectrum of T E Y is S U P P , T = Z { ~ : XXE and ( R ( x ) ) ( T ) = O } &X”’.

(“L(Y, Y)” designates the space of continuous linear functions, Y -+ Y, cf. “ YY’’ in Section 2.4). We have SUPP, = supp if X = A(r),Y = A’(T), and (R(q))(T)= Tq.The study of spectra of representations has been pursued by Feldman, Lyubich, and Matsaev (cf. the penultimate remark in [Feldman, I ] with [DeVi t o , I]). 1.4.4 A technical remark about S-sets. The elements in Y are synthesizuble if (1.4.9)

9-=Y-sll

(cf. (1.4.2)). We now relate this notion with those introduced at the very end of Paragraph I .3. This comes down to being more precise about Proposition 1.4.2b in the following way. Proposition 1.4.3. Let E E r be closed. a) j-)

is the smallest closed ideal and k( E ) the largest for which E = Z ( T ) )= Z(k(E)).

b) A&(E)is the smallest weak * closedsubmodule and A’(E) the largest for which E = SUPP A i ( E ) = SUPP A’(E).

P r o o f a) is Proposition 1.2.5. b) Since suppTq E suppTit is trivial to check that A’(E) and Ak(E) are weak * closed submodules. It remains to prove that A @ ) is the smallest weak * closed submodule M for which E = suppM. We have Z(I(M))= E from Proposition 1.4.2, so that I ( M ) c k ( E ) from a). Hence, using Proposition 1.3.9,

M

= M(I(M))2

M(k(E))= AH(E). q.e.d.

1.4 The spectral synthesis problem

63

1.4.5 Standard characterizations of S-sets. Define M,(E)= {p E M(E):cardsuppp < a). Note that the closure of M,(E) in the o(A'(r),A(T)) topology is AB(E) (as we proved in the proof of Proposition 1.4.3b). Summing up our above remarks, we obtain Theorem 1.4.1. Thefollowing are equivalentfor a closed set E E r : a) E is an S-set (i.e., A'(E) = AL(E)). b) j(E)= k( E ). c) E = ZI for a unique closed ideal I G A(T). d) E = s p y for a unique Y E L"(G). e) M,(E) is weak

* dense in A'(E) (with the induced weak * topologyfrom A'(r)). = 0.

f) For each T E A'(E) and for all cp E k(E), ( T , cp)

g) For each T E A'(E), where T is uniformly continuous, and for all cp E k ( E ) ,( T , cp) = 0. h) For each closed ideal I E A(T)for which ZI = E, we have I=

nM,, YE

E

where M y was defned in Exercise 1.1.6. Proof. The only parts that require any comment are g) and h). g) Take S E A'(E), cp E k(E),and $ E A(T). Then @is uniformly continuous and so (S$,cp) = 0. Thus (Scp, $) = 0 for all $ E A(T) and so Scp = 0. Consequently, (S, cp) = 0 by (1.3.12). h) For any closed set E E r, observe that k ( E ) =

My.

YGE

q.e.d.

We shall continue our characterization of S-sets in Theorem 1.4.2 and Theorem 1.4.3. These results require a different type of proof. 1.4.6 An approximation in G to determine S-sets in r. Consider the following approximation condition for a closed set E in a compact group r: V T E A'(E), V {r,,:n = 1, . ..} C R+ increasing to infinity, and V {x,,: n = 1, 3 p~ M,(E) such that V n = 1 , ...,

(1.4.10)

Ifl(xn)

...} E G,

- T ( X n ) I < r,,.

Theorem 1.4.2. Let valid.

r be compact and let E E r be closed. E is an S-set o (1.4.10) is

Proof. (e) Given T EA'@),!= cp E k(E),and E > 0. We shall prove that I (T, cp) I < E .

64 1 The spectral synthesis problem

By Dini's theorem in infinite series take {r,,:n = 1, ...} that

2rn IS(xm) I <

~

0

R+ increasing to infinity such

3

where {x,,:n = 1, ...} E G is precisely the set where f is non-zero. Choose E~ > 0 for which el 2 r,,lf[x,,)l c E, and lets,, = Then taking p E M,(E) as in (1.4.10) (with r,, replaced by s, there) we compute

I (r,CP>

I

= ( T , CP>

-

(

~

I I 2 (Rxn) - p(xn))f(xn) I < 2 sn lf(xn) I < E .

~9 p > =

Let Ebe an S-set. Assume there is T EA'(E),{x, :n = 1,. ..} E G, and{r,, :n = 1,. ..} C R+ increasing to infinity such that for each p E M,(E),

(3)

for some n. We shall obtain a contradiction. Define V n = 1,. . .,

p $ = fi(x,,)/r,, and

T: = T(Xn)/rn,

and set p* = {p;:n = 1, ...} and T* = {T,*:n= 1, ...}. Clearly, p*, T* E C,(Z) by the hypothesis on {r,,:n = 1, . ..}. From (1.4.1 I), llP* - T*I), 2 1,

and so

T* $ b*: P E MAE)}, where the closure is taken in the sup norm. Thus, from the Hahn-Banach theorem, choose {a,,:n= 1, ., .} E L1(Z) which annihilates {p* :p E M,(E)} and such that ({a,,},T*># 0. Set q ( y ) = 2 a,,(y,x,,), so that cp E A(T). By the above observation with the Hahn-Banach theorem we have that

(1.4.12)

and

Since cp E A(T),

and

+ E k(E) from (1.4.12).

1.4 The spectral synthesis problem

65

From the hypothesis that E is an S-set we take {I),,:n= 1, ...} s j ( E ) for which WII) - I)nllA = O * Thus 0 = (T, I),,)--f (T, I)),and this contradicts (1.4.12). q.e.d. Theorem 1.4.2 can be proved more generally with a proper modification of (1.4.10); in fact, with such a modification it is trivial to prove the sufficient conditions that E be an S-set for any LCAG r. 1.4.7 A characterization of S-sets in terms of principal ideals. If cp E A(f) the closure of the principal ideal generated by cp is denoted by I, and is referred to as the closed principal idealgenerated by q. Theorem 1.4.3. Let r be a-compact and metric (so that G is metric and a-compact) and let E c r be closed,

jv)

a) is a closed principal ideal. b) E is an S-set ofor each closed principal ideal I, for which ZI, = E, we have I,

=

OMy. VEE

Proof. a) will be proved in the sufficient conditions of b). b) (-) This is clear from Theorem 1.4.1. (e) We'll prove t h a t j v ) = I, for some q E A(T). This is sufficient since then ZI, = E and so k ( E ) = I,. Because r is metric, E is a closed G, set. Thus E" is an open F, set, and hence we can write m

E*

= UK,,, 1

,

where each K, is compact. We take K,,G int K, + for each n; this can be done by the "Baire category" property of locally, compact spaces. Consequently we can choose q,,E j ( E ) such that qn= 1 on K, and 0 < cp,, < 1. Set

so that cp EKE), and hence I, E J-). Proposition 1.2.5b.

Clearly, Z I ,

= E;

and therefore I,

=)-J

by q.e.d.

1.4.8 Non-synthesis and the principal ideal problem. If G is not a-compact then k(E) need not be a closed principal ideal. Generally, E = ZI, if and only if E is a G, set and E" is an F, set.

66

1 The spectral synthesis problem

If G is not compact then f contains non-S-sets as we’ll prove in Paragraph 3.1. With this in mind we see that if E is an S-set in a metric a-compact r then k(E) is a closed principal ideal (from Theorem 1.4.3a). On the other hand (and this is not the converse situation), using M a 11i av i n ’s technique to determine non-S-sets, R u d i n proved the existence of cp E A ( f ) such that the elements of {I@ : n = 1, ...} are all distinct, noting that ZI@ = ZIQ(e.g. Section 3.1.5). A special case of R u d i n’s approach was first given for r = a”, n 2 3, in [Reiter, 51 using the original counter-example to synthesis discoveredby [Schwartz,2]. Reiter’sworkcoupledwith [Varopoulos,7,Theorem 31 yield the fact that k(E) E A ( W ) can be a closed principal ideal whereas E is a non-Sset (e.g. Exercise 1.4.4). The above observations do not preclude the possibility that every closed ideal in A(f) is a closed principal ideal. For perspective, note, from Theorem 1.1.3 (Wiener’s Tauberian theorem) and Proposition 1.2.6, that A ( f ) is a closed principal ideal if r is a-compact (recall Exercise 1.1.1). [At zm on, 21 has settled this principal ideal problem by proving : f r is not discrete then A(f) contains a closed ideal which is notfinitely generated. The above remarks indicate that such non-principal ideals have non-S-zero sets in acompact metric f. 1.4.9 A characterization of C-sets. We mentioned C-sets earlier (“C” is for [Calderbn,

I]). Now let’s be more explicit. A closed set E E f is a C-set if

-

kf cp E 4% cp E vW.

Obviously each C-set is an S-set. We mention C-sets at this point, when we are giving characterizations of S-sets, since it is not known if every S-set is a C-set. We shall see in Section 2.5.2 that a very extensive generalization of Wiener’s Tauberian theorem is closely related to C-sets. For perspective recall that one version of the Tauberian theorem states that closed proper ideals are contained in maximal ideals (Theorem 1.2.5) whereas every S-set is characterized by Theorem 1.4.1h. For now we characterize C-sets as follows : Theorem 1.4.4. Let E E f be closed. E is a C-set o whenever T E A‘(T) and cp satisfy cp E k(E U supp T)

and

E

A(f)

supp Tcp c E

we can conclude that Tcp = 0.

m),

Proof. (c) Let cp E k(E). We shall show that Z* E i.e., M(IQ)z M m ) ) . If T E M ( 3 ) ) then for each $ E K E ) ,$ = 0 on suppTcp (by Theorem 1.3.1), and so suppTcp E E. We now use Theorem 1.3.1 to obtain that cp$ = 0 on suppT for each $ E K E ) . Consequently, if y E (supp T) \ E and $ E K E )doesn’t vanish at y we have cp(y) = 0. Therefore cp = 0 on (suppT) \ E. Thus we can use our hypothesis since cp = 0 on E. Hence Tcp = 0, i.e., T E M(I,).

1.4 The spectral synthesis problem

(*) Take cp E k(E) and suppTcp c E. We’ll prove that Tcp = 0. Since E is a C-set we can choose {$,, :n = 1, .. .} c j ( E ) for which limllcp - cp$,IIA n Thus if @ E A(f)

(Tv, $> = (T, v$>= lip (Tv, $ $ n )

67

= 0.

= 0.

q.e.d. Note that in the necessary conditions we really prove that if cp E k ( E ) and suppTcp c E then Tcp = 0. In fact, the proof of Theorem 1.4.4 shows that the conditions cp E k(E U suppT) and suppTcp E E, are equivalent to the conditions, cp E k(E) and suppTcp E E; that is, if cp E k ( E ) and suppTcp c E, then cp E k(E U suppT). 1.4.10 Local synthesis. We now discuss “local synthesis”. T E A‘(T) (resp. cp E A(T)) is synthesizable if for all cp E k (suppT) (resp. for all T E A‘(Zcp)) (T,cP)= O * Naturally this definition is consistent with the definition of synthesizable elements in F. By duality, the characterization of synthesizable pseudo-measures is intimately related to a characterization of A(f) (cf. the remarks accompanying Exercise 1.3.5). Clearly, Proposition 1.4.4. Given a closed set E E r. a) If each T E A’(E) is synthesizable then E is an S-set. b) Ifeach cp E k ( E ) is synthesizable then E is an S-set. Obviously, in general, each possible converse in Proposition 1.4.4 is false since S-sets may contain non-S-subsets. Note that if Zcp, cp E A(T), (resp. suppT, T E A’(r)) is an S-set then cp (resp. T) is synthesizable. Proposition 1.4.5 a) Given T E A’(T), where f’= @. Then (1.4.13)

A;(suppT)” = (Fa),*,

b) Given T E A‘(r), where f’= @. T is synthesizable o Fa= (Fa)sp. Proof. a) Clearly (F& E A;(suppT) since sp Q, = suppT (e.g. Proposition 1.4.1). If S E Ak(suppT) let Iipp, = S in the weak * topology a(A‘(f), A(f)), where pa E Mf(suppT). When y E sp @ then ( ~ ~ 6 E) (Fa)sp, ” and so p, E (.Fa)sp since sp@= supp T. b) (-) Let S E M T ;from a), we must prove that S EAk(suppT). We have l i p S, = S in a(A‘(r),A(T)), where S, = T@,and $, E A(T). If cp E k(suppT) then (T, cp) = 0 by hypothesis, and so (S,, cp) = 0. Thus S E A;(suppT).

68

1 The spectral synthesis problem

(e) T E A;(supp T) by hypothesis and a) ;hence (T, cp) = 0 if cp E k(supp T).

q.e.d. Proposition 1.4.5b ties in with the remark at the end of Section 1.4.2: the search for synthesizable pseudo-measures T E A’(T) is precisely the problem of finding elements @ E L”(G) which can be reconstructed as “weak * convergent Fourier series” 2a,(y,x) with frequencies y E sp @. Y

1.4.11 The role of uniformly continuous functions in L“(G). We’ll see in Paragraph 3.1 that there are non-synthesizable pseudo-measures T, ?= @ (as we mentioned in Section 1.4.8). Consequently, for such T, Fais “not determined” by sp @. Noting that we prove sp @ is a family of uniformly continuous functions in Fa,

Proposition 1.4.6. Given F E L“(G) and let FUcbe the weak * closed translation invariant subspace generated by the uniformly continuous elements in F . Then F = F”C.

Proof. Let & = F (recall Proposition 1.3.8). Since M(A(T))E M it is sufficient to prove that X,the weak * closure of M(A(T)),is M. If T E M \ X we employ the dual Hahn-Banach theorem (e.g. (1.3.19)) and have that 3 cp E A(T)

such that ( T , cp) # 0

and ( T , cp$> = 0.

V II/ E

We obtain a contradiction by letting limllcp - cp$.IIA

= 0.

n

q.e.d.

Another important feature of uniformly continuous functions is

Proposition 1.4.7. Given T E A‘(T). is uniformly continuous 0 there is a directed system {T,} E A’(r), for which each suppT, is compact, such that IimIIT- TallA.= O .

Proof. (-=) It is only necessary to check that ?’ is uniformly continuous if suppT is compact. is uniformly If cp E A(r)equals 1 on a neighborhood of suppT then T = Tcp and continuous.

6

(a) Let @ = ?‘be

V

(1.4.14)

E

uniformly continuous. Thus

> 0 3 U E G, a neighborhood of 0 E G, such that

V X E U ,l

E

l @ - ~ ~ @ [ 2l ~ < - .

1.4 The spectral synthesis problem

Take

{A} as in Proposition

69

1.2.2 so that T, = Tcp. has compact support. Recall that

f, 2 0 andjj.(x)dx= 1. Choose ao%othat (1.4.15)

V

o!

> ao,

j S,-(x)dx < &/(411q4,). U"

Then for all c1 2

tl0

(1.4.14) and (1.4.15) conclude the proof. q.e.d. Using Proposition 1.4.7 and an extension of the Cohen factorization theorem, [Curt i s and F i g a - T a l a m a n c a , 1, pp. 174-1761 have observed that the uniformly continuous functions on G are precisely the elements of {(Tcp)":T E A'(r)and cp E A(T)}= L"(G) * L'(G). Because of this we conclude that if FEL"(G) is uniformly continuous then supp T is 0-compact. Clearly,

Proposition 1.4.8. cp E A ( T ) is synthesizable formly continuous, we have ( T , cp) = 0.

0

for all T E A'(Zcp), for which

is uni-

1.4.12 Remarks on non-weak * analysis and synthesis. We have defined "spectral synthesis for the weak * topology". We chose to define the spectrum in (1.4.1) since we were able to solve the spectral analysis problem by guaranteeing that s p y # 8 if 9- # 0; consequently, our main effort in this book will be to see when we can synthesize this spectrum into the given phenomenon. There is, however, a next step in the study of spectral analysis and synthesis. Let X be a translation invariant subspace of functions on G. The problem of spectral analysis can be viewed as finding the strongest topology T, in which sp 8 # 0 (the closure being taken in 'T& where sp 8 is defined as in (1.4. I ) with 9- replaced by 8.The problem of spectral synthesis is to find the strongest topology T, for which X can be synthesized from sp R, i.e., such that X is contained in the T,closed translation invariant subspace generated by sp R. For example, [Beurli n g, 21 originally envisaged doing analysis and synthesis in terms of the narrow topology, which we'll discuss in Paragraph 2.2; and, also, because of the influence from integral equations to analysis and synthesis, he was led to introduce the b topology (e.g. [Beurling, 2, p. 1331). The topology has been widely developed, its fundamental properties being given in [Buck, 1 ; 21. Physical considerations warrant this "topological approach"; obviously, a given phenomenon might not be synthesized in a real world model by a weak * procedure since its construction in terms of its frequency components (its spectrum) might demand a more restrictive piecing-together process (toPologY).

70

1 The spectral synthesis problem

Translation invariant phenomena fall naturally into the realm of Fourier analysis. On the other hand, analysis and synthesis form a general framework in which to study many physical and social phenomena. We present some examples in Paragraph 1.5.

Exercises 1.4 1.4. I Ideals in A(T)f o r which ZJ = Z fl J

Let Z , J c A(T) be closed ideals and assume that Z contains a bounded approximate identity. Prove that

(cf. Remark 1 in Exercise 1.2.5). (Hint: The proof of (1.1.5) in [Cohen, 11 yields the statement that if X is a commutative Banach algebra with bounded approximate identity and cp E X then cp = $0 where $, B E Xand 8 EZ,. Thus if cp E Zn J l e t X = Z and note that loc I n J G J ) . Observe that (1.1.5) is a special case of ZJ = Z n J. Using this exercise it can be shown that if J, K E A ( T ) are ideals such that ZJ = Z K , where J E K and either J or K is closed, then there is an ideal Z for which J 5 Is K (e.g. [Diet r i c h, 13). When both J and K are closed we can also choose I to be closed; we shall discuss this phenomenon in Section 3.1.7. 1.4.2 Estimates with the Fej& kernel on G

Let U s f have positive Haar measure and define the continuous map p u : f + R+ as

r r

pu(x,y) = po(x - y ) defines a pseudo-metric on x which is invariant in the sense that pu(x z , y z) = pu(x,y). Take the FejCr kernel from Proposition 1.2.2 and

+

+

{L}

Theorem 1.2.1 ;and recall that

where U E r is a compact neighborhood of 0 E r. Writing fa exercise, prove that

=f u

and pa= 'pu for this

1.4 The spectral synthesis problem

71

1 (Hint: Sincef u ( x ) = - 1fU(-x)l2, mU

(E1.4.1)

For a), (. .

1 -Ifu(x-z)-fu(Y-z)I mU

<

is bounded by 2 and

For b), use Holder's inequality on (E1.4.1) and estimate the L2-norms of (...)l and (. * .)2).

1.4.3 All subsets of discrete r are S-sets

Let G be compact. Prove that every closed set E c r is an S-set. (Hint: Let Zc A(f) be a closed ideal with ZI = E; if cp E k ( E ) we have 21 c Zp. Apply Exercise 1.2.2). 1.4.4 The Fourier transform of a radial function

In Section 1.4.8 we mentioned that there are non-S-sets E G R3 for which k ( E ) is a closed principal ideal. Setting S"-'= (7 E R": IyI = l} for n > 1, it turns out that we can take E = S2 c R3 and that there is cp E A(R3) for which j ( P ) = 1+ s Zv = k(S2) [Varopoulos, 7, Theorem 31. In Exercise 2.5.5 we shall indicate the proof of the earlier (and weaker) result that I V 2 s Iv for some cp E k(S"-'), n 2 3 [Reiter, 5 , pp. 469-4701. Both results depend on properties of radial functions. A function cp :R" + C is radial if there is a function I// : [0, 03) + C such that

v Y E fin,

cp(Y) = $(lYl).

In this exercise we calculate the Fourier transform of a radial function. Our computation is found in [Bochner and C h a n d r a s e k h a r a n , 1, pp. 69-74]. a) Let f E L1(R"),n 2 2, be a radial function and write F(r) =f ( x ) =f ( x l , ..., x,) where r = ( x ; +. ..+ Prove that f is radial. This is equivalent to proving that if a linear map R:Rn+ R" is an orthogonal transformation, i.e., R preserves the Euclidean inner product and the determinant of R is 1, then f(Y)

=my).

Thus .we writef(y) = cp(yI,. .., y,,)

= @(p) where p = (y:

m

(E1.4.2)

Ftr)rn-lK(pr)dr,

@ ( p )= 0

+...+ ~.2)'/~.Prove that

72

1 The spectral synthesis problem

where x

(E1.4.3)

K ( r ) = c[eircoae(sin8)n-2de 0

and c is a constant we'll be able shortly to forget. The calculation is straightforward using the definition of the parametric surface describing Sn-' and the definition of surface integral. It is also true that iff€ L1(Rn)andfis radial thenfis radial. b) Letting r E C in (E1.4.3), prove that K(r) has a non-trivial power series expansion, m 2 a k r k ;in fact, note that IK(r)l < elr', i.e., K is an entire function of exponential type

k -0

(cf. Exercise 2.2.3b). c) Prove that (E1.4.4)

5 + r(T)ukpk. k+n

71n12e--pa'4 =

k- 0

(Hint: Let F(r) = e-r2 so that @(p)= 11"12e-p2'4by direct calculation (cf. Exercise 1 .l.la). For the right-hand side of (E1.4.4), use (E1.4.2), the expansion K(pr) = 2a k p k r k and , 0

the definition of the gamma function (as a Laplace transform)

1

T(z)=s' e-Jttz-l dt, 0

Re s > 0,

Re z > 0

(cf. Exercise 1 . 1 . 1 a)). d) Prove that

K(pr) = 271 ' I 2

2 k-0

1

4'T(k)T(/9+ k - 1)'

(Hint: Expand e-p214as a power series and compare coefficients with (E1.4.4)). e) Using the fact that JB(r),the Besselfunction of order /3 > -1/2, can be written as

k= 0

(-l)'rZk 4kT(k)T(B+ k

+ 1)'

/3>-,,

1

1.4 The spectral synthesis problem 73

Remark 1. It would be interesting to verify if there are non-S sets E E FI for which k(E) is principal. Even with the tensor algebra techniques of [Varopoulos, 81 and the validity of the result in R3,the problem presents difficulties. 2. With regard to Atzmon's result, Exercise 1.2.5 and these remarks, it is natural to ask if every closed ideal in A(R) is the intersection of the closed principal ideals containing it. 1.4.5 The classification of closed ideals in algebras of distributions

In the spirit of the "finite dimensional" Exercise 1.2.5 we now consider infinite dimensional Banach algebras X which are synthesizable in the sense that every closed ideal Z E X is an intersection of maximal ideals (e.g. Theorem 1.4.1h); such Banach algebras are sometimes called N-algebras, e.g. [Sh i 1 ov, 11. General criteria for a Banach algebra to be an N-algebra are found in [Shilov, 1, p. 108; Rickart, 1, pp. 92-96]. a) Let X = C o ( f )and prove that every closed proper ideal in the sup norm algebra C , ( f )is an intersection of maximal ideals. (Obviously, f is only required to be a locally compact Hausdorff space). This result was first established by M. S t o n e in 1937 and there are several well-known proofs, e.g. [Hewitt and S t r o m b e r g , 1, pp. 365-366; Loomis, I , pp. 57-58]. b) For each p E Mc(R) let F:FI + C be the unique continuous function of bounded variation vanishing at -co with distributional derivative F' = p (cf. Exercise I .I .4b and Exercise 1.3.6). If F corresponds to p E Mc(FI)and G corresponds to v E M@) then (FG)' E MAR)

defines a multiplication under which Mc(R) is a Banach algebra normed by

v P E Mc(FI),

1lPIl = IlFllm + IIPlI1.

Prove that with this norm and multiplication Mc(FI)is an N-algebra. Spectral synthesis properties of Helson sets have been studied in this context in [Benedetto, 41, cf. Exercise 3.2.3e. c) Let C"(T), n < co, be the space of n-times continuously differentiable functions 40 on [0,2n]such that ~ ( ~ ' = ( 040(~)(2rc) ) for 0 < k < n. C"(T) is a Banach algebra under pointwise multiplication and with norm

Cm(T),considered in Exercise 1.1.4b, can never be a Banach algebra, e.g. [Gelfand, Raikov, and Shilov, 1, p. 671. Prove that C"(T) is not an N-algebra for n 2 1 ; in fact, for all y E T,j(y) 5 k(y) andj(y) is a closed primary ideal (obviously,jand k are defined in the context of C"(T)).On the other hand Shilov(1940) proved that each closed ideal in C"(T) is an intersection of closed primary ideals. The extension of this theorem to

74

1 The spectral synthesis problem

to regions in Fi'was made by [Whitney, 11, cf. [Glaeser, 11. Given this S h i l o v Whitney result, what can be said about the C" case? Remark 1. The ideal structure of the space of functions analytic in the open unit disc and continuous on the boundary has been characterized by Beurling (unpublished) and [Rudin, 21, e.g. [Gelfand, Raikov, and Shilov, 1, pp. 235-239; Hoffman, 1, pp. 82-89]. The study of closed ideals in locally convex spaces of analytic functions is presently a popular sport. 2. The problem to classify the closed ideals in algebras of distributions goes back to

[Schwartz, 1). This problem is closely related to questions about ideals in spaces of analytic functions (e.g. Exercise 1.3.6 and the Laplace transform operator L) and Delsarte's theory of mean-periodic functions. Let F(G) be a topological algebra of functions defined on G. V s F(G) is a variety if it is a closed translation invariant subspace of F(G); V , will designate the variety generated by @ E F(G). For example, if F(G) = L"(G) is taken with the weak * topology then V , = F,. @ E F(G) is meanperiodic if the variety V, is properly contained in F(G). Expositions of the theory of mean-periodic functions are found in [Kahane, 5; Meyer, 51; the latter reference follows the outline of the former and contains the more recent work of [Beurling and Malliavin, 13. For the duality (C(R), D,O(R)), [Schwartz, 11proved that every closed ideal in D,O(R) is an intersection of primary ideals. If V G C(R) is a variety then V contains exponential monomials @, i.e., @(x)= Yesx,where m E Zf and s E C; this is the spectral analysis of V. Sch wartz's theorem can be rephrased as the corresponding spectral synthesis result: if V E C(R) is a variety then the set of exponential monomials (in V) is total in V. An exponentiafpofynorniaI has the form @(x) = P(x)esx where P(x) is a polynomial. It is interesting to note that D,O(R) has non-maximal primary ideals. @

E

C(R") (resp. @ E Cm(R")) is mean-periodic if and only if

3 T E D,O(R") \ (0) (resp. T E D,(R") \ (0)) such that T* @=O.

In this context Sch wart z ' s investigation is closely tied in with the theory of partial differential equations since many partial differential operators are elements of DJR"). [Malgrange, 1, p. 3101 proved that any solution, @ E Cm(R"),of (E1.4.6)

T * @ = 0,

for a fixed T E D,(R"), belongs to the closure of the set of solutions (of (E1.4.6)) each of whose elements is a linear combination of exponential polynomials. Related work is found in [Ehrenpreis, 1; 21.

1.5 Classical motivation for the spectral synthesis problem

75

In these questions it is important to note that the maximal ideal space of Dc(R), say, is identified with C. In fact, if @: D,(R) -+ C is a continuous homomorphism then for each T E Dc(R), (6’ * T, @) = ( T , @ ) ( 6 ‘ ,Qj); thus Qj’ = @(O)@ and so @ ( x ) = esx for some s E C. Functions of the form @(x) = esxare mean-periodic. Also, every primary ideal I = I,. , s Dc(R)has the form Is,,, = { T E

Dc( R) : Vj

= 0,.. .,n,(L(T))(’)(s) = 0).

There are many open problems in this area associated generally with extensions of 1dimensional results to higher dimensions, cf. [Gilbert, I]. For example, if S, T E Dc(R) and L(S), L(T)have disjoint zero sets then the closed ideal generated by S and T is D,(R) (the converse is obvious for R”, n 2 1); the result is not known for R”, n > 1. On the other hand, Schwartz’s result remains true in the setting of an LCAG G if the added hypothesis is made that the annihilator ideal, V L = {T E @!(G):V @ E C(G), (T, @) = 0}, of the variety V is a closed principal ideal [Elliott, I]. The techniques related to annihilator ideals were developed by [Ehrenpreis, 1; 21. In thecase that G is discrete Schwartz’s result remains true without the hypothesis concerning annihilator ideals [Elliott, 21.

1.5 Classical motivation for the spectral synthesis problem 1.5.1 Analysis and synthesis. The process of analysis and synthesis has proved to be a significant intellectual pattern in the arts and sciences. Generally, the analysis of a phenomenon is the determination of its fundamental components, i.e., its spectrum; and the synthesis is the reconstruction of the phenomenon in terms of these components. The significance of this reconstruction becomes apparent if one views it as a study of the manner in which the given phenomenon is a combination of elements from its spectrum. The basic problem of specifying what this manner should be or how one should define the notion of spectrum is, of course, intimately related to the process of analysis and synthesis itself. 1.5.2 Examples 1.5.2.1 The theory of tides The ancient mariner was aware that tidal phenomena were related to astronomical factors. The study of tides received its first scientific basis with Newton’s law of gravitation for the sun,moon, and earth. Laplace was able to separate various cyclic influences of the sun and moon (on tides) by defining a model of the sun, moon, and earth having a number of tide-affecting satellites ; the Newtonian solution to this model associated an elementary tidal constituent with each satellite in such a way that the tide was viewed as a combination of these constituents. Lord Kelvin (1824-1907) systematized this

76

1 The spectral synthesis problem

method and initiated the design of mechanical analyzers and synthesizers which determined the constituents and reconstructed the tides, respectively. [Godin, 1 ; M a c mill an, 11 and exhibits of mechanical harmonic analyzers and synthesizers at various science museums present interesting introductions to tidal analysis. 1.5.2.2 Cubism The development of cubism dating from the influence of CBzanne and African art on B r a q u e and Picasso to the latters’ introduction of collages and papiers collds (in 1912) forms the background for a pattern of synthesis in the “synthetic” cubism of Gris. In his early cubism of 191 1-1912, G r i s dissected his subject, examined each part from a different angle, and then reconstructed the total image again from the various parts [Golding, 1, p. 1011. Later, after his own work with papiers coIl&s,his explicit “synthetic” cubism evolved to the point where he began a painting by first planning its structure and then superimposing the subject in this framework [Read, 1 , p. 861. The actual composition was made up of “flat colored forms” [G 01 d i n g, 1, p. 1361. In our language the above structure or framework is the “topology” in which we wish to synthesize, and the colored forms represent the elements of the spectrum ; the actual synthesis of the given subject, then, is the painting. 1.5.2.3 The theory offilters (in communication theory) The theory of filters in communication networks [Cuccia, I] and general information theory [Goldman, 11 provides our third example. A filter is a circuit which is chosen to produce specific frequency responses F(o)given a certain class of inputs (to the circuit). Thus, a filter might be designed to attenuate certain elementary waves (corresponding to given frequencies o)which go into the composition of an input signal. The design of such filters can be viewed as a synthesis problem in which Fis to be synthesized by the circuit in terms of some interval of frequencies. [Wiener, 9, pp. 88-96] referred to such problems in the context of synthesis and posed the corresponding problem of analysis, e.g. [Wiener, 9, pp. 97-1 001. Wiener’s first and fundamental work, [Wiener, 111, on prediction and filtering problems dates from 1940-1943; and in subsequent years he developed his basic notion of filter t o the context of cybernetic problems. In [Wiener, 10, pp. 69-1251, M a s a n i has written an important essay, remarkable both in its clarity and scope, which deals with Wiener’s filter theory. 1.5.2.4 Music

An origin of spectral analysis and synthesis is the study of the vibrating string problem: the sounds produced by a bowed violin string can be considered as combinations of pure sounds or harmonics. (Actually there are non-trivial non-linear effects associated with bowing as opposed to plucking; we shall not discuss these.) The spectral analysis is the search for these harmonics; the synthesis is the reconstruction of the original sound in terms of its harmonics. With regard to the discussion of filters in Section 1.5.2.3 we note that an important spectral analysis and synthesis result in non-linear filters was given by [Wiener, 9, pp. 88-1001 with white noise as the input. In music theory,

1.5 Classical motivation for the spectral synthesis problem

77

white noise is sometimes considered as a mathematical model for snare drums or cymbals; actually, given the purely stochastic nature of white noise, such a model is really quite inaccurate. The propagation of sound in one space dimension was characterized in terms of the wave equation by E u l e r in 1759; of course, the mathematical model for the motion of the vibrating string with certain restrictions on its behavior is given by this equation. The subsequent analysis led not only to the representation problem in Fourier series (cf. Section 1.5.3) but to a precise definition of function. There was a good deal of intellectual bloodshed before the final resolution of these two intimately related issues. It was D a n i e 1 Bern o u 11i ( I 700-1 782) who suggested a Fourier series solution to the wave equation. He also did major experimental spectral analysis and synthesis; for example, he showed that in closed organ pipes the frequencies of the overtones are odd multiples of the frequency of the fundamental. H e l m h o l t z devised effective mechanical devices to synthesize complex tones, analogous to the mathematical synthesis associated with solutions to the wave equation in terms of Fourier series. These are described in [Josep hs, I , pp. 70-721. There are, of course, startling recent developments to H e 1m h o 1t z ’s research in the form of electronic music. Now we can listen to the Toccata and Fugue in D Minor not on the organ as Bach (1685-1750) originally planned but on the Putney V.C.S.3 synthesizer [Hankinson, 13. 1.5.2.5 The interferometer The origin of W i e n e r ’s work in generalized harmonic analysis was in the study of beams of white light. The well-developed area of Fourier spectroscopy, of which Wiener’s work can be considered a part, has its origins in Michelson’s invention of the interferometer [Loewenstein, 1 ; Michelson, 11. Basically, an interferogram [Loewenstein, 1, p. 8461 is the record of a detected signal and corresponds (for us) to some @ E L”(G). The interferometer synthesizes the spectrum experimentally to produce the interferogram; and the problem of Fourier spectroscopy is the problem of spectral analysis, viz. to determine the spectrum E of @. In optics, the spectroscopy problem is sometimes posed so that the spectral distribution p E M ( f ) is sought (cf. Chapter 2.1). In our terminology this is equivalent to finding E since supp ,u = E.

1.5.2.6 The Heisenberg uncertainty principle The wave theory in quantum mechanics arose since electron beams diffracted through crystals produced an effect analogous to Newton’s spectral theory of white light diffracted through a prism (cf. Section 1.3.10). This experiment with electron beams actually followed d e Broglie’s suggestion (1924) that matter has both a wave and particle representation related by

(1.5.1)

p =fix,

where h is Planck’s constant, p is the momentum of the given particle, and x is the frequency of the corresponding wave. The wave function Y(y,t ) , normalized so that

78

1 The spectral synthesis problem

1IY(y,t)12dy

= 1, was introduced by S c h r o d i n g e r (1926) to describe the wave inter-

pretation of matter; and an important aspect of its physical significance is that for a fixed time t, (1.5.2)

jj’J’(y,t)JZdy 1

is the probability that the given particle is in A c FI [Schiff, 1, Chapters 1 and 21. Also, as in the spectral synthesis associated with classical problems characterized by the wave equation, we have formally that (1.5.3)

Y(y,t)=If(x,t)e‘Yxdx. R

[ i,i]then

Because of (1.5.2), if at time t the particle is in -

*r

IY(y,t’)12dy=1.

-A12

Consequently, in this case, the support of the wave function is “concentrated” in

[-;, ;] at time t. Suppose that the support off is “concentrated” in [-?

a a

in the

sense that

“f If(x,t)I2dx - 1 -a12

(i.e., Plancherel’s theorem) and Y(y,t )

- I’

f ( x , t ) elVxdx.

-012

Then, by Holder’s inequality, 112

(1.5.4)

1-A12

1

a12

f ( x , t ) eivxdx12dy< A( T l f ( r , t ) l d x ) 2 -a12

-a12

G Aa

‘f If(x,t)12dx

N

la;

-all

and so a must satisfy the inequality (1.5.5)

a

2 l/A.

[4

From (1.5. l), (1.5.4), and the fact that - -, - is an interval of frequencies, we have the

1.5 Classical motivation for the spectral synthesis problem

79

Heisenberg uncertainty principle, A Y A p 25,

where we’ve written A y = 1 and Ax = a. It is important to compare this phenomenon and (1 5 5 ) with calculations such as Exercise 1.1.1a and Proposition 1.2.2. Thus, “if a function f E L2(R)is close to 1 on a long interval of R, then f will not only have its support “concentrated” in a small set but also will attain large values there”. Because of the previous remarks, this “fact” concerning the relation betweenf and f could be alluded to as the Heisenberg uncertainty principle. 1.5.3 Synthesis and the representation problem in harmonic analysis. We have defined a precise spectral synthesis problem in Paragraph 1.4, but let us now observe, for a moment, that the two major problems in harmonic analysis can be viewed in terms of spectral analysis and synthesis. These two problems are, respectively, to compute Fourier coefficients, for distributions T o n 1,say, and then to construct Tin some way in terms of these coefficients. This latter problem is usually referred t o as the representation problem. A typical situation is that we are given a function cp (on 1)and are able to solve the spectral analysis problem by computing the Fourier coefficients @(n); the problem of representation or synthesis is to investigate if there are any ways (i.e., convergence criteria) of representing cp in terms of 2 @(n) e? For example, [Schwa r t z, 5, pp. 225-2271 gave a spectacularly simple solution to the representation problem for any distribution on 1”in terms of the topology on D(T“);whereas [Carleson, 21 (cf. [Fefferman, 11) gave a spectacularly difficult solution to the representation problem for any cp E C(T) in pointwise convergence a.e.

1.5.4 Integral equations. Many of the phenomena of classical mathematical physics are approximately and realistically described by linear operators Kin the form of ordinary and partial differential equations and integral equations, e.g. [Schwartz, 4; Tricomi, 11. Note that on R, df -=d’*f, dx where “dldx’’ is the ordinary operation of differentiation for a continuously differentiable function$ Generally, large classes of the above-mentioned equations can be written in the form (1.5.6)

K*f =g;

where K,g E D(R”)are given andfis to be found. Naturally, a major problem in the solution of (1.5.6) is to determine the proper range for K .

A special case of (1.5.6) and one for which a very fruitful method exists, e.g. [F. Riesz and Sz.-Nagy, 1; D u n f o r d and Schwartz, 11, is the eigenvalue problem where K

80

1 The spectral synthesis problem

has the form Ad - H for a compact operator H . This problem can be viewed in terms of spectral analysis and synthesis. The spectral analysis problem has a positive solution since there is a non-empty set of eigenvalues; and for the synthesis problem, solutions are given by eigenfunction expansions where eigenvalues play the role of frequencies in the eigenfunction. Because of our discussion of Riem a n n ' s l-function in Paragraph 2.3, we note that C a r l e m a n (1934) expressed the Green's function of the heat equation, when dealing with a two-dimensional membrane, in terms of a generalized lfunction ;and using standard c-function arguments he was able to calculate asymptotic formulas for the eigenfunctions [Kac,l ; Weyl, 11. The study of (1.5.6) for more general kernels K than those of the form K = ,Id - H presents formidable problems. One approach which takes advantage of the convolution in (1.5.6) is to use the Fourier transform and therefore deal with Rf=g. Thus if K,g E L'(2) and we wish to find a solutionfe L'(2) we need only assume that doesn't vanish and then apply Proposition 1.1.5 to assure the existence of such anf. Obviously Zg plays a role in any refinement of this technique. Such refinements and the corresponding study of zero sets have been one influence for the development of the spectral synthesis problem as stated in Paragraph 1.4. If we take g = 0 in (1 3 . 6 ) we see quite precisely the connection with spectral synthesis; in fact, in order to determine if a given K E L"(R) is synthesizable we ask if K * f- = 0 for eachf E k(spK) (cf. Section 1.4.2).

2 Tauberian theorems 2.1 The Wiener spectrum and Wiener’s Tauberian theorem 2.1.1 Introductory remark. In this chapter we shall see how the notion of “spectrum” has evolved in the context of Tauberian theorems (Paragraphs 2.1 through 2.3), and then study the intimate relation between Tauberian theorems and spectral synthesis (Paragraphs 2.4 and 2.5). As such, our setting will be G = R and r = FI for part of the chapter, although much of what we say can be developed in a more abstract framework; for example, the notion of the “Wiener spectrum” is studied generally in [Benedetto, 11 ; Herz, 5, p. 1951.

2.1.2 Heuristics for generalized harmonic analysis. In the late 1920’s W i e n e r took up the problem of characterizing rigorously the spectral analysis of a white light signal @ E Lm(R) (cf. Section 1.5.2.5 and Section 1.3.10). Two mathematical problems arose immediately; and these led him to his theory of generalized harmonic analysis [Wiener, 51. First, Newton’s experiment with sunlight demanded a notion of a continuous spectrum ; as such, Wiener was forced to abandon the theory of Fourier series as a tool since series can only yield a discrete spectrum of frequencies. Second, sunlight is conveniently considered as a signal which is supposed to last for an indefinite time and which does not decrease to zero with time; as such, Wiener could not use the classical L’ or L2 Fourier transform theory on R as a tool since i+a

when @ € L 2 ( R ) (even though the spectrum of @ could be defined reasonably as “supp P ,a non-discrete set). Heuristically it is desirable to associate an energy to each frequency of a given signal @ E Lm(R)so that the total energy is distributed among all of the frequencies making up the signal. Thus, if n

@(x)=

2upel-, 1

then the spectral distribution of energy for the signal @ is

82

2 Tauberian theorems

where (as would be expected physically) each coefficient la, Note that in this case,

Iz

represents an energy.

-R

Consequently, for @ E L"(R), Wiener defined the autocorrelation or covariance function

1G) P

1

W

@(x) = lim -

R-2R

@(x

+ y ) dy,

-R

when the limit exists pointwise everywhere. From the point ofview of optics, Wiener's generalized harmonic analysis of an arbitrary signal @ E L"(R) was designed to define reasonably the spectral distribution of energy for 9 and to relate analytically the brightness (or power) of the signal @, viz.

(assuming the limit exists), to this spectral distribution; this program was accomplished in terms of the autocorrelation function The statistical terminology in defining 8 i s justified in [Wiener, 11, pp. 4-51, and some physical examples of the Wiener theory are given in [Arsac, 1, Chapter 111.

8.

2.1.3 The Wiener spectrum and generalized harmonic analysis. We now present a formal, brief, and non-historical introduction to generalized harmonic analysis. Given T E A'@), where ? = @. Define

W

The set, {QR:R > 0}, is obviously 11 ll,-bounded. Let Y E Lm(R) be a weak * limit W point of {@R: R > O}. We shall show that (2.1.1)

Y

=p,

where p E M(FI) and p 2 0. We first make some remarks on positive definite distributions. It is a routine calculation, using Theorem 1.3.2b, to prove that a continuousfunction @: R --f C is positive definite if and only if V f E C?(R),

]@(x) f *.f(x)dx 2 0 R

2.1 The Wiener spectrum and Wiener's Tauberian theorem

83

[Schwartz, 5, pp. 274-2751. As such, a distribution P E D ( R ) is defined to be positive definite if

v f E C;(R),

V , f*f>2 0.

[Schwartz, 5, pp. 276-2771 has proved Bochner's theorem for positive definite distributions which, when combined with Proposition 1.3.3, yields the following statement of Bochner's theorem (cf. [ C r a m t r , I]): if @ E L m ( R ) is a positive definite distribution then there is v E M(R), for which v 2 0, such that P = @ a.e. (in particular, there is a continuous positive definite function which equals @ a.e.). In order to verify (2.1.1), we take f E C?(R), consider it as an element of M ( R ) , and W approximate it in the weak * topology by elements of M,(R). Then, since aRis a continuous positive definite function, we have that

v R > 0.

j&(X)

f *f"(x)dX 2 0.

R

Hence, Y is a positive definite distribution and (2.1.1) is true. By standard measure theoretic results, e.g. [Bourbaki, 1, Chapitre 11, Section 1, no. 3, Prop. 1 and Chapitre 11, Section 2, no. 2, ThCorkme 11, we can conclude the following from (2.1.1) for a fixed pseudo-measure T E A'(R), where f'= @: there is a unique element pa E M(R), pe 2 0, such that for each a (L"(R), L1(R)) limit point Y = of {JR:R > 0}, we have

P

2 0, and if p - v 2 0 for each such p then - P'e

Po-V20

(where v E M(R), v 2 0). pa is the spectral distribution of energy for @. The Wiener spectrum of f'= @ E L"(R), sp, @, and the Wiener support of T E A'(R), supp, T, are defined as (2.1.2)

sp, @ = supp, T = supp p'e.

For technical convenience in our presentation we say that T E A;(R) G A'(R), where f'= @, if { &R :R > 0} converges (as opposed to convergence of a sub-directed system) in the a(Lm(R),L ' ( R ) ) topology. 2.1.4 Examples 2.1.1 a) The space, A:@), of continuous pseudo-measures consists of pseudo-measures

T E A'(R) for which a

(2.1-3)

R+m2R lim

!

!

J I?(x)J2dx = 0. -R

84

2 Tauberian theorems

Clearly, Ai(FI) E A;(FI). Because of Holder's inequality, (2.1.3) is equivalent to the condition that

-R

Thus if T E A:(R) then

and so supp, T = 0. b) Define

T(x) = ~ ( x=) eix' (which can be considered as a solution to a S c h r o d i n g e r equation). Clearly, T4 A m . On the other hand, for x = 0, W

-R

-sin2Rx 2Rx

for x # 0

and so supp, T = 0. The same result is valid for T(x) = el"', r > c) Define the discontinuous measure

If ?= @ then

y#a

Thus, supp, T = supp T = R if

For example, this series converges if X =

c

I

- : n = 1, ...

and a,, = 1/2". Actually, a much

stronger result is true, e.g. Remark 3 after Theorem 2.1.1. Remark. Assume that the sequence { p ( N ) > 0 :N

lim p ( N + I ) / p ( N ) = 1 ;

N+m

= 1,

. ..} satisfies the condition that

2.1 The Wiener spectrum and Wiener’s Tauberian theorem

85

and define N

W

for T = @ E L“(R). Starting with the sequence { @ p ( N ) := N l,.. .} (instead of W { @ , : N = 1, ...}) and using an ingenious trick due to M. Riesz, [Bochner, 1, pp. 328 ff.] was able to prove the existence of p; E M(R), p; 2 0, in the same way that pa was determined. 2.1.5 Outline of the following sections. Given M c A ’ @ ) . Analogous to (1.3.18) we define (2.1.4) R

(“d,” was defined in Exercise 1.2.6a). It is easy to check that Iw(M) c A(FI) is a closed ideal and that ZI,(M) G ZI(M). Setting

we shall see in Theorem 2.1.2 that (2.1.5)

V M c A;(R),

SUPP,

M

= ZI,(M),

and so s u p p , M ~ suppM(cf. [Benedetto, 11, Section 31). It is in the verification of results centering around (2.1.5) that W i e n e r was led to introduce his Tauberian techniques (although presumably he did not think of (2.1.5)); as such we shall give the background for the use of Tauberian results in Theorem 2.1.3 as well as outlining the proof of (2.1.5). Our outline is taken from [Meyer, 11; modifications and extensions are found in [Benedetto, Ill. 2.1.6 A characterization of the Wiener spectrum in terms of summability kernels. If T E A’(R) we set e-IYxf(x)dR(x)dx.

tR(y) = R

It is easy to check that limt,

= Tin

the weak * topology a(A‘(R), A(R)). Next, by some

R-rW

routine calculations, we see that if T E A’(R) and cp E A,(R) then

86

2 Tauberian theorems

and

Consequently, cp E A ( A ) is an element of Iw(MT)if and only if 1 A

Armed with (2.1.8) we have the following characterization (Exercise 2.1.4) of the Wiener support. Theorem 2.1.1. Given M G A'@). ZI,(M) is the smallest closed set such that for any compact neighborhood F which satisfies (2.1.9)

F n ZI,(M)

= 0,

we can conclude that

or equivalently, that

Remark 1. In order to prove Theorem 2.1 .l, proceed as follows: a) Observe (from Proposition 1.2.5) that (2.1.12)

.W,(M)) E M M ) ;

b) Use Plancherel's theorem to prove that

where cp E A,@) is 1 on F; c) Deduce (2.1 .lo) from (2.1.8), (2.1.12), and (2.1.13). d) Prove that cp E Iw(MT)if and only if (2.1.14)

lim I(--!-tRcpll Rfm

m

by using (2.1.8) again.

= 0, 2

2.1 The Wiener spectrum and Wiener's Tauberian theorem

87

e) Suppose that ZIw(M) is not the smallest such set and obtain a contradiction by employing (2.1.14). 2. Results such as Theorem 2.1.1 were developed by [B e u r 1i n g , 5 ] forZZ(M,) (instead of ZIw(MT))and used by him in [Be u r 1i n g, 6, p. 2291 to prove that all pseudo-measures are synthesizable for a certain weaker than weak * topology. [Pollard, 21 then used similar results to characterize a class of synthesizable elements in A ( f l ) , e.g. Section 3.2.5. The technique has also been refined in [Herz, 2; K i n u k a w a , 3; Pollard, 11. We refer to Exercise 2. I .4 for further remarks on this type of characterization of the support. 3. Using a result similar to Theorem 2.1.1 as well as the basic properties of almost periodic functions (e.g. Theorem 2.2.3) we can show [Benedetto, 1I] that ZI(M,) = ZIw(M,) if is almost periodic. In light of Theorem 2.2.3, this strengthens Example 2.1. lc. 2.1.7 A characterization of the Wiener spectrum in terms of ideals

Theorem 2.1.2. Given M c A;(R). (2.1.5) is valid. W

Proof. i) Given T E M, where = @, we know that lim @, = /I,in the weak * topology R+m

a(L"(R), L1(R)), and we must prove that supp p, = ZIw(MT).

ii) Let

It is easy to check that lirnd, = 0,uniformly on compact sets, and so R+m

in the weak * topology, a(Lm(R),L1(R)). Thus, (2.1.15)

1

lim-ltt,lZ R-m2R

= p,

in o ( A ' ( f l ) , A ( f l ) ) .

iii) If F G (suppp,)- is a compact neighborhood then we take cp = 1 on F and cp E k(suppp,); from (2.1.15) we compute that

and so F G (ZZw(MT)) by Theorem 2.1.1. Hence, ZIw(MT)c suppp,. iv) Conversely, if F s (ZIw(MT))" then (2.1.10) holds and we wish to show that p, restricted to F is 0.

88

2 Tauberian theorems

This follows by (2.1.10), (2.1.15), and by restricting the measures

- IfR(':

(2!R

R > O] to

F. q.e.d. Remark. An interesting fact about Iw(M), where M c A'@), is that if E = ZZ(M) = ZIw(M) then ME)= M

W;

and so the elements of A4 are synthesizable if and only if I ( M ) = Iw(M) [Benedetto, 1I]. 2.1.8 The primitive of a pseudo-measure on R . Given a distribution S E D(R). As in Exercise 1.3.6, we define the distributional derivative,

S':C:(R)

+

c,

of S by

v cp E C,m(R),

( S ,rp>

=

4 s q'>.

Auspiciously enough, S' E D(R). Note that if F E Ltoc(R) (i.e., F E L'(K) for each compact set K c R) then F E D(R) and

v cp E Cm,

( F , rp>

=

J F(Y)cpw dy.

Proposition 2.1.1. Given T E A'(R), where'f = @. a) Define (2-1e16)

@(XI

F I ( Y ) = ~- ~ R \ L - ~ , ~ -e-lVXdx, J(x) 1x

R

(2.1.17)

F2(y) = -

1

[e-lY"- Ildx,

-1

and (2.1.18)

F(Y) = Fdr) +

my).

Then F E L:,,(R) and T is the distributional derivative of F. b) Define (2.1.19)

1 F*(Y)=+F(Y+l)-F(Y

Then FAE L2(R) and

(2.1.20)

1 sinlx Fy(x)= --@(x). f i x

-A)).

2.1 The Wiener spectrum and Wiener’s Tauberian theorem

89

Proof. a) Fl E L2(fl)by the Plancherel theorem and F, E L“(FI) by the differentiability of the exponential function at the origin; consequently, F E L?Jfl) 5 Lioc(fl). I f f = cp E C:(fl) we compute, using Theorem 1.1.2, that

1

(C9cp> = - F,(Y)cp’(~) dy = n

1 @(x)f(x)dx. IXl>l

Because of Fubini’s theorem and the fact that \cp’(y)dy = 0 (cf. [Benedetto, 12, ri Theorem 4.161) we compute

b) (2.1.20) is immediate by formally comparing the terms of the difference,

using the identity sin Ax = (elAx- e-’”)/2i, noting the cancellation of the “constant” terms when dealing with F,, and applying the definition of the L’-Fourier transform. q.e.d. We shall have more to say about the primitives of pseudo-measures in Paragraph 3.2. 2.1.9 Generalized harmonic analysis and Wiener’s motivation for his Tauberian theorem. (2.1.22) establishes the relation between the power of a signal and its spectral distribution of energy; in order to see this, let v = 6 in (2.1.22) (although technically we shouldn’t) and recall (2.1.15) and the discussion after (2. I . 1). Wiener had to introduce Tauberian arguments to prove (2.1.25), which in turn yields (2.1.22) in a relatively easy way. He was led to these techniques by A. E. I n g h a m [Wiener, 6, p. 6; 8, p. 1151 in 1926. Instead of following the results of H a r d y and L i t t l e w o o d to which I n g h a m directed him, W i e n e r devised a general Tauberian theorem which combined some of his own trigonometric methods with a new general Tauberian method due to R. Schmidt. S c h m i d t further suggested the problem to prove H a r d y and Littlewood’s Tauberian theorem for Lambert series (1921) independently of the prime number theorem; and Wiener’s solution was considered one of the main successes of his Tauberian theory. We shall discuss this in detail in Paragraph 2.3. In any case, the raison d’&trefor Wiener’s Tauberian theory is (2.1.22) [Wiener, 5, pp. 141-1 53,esp. p. 1521. In Theorem 2.1.3 we’ll provide the details only for the implication which requires the Tauberian theorem; this is the more difficult direction.

90 2 Tauberian theorems If T EA’@), f’= @, and FAis defined as in (2.1.16)-(2.1.19) then

is bounded since (2.1.21)

‘1 I

I):(

IT@ sin x dx

14(y)12dy=1

n

2

R

because of Plancherel’s theorem, (2.1.19), and a change of variable. Consequently, by the Alaoglu theorem,

1:-

1

IFAI2:O < I < 1 has weak * limit points in M(FI).

Theorem 2.1.3. Given T EA’@), where p = @. T E A ; @ ) ifand only if

In this case we have (2.1.22)

V

Y E Mo(R),

‘s

J1312dy,= limA+O

n

3,

I&(y)i(y)I2dy

I

=

lim ’ J R-fm2R

I@ * v(x)I2dx.

-R

1

Proof. i) W e shallassume rhar lim - IFAI2 = p@in a(A’(FI), A(R)) and uergy (2.1.22). A+OI

By Plancherel’s theorem,

and

2.1 The Wiener spectrum and Wiener's Tauberian theorem

91

Also, it is straightforward to check (e.g. [Meyer, 1, p, 1941) that the difference between the right-hand side of (2.1.23) (resp. (2.1.24)) and

tends to zero. Thus, setting Y(x) = we show that (2.1.25)

I@ * v(x) I2 + I@

* v(-x) 12,

sin21x Y(x) -

lim1-0 A 1

R-+m 0

2R

we shall have verified (2.1.22) once

0

ii) By the above remarks, our hypothesis is that

sin21x

dx = r ,

0

where Y E Lm(R). iii) Let R = 1/A = eY and let x = e' so that (2.1.25) becomes

Note that in order to find the upper bound of integration in the second integral it is necessary to consider first the substitution x = e'. iv) Setting 2

f ( x ) = - exsin2(e-x), X

we now assert that (2.1.28)

Vy

E

f(y) # 0.

F1,

The verification of (2.1.28) is left as Exercise 2.1.5. Also, by an easy residue argument, m

(2.1.29)

2

sin2x f ( 0 ) = - 1-d. 7t x2 0

= 1.

92

2 Tauberian theorems

1

1-

0

0

m

Note that

(0

sin2x dx =

sin x X

dx.

v) Consequently, if we write@(?)= Y(et) - r then (2.1.26) becomes (2.1.30)

*f@) = 0.

Q;$

Let Z consist of all elements g E L'(R) such that lim 0 * g(y) = 0. y-rm

Then Zis a closed translation invariant subspace ofL'(R). Thus, from Proposition 1.2.6 and Theorem 1.2.3 (Wiener's Tauberian theorem), I = L'(R). Hence, if we let g(x) = e-"x (x), the right-hand side of (2.1.27) becomes lim 0 * g(y) + r y+m

m

e-=dx = r, 0

and we are done. q.e.d. 2.1.10 Riemann's summability methods, Tauberian theorems, and generalized harmonic analysis

Remark. The basis of R i e m a n n ' s theory of trigonometric series is the following m

result, e.g. [Zgymund, 2, I, pp. 319-3201: assume that 2 an = s; then 1

(2.1.31)

lirn A+O

$ (x)

lim

2

an

and (2.1.32)

h+O A

2

sinn1

=s

sin2nl = s, n21

Sn-

1

n

where s,,

=

2a,. The summability technique embodied

in (2.1.31) (resp. (2.1.32)) is

1

denoted by (R,2) (resp. R,); it is (R, 2) to which we referred in our reference to Riemann after (1.3.15). In 1935, [Marcinkiewicz, 11 showed that a trigonometric series may be summable (R,2) without its series of partial sums being summable R,, and that the series of partial sums of a trigonometric series may be summable R, without the original series being summable (R,2). In 1947, H a r d y and Rogosinski proved the same result for Fourier series. In the verification of (2.1.25) (in the direction we proved) Wiener really showed that if

'I

Y is bounded and x

Y(x)dx is R, summable to 2r then the arithmetic means,

2.1 The Wiener spectrum and Wiener’s Tauberian theorem

(‘.$

93

I

Y(x)dx: R > 0 , converge to 2r. This is the classical format of any Tauberian

theorem :a boundedness (or some such) condition and summability for a certain method yield summability by other means (e.g. Paragraph 2.3). Note that the R, kernel in (2.1.25) is the Fejtr kernel f A (e.g. Exercise 1.2.6a) but that in Theorem 2.1.3 we let “I + 0” (instead of “1 + a’’)and that we have an integral of a product (instead of a convolution). 2.1.11 Wiener’s original proof of his Tauberian theorem. Our proof of (2.1.25)

followed directly (after some indirect substitutions) from Theorem 1.2.3. The first proofs of (2.1.25), including the non-Tauberian implication, are found in [Bochner and H a r d y , 1 ; Wiener, 2; 31. Wiener’s original proof [Wiener, 4, pp. 163-1721 (I 928) of (2.1.25) in the general form of Theorem 1.1.3 did not have the algebraic flavor one finds in the statement of Proposition 1.1.5 (cf. Section 1.1.8 and Paragraph 2.4). On the other hand this first proof clearly shows the necessity of the condition that “cp never vanishes”, whereas the elegant intrinsic algebraic property of Proposition 1.1.5 was replaced by added integrability conditions on f (e.g. (2.1.33)), for 3=cp. The following outline of Wiener’s original proof is due to [Levinson, 2; 31. We say that f E L’(R) is normalized if f(x)dx = 1.

I

Theorem 2.1.4. (i.e.,Theorem 1.1.3)Let f E L’(R) have anon-vanishing Fourier transform

3=cp and let @ E L“(R). Assume that (2.1.33)

j l x f ( x ) I dx < a. R

If (2.1.34)

lim f

X-m

* @(x) = r

I

f ( y )dy,

R

then for each g E L1(R),

(2.1.35)

lim g

x-m

* @(x) = r

I g ( y )dy.

R

Proof. i) Let f be normalized and choose a normalized element h E L’(R) which is “close” to 6 ; to be specific, suppose h isfAfor some “large” I (fAis the FejCr kernel and 3 A =AA).

We’ll show that (2.1.36)

3 w E L’( R), normalized, such that w * f

= h.

Let I(/ = AJcp, which, by our hypothesis on 43, is an element of Cc(fl). Because of (2.1.33), cp’ exists and is a continuous function.

94

2 Taubeflan theorems

Thus $’ E Cc(fi). Consequently, 111/1112 c (2.1.37)

w

and 111/1‘112 < w ; and so, by Plancherel’s theorem,

jlw(x)12dx < w

and

IIxw(x)12dx< w , R

R

where w(x) =

I

R

$(y) e-lXydy.

Therefore, by Holder’s inequality and (2.1.37),

and we have (2.1.36). ii) With (2.1.36) it is easy to verify (2.1.35) heuristically. From the definition of h, (2.1.38)

g

* (h * @) x g * @,

and because of (2.1.34) and (2.1.36), (2.1.39)

lim h

* @(x) = r.

x-a0

By the normalization of g, lim g * (h x-m

* @)(x) = r,

and thus, by (2.1.38), we can conclude with (2.1.35). “q.e.d.”

Remark 1. We’ll discuss the classical distinction between Abelian and Tauberian theorems in Paragraph 2.3. Let us note now that the implication “(2.1.34) =, (2.1.39)” is Abelian since the asymptotic behavior (2.1.34), implies the same asymptotic behavior for the mean w * cf* @) off* @, e.g. Exercise 2.1.3~. 2. Motivation for Theorem 1.1.3 in terms of K a r a m a t a ’ s extension process is given in [Bochner and C h a n d r a s e k h a r a n , 1, pp. 171-1741. K a r a m a t a ’ s work was published in 1930-1931. An enlightening observation about Theorem 1.1.3 is that q(x) somehow approximatesf* cp (x) (in (2.1.34)) sincefis normalized, integrable, and the weight off is at x in the convolution. Thus when the limit in (2.1.34) exists it is reasonable that (2.1.35) would also be true for all normalized elements of L’(R).

2.1 The Wiener spectrum and Wiener’s Tauberian theorem

95

2.1.12 An algebraic proof of Wiener’s Tauberian theorem. Using results from Paragraph 1.2 we’ll now prove Theorem 2.1.4 without the hypothesis (2.1.33); we give a slightly jazzed-up statement for use in Paragraph 2.3. Theorem 2.1.5. Let f E L1(G)have a non-vanishing Fourier transform$= cp (and SO f is a-compact) and let @ E Lm(G).Assume that limf* @(x) = r J f(y)dy.

x+m

G

Thenfor each sequence {g,, :n = 1,. . .} G L1(G) which satisfies the properties that

lim gn* N x ) = L(x) a.e.

R-W

and

lim

n+x

g,,(x)dx = L, G

we have that

lim L(x) = rL.

x-m

Proof. We prove the case that g, = g for each n. It is then trivial to prove the result as we’ve stated it. Without loss of generality (cf. (2.1.30)) assume that r = 0. Let I = { g E L1(G): 0 * g E C,(C)}. Clearly, l i s a closed ideal by Proposition 1.2.6(cf. the last part of the proof of Theorem 2.1.3) and f E I. Since cp never vanishes we obtain I =L1(G)by Theorem 1.2.3 and this completes the proof. q.e.d. Remark 1. If g,, = g for each n we have Wiener’s original result [Wiener, 4; 6, Theorem VIII, p. 25; 7, Theorem 4, pp. 73-74]. Wiener derived this asymptotic form of the Tauberian theorem from Theorem 1.1.3. 2. Recall that “lim Y ( x ) = s” (in C) means that Y - s vanishes at infinity. x-rm

2.1.13 Concluding remark. From the Hahn-Banach theorem and Proposition 1.2.6 we see that the condition I*

=

A(f)

is equivalent to the statement, (2.1.40)

whenever T q = 0 we can conclude that T = 0.

96

2 Tauberian theorems

Clearly, a spectral synthesis result would have the form (2.1.41)

if cp = 0 on suppT then T ~=J 0.

We shall see in Section 3.2.5 that if rp’ E A(FI) then (2.1.41) is valid. We mention this now because the condition (2.1.33), in the statement of Theorem 2.1.4, tells us that 40‘ E A @ ) .

Exercises 2.1 2.1.1 Sets of strict multiplicity

In Exercise 1.3.1 we saw that if p E M,(T) then the arithmetic means of P(x) converge to 0 at infinity. In light of this fact and the Riemann-Lebesgue theorem, a natural problemis todetermine ifM,,(f)\L’(T), or evenAA(T) \L’(T),can be empty(cf. Exercise 1.1.4~).A strong response in the negative is given below in part b). A closed set E c r is a set of multiplicity (resp. strict multiplicity) if

n A Y E ) + (0)

A;(E) =

(resp. Mo(E)# {O}); E is a set of uniqueness or U-set (resp. uniqueness in the wide sense) if it is not a set of multiplicity (resp. strict multiplicity). The study of such sets began with Rieman n; and historical-bibliographical notes and detailed proofs of some of the major results are found in [Bary, 2, 11, Chapter 14; Benedetto, 6, Chapter 3; 12, Appendix 3.11. a) Prove that if J./ is a discontinuous measure (i.e., p = ~~~8~ and 2 lay[< a)) then p 4 Mo(T)\{O}. Also, show that if p E M(T) is discontinuous and limD(n) = a, then n +m

p = ad, e.g. [Benedetto, 12, Exercise 5.16bI.

b) Prove Men’shov’s result (1916): there is a closed set E s T and p E Mo(E)\(O} such that m E = 0 and P(n) = O((logln~l)-1’2),In1 --+ a). (Hint: Let E G T be the perfect 1 k symmetric set determined by ‘ from Exercise 1.1.4b we see that m E = 0.

< -Zk+l’ +

After k steps in the construction, 1 2l +...+2k-1 = 2‘ - 1 contiguous intervals have been removed and we list them from left to right in [0, 2x1 as 11,..., /2k - 1. Define the continuous function Fk: [0,2x] += R by the conditions that Fk(0)= Fk(2x) = 0, 2xj Fk(Y)

=

lL

2@’- j ) 2k2k-11

for y E IJ IJand j < 2’-’ for y

E ZJ

and j 2 2’-l,

,.

and Fk is monotonic and linear between each I, and IJ+ The sequence {Fk:k = 1,. ..} converges uniformly to a continuous function F and we let p = F‘ distributionally (cf. Exercise 1.1.4b)).

2.1 The Wiener spectrum and Wiener's Tauberian theorem

97

In 1927, Bary posed the problem to see if there was any lower limit to the rate at which the sequence {fi(n):n E Z, p E M(T), and m(supp p ) = 0) can tend to 0. Obviously we can not have P(n) = O(l/ln1(1/2)+E), In1 -+ co, when E > 0, for such p because of the Plancherel theorem. In 1936, L i t t l e w o o d showed the existence of p E M(T), for which m(suppp) = 0, such that P(n) = O(l/lnle),In1 -+ co, for some E > 0; and, in !942, [Salem, I ] proved that for each E > 0 there is p E M(T),for which m(supp p) = 0, such that (E2.1.1)

fi(n) = O(l/ln1"/2)-e)), In1 -+ co.

Later, [Salem, 21 extended this result in the following way: given d E (0,l) and E > 0, 2 there is p E M(T),for which m(suppp) = 0, such that E Lp(Z)ifp 2 - + E and d is the d Hausdorff dimension of suppp. This theorem shows that Beurli ng's upper bound, mentioned in Remark 1 after Exercise 2.1.6, can not generally be replaced by a smaller number, cf. [Donoghue, 1 , Section 521. (For properties of Hausdorff dimension see [ K a h a n e and Salem, 4, Chapitre 23). It is interesting to note that in 1938 [Wiener and Wintner, 11, cf. [Zygmund, 2,11, pp. 146-1471, found measures p = F' (distributional derivative) which satisfied (E2.1.1) and the condition F' = 0 a.e. (ordinary derivative) but for which Fwas strictly increasing (thus m(supp p) = mT) ; in a sequel, which contains an error in the first part related to Salem's 1942 result, [Wiener and Wintner, 21 show that every positive bounded measure on FI is the spectral distribution of energy for some signal (not necessarily essentially bounded) on R. The most complete results on Bary's problem are due to IvaSev-Mucatov (1957) [Bary, 2, 11, pp. 404 ff.; K a h a n e and Salem, 4, pp. llO-lll] who proved, in particular, that E = 0 in (E2.1.1) still produces a valid result. 2.1.2 Continuous pseudo-rneasures As in (2.1.3) we say that T E A:(T) if r

N

and we also define I

Assume for this exercise that p(0) = 0. Thus, if

N

98 2 Tauberian theorems

we have F E L2(T)(even more is true by the Hausdorff-Young theorem as we'll discuss in Section 3.2.10) and F' = T distributionally. a) Prove that T E A:(T) if and only if

1IF(y

A

+ j . ) - F(y)I2dy= o(j.),

-+

0

T

(cf. Theorem 2.1.3, Exercise 1.3.1, and Wiener's original proof [Wiener, 1; 7, p. 140 and pp. 146-149; Zygmund, 1, p. 2211). b) Prove that T E A f ( T )if and only if 1.

/IT-?,TI("=o(l),

-+

0

(cf. [Goldberg and Simon, 11). c) Prove that T E AA(T) if and only if

A -+ 0.

IIT-TAT~~~. =0(1),

d) Prove that AA(T)c A:(T). e) Prove that T E A:(T) if and only if (E2.1.2)

ZI-1fw = O

lim(2N+1) N -cm

Inl>N

n

or, equivalently,

(Hinr : (E2.1.2) follows from a technique with dyadic sums which goes back to S. Bernstein and 0. Szasz(cf. [Kahane, 13, pp. 13-14]). (E2.1.3) follows since

2.1.3 An Abelian theorem

a) Given @ E Lm(R)and f E L1(R). Prove that for each b > 0,

5

b

lim x +m

f ( x - r ) @(t)dr = 0.

-m

b) Prove that if lim @(x) = A and @ E L"(R) then x+m

Vf€L'(R),

ik f * @(x)= A Ifly)dy. R

2.1 The Wiener spectrum and Wiener’s Tauberian theorem

99

Consequently, if g =f * @ then the behavior of g at infinity is determined by the behavior of @ at infinity (g is a mean of @).This result is the converse of Theorem 1.1.3 and is an Abelian theorem (cf. Paragraph 2.3). (Hint: Let A = 0 and consider the two integrals Z,(x) = r / ( x

- t ) Qr(r)dt

-m

and m

I,(x) =

/ f(x - t ) @(t)dt). XI

2

2.1.4 A comparison of suppT and supp, T

a) Provide the details for the proof of Theorem 2.1. I using Remark 1 which follows its statement (cf. [Benedetto, 11, Section 3; Meyer, 11). b) Prove the following characterization of the support: given M E A ’ @ ) ; suppM is the smallest closed set such that for any compact neighborhood F which satisfies F n suppM =@, we can conclude that V T E M,

lim l/rRIILa(F) = 0,

R-m

or, equivalently, that V T EM,

lim/~fR~~L~~F)=O

R+m

(e.g. [Benedetto, 11, Theorem 2.11). c) Prove that part b) is not true if fR(7) is replaced by

R

/

f ( x ) eciYx dx

-R

(e.g. [Benedetto, 11, Section 11). 2.1.5

J exp x(1 + iy)sin2e-”dx

Prove that

and so (2.1.28) is valid (e.g. [Wiener, 7, pp. 142-1431).

100

2 Tauberian theorems

2.1.6 The Wiener closure problem for LP(G)

a) Let f E L2(G).Prove that the variety generated by f (in the Lz-norm) is L2(G)if and only if (E2.1.4)

m{y E f :{ ( y )

= 0} = 0.

From Remark 2 after Exercise 1.4.5 we see that this is equivalent to the statement that f E L2(G)is mean periodic if and only if m(Z]) > 0. (Hint: Analogous to the verification of (2.1.40) we see that the variety generated by f is L2(G)if and only if whenever fg= 0, for g E L2(G),we can conclude that

2 = 0 a.e.

This is obviously equivalent to (E2.1.4)). b) Given r > 1 and define

fW=

2 for x E (0, l), 1 forxE(l,r), 0 otherwise.

i

Prove that V, = L'(R) if and only if r is irrational ; and that V, = L2(R)for every r > 1. (The closures in V, are taken in the L' and L2-norms, respectively.) c) LetfE L'(R)\{O}.Prove that V, = L'(0, a),e.g. [Newman, 21. Remark 1. Because of Exercise 2.1.6a and Theorem 1.1.3, W i e n e r posed the problem to find conditions that V, = P ( G ) in terms ofZfwhen 1 < p < 2 [Wiener, 6, p. 931. [Segal, 21 proved that for each p E (1,2) there is f ELP(R) with m ( Z f ) = 0 such that

v/ 5 L P ( R ) , where the closure in V, is taken with theLP-norm (cf. the Remark in Section 3.1 3).Note that iff E fl Lv(R) and V , s LP(R)then

v r E [l,~),

5 L'(R).

Using this observation [Beurling, 81 defined the closure exponent c = sup{p: V, 5 Lp(R)}off E n LP(R)and was able to determine an upper bound for c in terms of the Hausdorff dimension of@ [Newman, 11 gave an extension of Beurling's work. Anotheraspect o f w i e n e r ' s translation problem was takenup by P o l l a r d [Beurling, 3; Pollard, 1; K a h a n e and Salem, 4, pp. 111-1121 (also, see the discussion in [Segal, 31). Along with the closure property (2.1.40) and the synthesis property (2.1.41), consider the uniqueness property cp=OonsuppT

=>

T=O

and the Tauberian property (Theorem 1.3.1~) Tq=O

a

cp=OonsuppT.

2.2 Beurling’s spectrum

101

Thus the uniqueness (respectively, the closure) property implies the closure (respectively, the uniqueness) property because of the Tauberian (respectively, the synthesis) property. The extension of these results to the Lp case is the subject matter of [Herz, 3; Kinukawa, 3; Pollard, 11. Instead of the condition “9 = 0 on suppT”, P o l l a r d considered criteria for sets of uniqueness in the form of pointwise convergence to 0 on (Zrp)“ of an Abelian mean of T. The equivalence of these points of view obviously plays an important role in his proofs, as do certain smoothness conditions on rp to ensure that the synthesis property (2.1.41) is valid for the implication, “closure property implies the uniqueness property”. [Beu r 1 i ng, 4; 5 ;61 form a background for this work. 2. Besides the sort of question we discussed in Remark 1, we can also ask if every variety of LP(R), 1 6 p < 03, is generated by a single element. In light of the discussion in Section 1.4.8 we know that the answer to this question is negative for p = 1. On the other hand the answer is affirmative for p = 2; and [Atzmon, 31 has shown that the answer is negative f o r p E [1,4/3)!

3. Wemention [Carleson, 1, pp. 341 ff.] and [Beurling, 131for two closure problems which are related to but not directly concerned with the above discussion. Further, [Beurling, 71 is the source of a class of closure problems related to function theory; [Helson, 41 provides both background and bibliography.

2.2 Beurling’s spectrum 2.2.1 Fundamental properties of the narrow (i.e., a) and strict (i.e., B) topologies. Let Cb(G)be the space of C-valued bounded continuous functions on G. For each 0 E Co(G) we define the semi-norm

and the pseudo-metric (2.2.2)

V @,YE Cb(G),

pe(@yp) =)I@- YIIo+

III@IIm

-(I

‘IImI.

The locally convex topology on Cb(G)generated by the family {I] lle:0 E Co(G)} of semi-norms is called the strict or p (for Beurling) topology on Cb(G).If 0 E Co(G) never vanishes then the metric topology on Cb(G)generated by p e is called the 0narrow or ae topology on C,(G). In the space C(C) we let K denote the topology of uniform convergence on compact sets (of G). Proposition 2.2.1 a) Cb(G)is complete in the fl topology and in any Cle topobgy. b) Given a sequence { @,, :n = 1,. ..} E C,(G). { @” :n = 1,. ..} converges to @ E Cb(G)in the /3 topology if and only i f { @, :n = 1,. . .} converges to @ E C(G) in the K topology and is uniformly bounded.

102

2 Tauberian theorems

c) Given a sequence {a,,:n = 1,. ,.} E Cb(G).{ @,, :n = 1,. ..} converges to @ E Cb(G)in the tle topology if and only if{ @,, :n = 1,. ..} converges to @ E C(G) in the K topology and ~ ~ ~ l l @ n l l m ~ l l @
ii) We now prove that the fl topology and the topology of uniform convergence yield the same class of bounded sets in Cb(G). Clearly any uniformly bounded set is /?bounded since II@plle< ~ ~ @ ~ ~ man~ estimate ~ @ ~ ~ m which also tells us that the fi topology is weaker than the topology of uniform convergence on G. If B E Cb(G) is fl bounded and not uniformly bounded we choose sequences {@,,:n= 1,. . .} E Band {x,:n = 1,. ..} E G such that l@,,(x,,)l = A,, +- co as n -+ a. From the definition of p boundedness, limx,, = co in the sense that V

K c G, compact, 3 N such that V n 2 N , x,, E K-. Thus

Next choose 0 E C,,(G) for which O(x,,) =

II @nil@ 2 L,"*, and this contradicts the hypothesis that the sequence { @,, :n = 1,. ..} is /?bounded. iii) We now show that the u and p topologies are identical on uniformly bounded sets B E Cb(G). There is R > 0 such that for all @ E B, 1 @llmQ R . Taking Y in the K closure of R, a non-vanishing 0 E C,(G), and E > 0, it is sufficient to prove that (2.2.3)

such that 1 @ -

3@EB

<E

< R since B is uniformly bounded by R). (note that Y E Cb(G)and 11 Choose K E G,compact, such that

and, by hypothesis, take @ E B for which (2.2.5)

SUP XEK

I@(x)

&

-

Y(x)~< 2110llm'

,

2.2 Beurling's spectrum

103

From (2.2.4) and (2.2.5) we obtain

and (2.2.3) follows. iv) We prove that Cb(G)is a complete locally convex space in the p topology.

If {QU} E Cb(G)is a Cauchy directed system, then for each 0 E Co(G),{@@,} c Co(G) is Cauchy in the uniform norm, and so converges to some Ye E Co(G).Also, by choosing 0 E Co(G)equal to 1 on compact sets we see that QU converges to @ E C(G) in the K topology. Thus, for each 0 E Co(G), @ @ = Ye E Co(G). If @ 4 Cb(G)we choose {x,:n = 1,. ..} c G such that for each n, I @(xn)I2 n. Without loss of generality we assume that if n # m then @(xn)# @(x,). Consequently, {xn:n= 1 , . ,.).is discrete since @ is continuous. Since G is locally compact it is straightforward to construct 0 E Co(C) such that @(x,) = l / n for each n. Thus lYe(x,,)l 2 1 for each n, a contradiction. Therefore Cb(G)is /3 complete. The ae case is clear since 11 @,,I( < A4 for an Cauchy sequence { @,, :n = 1,. . .}. v) We now prove part b). If {@,,:n= 1,. . .} c Cb(G)is p Cauchy then lifi"a,,= @ E Cb(G) in the p topology by part iv). By part i), lifi" @,, = @ in the K topology; and since { @,, :n = 1 , . . .} is a fl convergent sequence we apply ii) to obtain its uniform boundedness. For the converse, first note that @ E Cb(G)since { 11 @, 1 :n = 1,. ..} is bounded (by M). Let 0 E Co(G). Take E > 0 and choose a compact set K G G such that V x $ K,

I@(x)I < & / ( 2 M ) .

Then, by hypothesis, 3 N such that V n 2 N and V x E K, I @.(x) - @(XI[ <

@\Im;

and so limll@(@,, - @)[I = 0. n+m

vi) Part c) follows by an argument analogous to that in part v) and by the definition of Pe-

q.e.d.

2.2.2 Remarks on the a and /? topologies Remark 1. We make some comments on the ae topology. Any ae (sequential) convergence criterion is stronger than p sequential convergence. For each subset X G Cb(G)we let cl X consist of those elements @ of cb(c) for which there is {@,,: n = 1,. ..} such that = @ in the K topology and = 11 911,.

!em@,,

hill

104

2 Tauberian theorems

Clearly, cl0 = 0, X c clX, and cl(X U Y) = clX U cl Y, where X, Y E Cb(C). Take XECb(G)and @Ecl(clX).Choose(@,:n=l,...} c c l X s u c h t h a t nl +i m @ , = @ i n the m K

topology and lim 1 @,,I1 n+m

that lim @,, m+m

=

1) @I/

; and for each n choose {a,,,, : m = 1,. ..} G X such

,, = a,, in the K topology and lim 11 @,, , 11

=

m-bm

11 @,, 1 m. Fix any 0 to define Pe

and for each n choose m,,by Proposition 2.2.lc, such that Pe @ (,,, 0,) < l/n. Pick e > 0. Thus, there is Nl > 0 for which Pe(Grn,,,@,) < 4 2 when n 2 N , . There is also N2 such that Pe(Qn,@)< ~ / 2if n 2 N2. Let N = max(N,,N2). Then if n 2 N we have Pe(@m,,. n , @) < E ; and SO @ E cl X. Hence cl(c1 X) = cl X. Therefore “cl” is a closure operator, and Ti = {UE Cb(C):cl(Cb(G)\ U)= Cb(G)\ U} is a topology on C,(G) with the property that clX, where X E Cb(G)and “-” represents closure in Ti. Clearly, Ti is independent of any 8.

x=

We now observe that each metric space (Cb(C),Pe) generates the same topologyT, on Cb(G).In fact, if X E Cb(G)is ae, closed and @ is in the ae, closure of Xthen it is easy to check, using Proposition 2.2. lc, that @ E X. From Proposition 2.2.1 we see that the a(L“(G), L’(G)) topology on C,(G) c L“(G) is weaker than T,. Compare T, and T: (obviously, T, is weaker than Ti). 2. The K (resp. the sup norm) topology on Cb(G)is formed in the same way as the fl topology except that the family C,(G) (resp. Cb(G))is used instead of C,(G). A systematic treatment of spectral synthesis in terms of the fi topology is given by [Herz, 51. We also mention [Domar, 41. In light of Remark 1 we refer to any ag topology as the narrow or a topology. Recall the remarks made in Section 1.4.12 on the narrow and strict topologies.

2.2.3 Examples Example 2.2.1 All of the functions in this example are defined on R. a) From the above discussion we know that the topology of uniform convergence is 1 x stronger than the a topology. Note that if @,,(x)= -sinlim @, = 0 in the sup n n then n-m then l$m@n = 0 in the fl topology, but since n 1 @,,I1 = 1, { @,, :n = 1,...} does not converge in the a topology. As another example of this type, define Y

norm topology. Now if On(x)= sin:

2(x - n) -2(x-n-I)

@,(x)=

[

o

+

for x E [n,n +], forxE[n++,n+ll, elsewhere.

Finally we show that the a topology is strictly weaker than the supnorm topology. Let

2.2 Beurling's spectrum

105

{yn:n= 1,. ..} c S l be a sequence of distinct points which converges to y E fl, and define @,(x) = eirym.

Clearly, lim @,,(x) = elxy= @(x)in the u topology. On the other hand, n+m

X

I

JeWm- eixy = I2sin$,

- Y)l,

and so 11 @,, - @/Im= 2. b) If @(x)= sinx then Y ( x ) = sinx + cosx belongs to the translation invariant sub-

(

'2")

space generated by @ since cosx = sin x - - . On the other hand, if @(x)= elx then Y(x)=eZix does not belong to the translation invariant subspace generated by @. Now if @(x) = e-lxl then Y ( x ) = (1x1 i)e-lxl E V@where the closure of the variety is taken in the supnorm topology (note that @ * @ = Y ) . In this regard see Proposition 2.2.2. Also, it is clear that if @ E C,(G) then

+

where we are again considering the supnorm closure of the variety. c) Because of Beurling's theorem, which we shall prove shortly, let us recall some examples concerning uniformly continuous functions. @(x)= x is unbounded and uniformly continuous on R, whereas @(x) = xz is neither bounded nor uniformly continuous. @(x) = sinx is bounded and uniformly continuous, whereas @(x)= sineX is bounded but not uniformly continuous. d) In the proof of Theorem 2.2.1 we'll also use the Arzela-Ascoli theorem (e.g. [Bened e t t 0,121) and so we make the following observations. If @:(x) exists for each x E R and each n = 1,. . ., and if sup11 @illm< 03, then { @ , , : n = I , . . .>is an equicontinuous n X

family. Thus, {@,,(x) = n sin-:n = 1,. . .} is an equicontinuous family. Now, {@,,(x) = n . x sin - : n = 1, . . .} is an uniformly bounded equicontinuous family which converges n pointwise to @ = 0; but { Qn:n= 1,. . .} does not converge uniformly on R. 2.2.4 Beurling's theorem: supp,T = suppT. Take T EA ' ( f ) for which p= @ E CdG), and let V , be the u closed variety generated by @. We define

supp.T= {y E r : ( y , * ) E V,}. Proposition 2.2.2. Let @ E C,(R) \ (0) be uniformly continuous and take f E L1(R). Then

f * @ belongs to the sup norm closed variety generated by @.

106

2 Tauberian theorems

Proof. Fix E > 0. Since @ is uniformly continuous, 3 6 > 0 such that V y

E

+ 1)6] and V x E R, I@(x - y ) - @(x - k6)I < .s/(211flll).

[k, (k

Define ( k + 1)d

a

2

Y(x)=

J

@(X-kS)

k--a

f(t)dt-

kb

This series converges uniformly on R by the Weierstrass M test. Note that

f * @(XI - W x ) =

z

(k+ l ) b

[@(x- Y ) - @(x - Wlf(.Y)dY,

j

k--m

and so llf* @ - Y 1 < 42. Set Y,,(x) =

$

(k+l)b

@(x-

k6)

k--n

f(t)dt, kb

and thus 3 N such that V n 2 N,

Consequently, 1 Y,, -f *

@[Irn

11 Y,, - Y 1Irn < 42.

< E for all n 2 N. q.e.d.

Theorem 2.2.1 ([Beurling, 21). Let T = @ E C,(R) be ungormly continuous. Then (2.2.6)

SUPP.

T = SUPP T.

Proof. i) If T # 0 we know suppT# (2.2.7)

supp.T

0 by Proposition 1.4.2; also,

c supp T,

since the c1 topology is stronger than the weak * topology. Take 1 E supp T. Because of Proposition 2.2.2 it is sufficient to find a sequence {f.:n = 1,. ..} c L1(R) such that lim J , * @(x) = elXAin the a topology. n+m

ii) Using the notation of Exercise 1.2.6 let p,(y) = T~ Al,,,(y), so that g,, = cp. where gn(x> = e-’”fI/n(x)* Note that Tp,, # 0, for if it were we’d have pn= 0 on suppT by Theorem 1 . 3 . 1 (but ~ I E suppT and p,,(I)= 1). Thus @ * g,,- is not identically zero and so we can choose x,, E R for which

(2.2.8)

I@* gn-(x,>) 2

2.2 Beurling’s spectrum

107

Define

Clearly (2.2.9) and (2.2.10) since

iii) We now prove that { @ * h,- :n = 1 , . ..} is a uniformly bounded equicontinuous family so that we can apply the Arzela-Ascoli theorem. The uniform boundedness is clear from (2.2.10). If g = T~ V,, where V, is the de la VallCe-Poussin kernel, we see that

v n 2 2,

g- * (@

* h,-) = @ * hn-.

Thus,

j

I @ * h,-(x) - @ * h,-(y)( < n-1

Ig(t - x) - g o -Y)I

dr

R

(2.2.1I )

by a basic property of the Lebesgue integral. (2.2.11) obviously entails the equicontinuity. Therefore, by the Arzelh-Ascoli theorem, there is a subsequence {fn :n= 1 , . . .} of {An- :n = 1 , . . .} such that @ * fn converges in the K topology to a continuous function Y .

E CJR) by (2.2.10) and IIYllm= limll@*fnllmbecause of (2.2.9) and (2.2.10). Thus @ * fn converges to Y in the c( topology. iv) Since Y(0) = 1 and Y is continuous, we know that sp Y # 0 (Proposition 1.4.2). We’ll prove that suppS = {A}. Thus, Y ( x )= elx’ because sets consisting of a single point are S sets (cf. Exercises 1.2.3, 1.2.4, and 1.3.7). Take y E R, where y # A. Hence we can choose E > 0 and N > 0 so that $,, cp = 0, where f = cp = T, A , and n 2 N (note that cp(y) # 0). Therefore (7‘$,,) cp = 0 if n 2 N . =Y

108

2 Tauberian theorems

n Because of the a convergence of {@* f , : n = 1, ...} and the fact that (T$,)cp = ( T * h,,J * f- we can use the dominated convergence theorem to conclude that Scp = 0 (obviously we do not use the full strength of a convergence at this point). Consequently, if y E suppS we deduce that cp(y) = 0 by Theorem 1.3.1~;on the other hand, as we pointed out above, ~ ( y #) 0. q.e.d.

2.2.5 Wiener’s Tauberian theorem and Beurling’s theorem. Be u r 1i n g ’s original proof of Theorem 2.2.1 used complex variable techniques and did not use the Tauberian theorem; a detailed presentation of this proof is found in [Benedetto, 2, pp. 58-65 and pp. 70-801 (in this reference, the word “strict” should be replaced by “narrow”). The extension of Theorem 2.2.1 to a LCAG with a proof dependent on Wiener’s Tauberian theorem was first stated in [Godement, 11; unfortunately, an error was involved on this point [Godement, 1, Section 4; 2, p. 1311 which was not cleared up until the mid-l960’s, e.g. Section 2.2.7. The above proof of Theorem 2.2.1 is due to [Koosis, 2, p. 1221 and depends on Wiener’s Tauberian theorem and the fact that points are S-sets. The latter fact, although proved at the time of Beurli ng’s paper in 1945, had not yet been properly advertised (cf. Section 2.5.1). Other proofs of B eu r 1i n g ’s theorem, including generalizations to the LCAG case, are found in [Domar, 1; 2; G a r s i a , 11.

In light of the fact that Beurling’s original proof did not use Wiener’s Tauberian theorem, it is interesting that Wiener’s Tauberian theorem (Theorem 2.2.2b = Theorem 1.3.1~)can be deduced from the statement of Theorem 2.2.1. Thus, in the following result we only assume the statement of Theorem 2.2.1 and the elementary definitions of certain operations with pseudo-measures ; in particular, we do not use Theorem 1.3.1~ in the proof. Theorem 2.2.2. a) Let f’=@ E C,,(R) \ (0) be uniformly continuous and take cp E A @ ) . If Tcp = 0 then cp = 0 on supp, T. b) Given T E A’@) and cp E A(R).

If Tcp = 0 then cp = 0 on suppT.

Proof. a) Since Tcp = 0 we know that if scp = 0.

s = Y E V,,

the a variety generated by @, then

From the statement of Theorem 2.2.1, there is an element (7,

a )

E V,,

and so, for each x ,

( y , x) q ( y ) = 0. Thus, cp = 0 on suppa T.

b) Since Tcp = 0 we have that (Tt,b)cp = 0 for each t,b E A @ ) . Therefore, since @ is uniformly continuous, we apply part a) and conclude that cp = 0 on supp, T$ = suppTt,b(we’ve used Theorem 2.2.1 again). Consequently, cp = 0 on suppT because t,b is arbitrary. q.e.d.

2.2 Beurling’s spectrum

109

[Pollard, 2, Section 3 and Section 5 ; K o r e v a a r , 21 give other proofs of Theorem 2.2.2b. 2.2.6 Spectral synthesis in the a topology. Since Theorem 2.2.1 is a positive solution for a strong form (i.e., the ci topology) of spectral analysis on the uniformly continuous elements of Cb(R),it is natural to investigate the corresponding spectral synthesis problem in the ci topology. Because of (2.2.6) and the results of Chapter I , we have

Proposition 2.2.3. Let

f = @ E C,(FI) be uniformly continuous. If card supp, T < to then

2

T= YE

a?&

ww,T

(cf. Theorem 2.2.4). We also note Beurling’s and G a r s i a ’ s proofs of Proposition 2.2.3 [Benedetto, 2, pp. 90-91 ; Beurling, 2, p. 132; G a r s i a , 1, Section 31. 2.2.7 Unbounded spectral analysis. Remark. One of the important features of Beurling’s spectral analysis for the a topology is that an element of supp, T can be approximated by a sup norm bounded sequence of linear combinations of translates o f f . (Norm boundedness, in one form or another, is the crucial assumption in problems of “switching limits”, e.g. MooreSmith or Lebesgue dominated convergence theorems.) The weak * spectral analysis of Lm(R)which comes from the Tauberian theorem does not provide any such boundedness property for those elements of L“(R) which are not uniformly continuous (we have the boundedness for uniformly continuous functions because of Theorem 2.2.1). [Benedetto, 12, Appendix I ; K a h a n e and Salem, 4, Appendix IV] describe what the weak * topology will and will not do for you. Generally, it will never provide you with norm boundedness when you need it because directed systems of linear combinations of translates of @ are required to find elements (7, .) E ’Fa.

In light of Theorem 2.2.1 it is reasonable to try to prove the following: i f @E Cb(G)and

( y , .) E T othen ( y , .) is the weak * limit point of a sup norm boundedfamily of linear combinations of translates of @. [Koosis, 2; K a h a n e , 91 have provided counter-

examples to this statement; K o o s i s notes that such counterexamples also provide counterexamples to [Godement, 2, Th6orkme D, p. 1311. 2.2.8 The spectral analysis and synthesis of almost periodic functions. We’ll now make a refinement of Proposition 2.2.3 (in Theorem 2.2.4), to do this it is convenient to outline some facts about almost periodic functions. We consider Cb(G)with the supnorm topology. The orbit of @ E Cb(G)is the set O(@)= G}. Generally an orbit of Cb(G)is a translation invariant set. @ E Cb(G)is almost periodic (resp. weakly alniost periodic) if O(@) is relatively compact (resp. { T@ ~ : xE

1 10

2 Tauberian theorems

relatively weakly compact) in C,(G); in this case we write 9 E AP(G) (resp. 9 E WAP(G)).We state the main facts that we need about AP(G) in the following theorem. Theorem 2.2.3. Given generated by 9.

T = 9 E AP(G) \ { 0) and let

a) (Spectral analysis) SUPP,,

V, be the sup norm closed variety

T = {y E r :(y, .) E V,} is countable and non-empty.

b) (Spectral synthesis) @ can be written as a uniformly convergent series (2. I . I 2)

2

Y E S U P P , ~T

aY(Y,X),

where each a, # 0.

c)

SUPP,,

T = SUPP T.

Part c) follows immediately by the definition of support and part b). It is easy to check that finite linear combinations of characters are almost periodic, and that uniform limits of almost periodic functions are almost periodic. Clearly, supp,, T = 0 if T E Ah(r); and, in fact, i f f ' € AP(R) and T E A:(FI) then T=O. On the other hand it is possible to find a uniformly continuous function T = 9 E Lm(R)\ AP(R) for which card SUPP,,T> w,, [Eggleston, 1; Wallin, I]. Expositions of B o h r ' s theory of almost periodic functions, including a proof of Theorem 2.2.3, are found in [Katznelson, 5; Loomis, 11. The technique developed after (2.1.l) to define the spectral distribution of energy can be used to verify Theorem 2.2.3a in a very neat way [Katznelson, 5, p. 1631. Also, as we indicated in Remark 3 of Section 2.1.6, supp,,T c ZIw(MT).

2.2.9 The extension to G of Bohr's differentiation criterion for almost periodicity. If = 9 E AP(G) we say that T is almostperiodic. T E A'(r)is almostperiodic at y E r if there is a non-negative function cp E A(r)which is equal to 1 on a neighborhood of y and which has the property that Tqo is almost periodic. Clearly, {y

Ef

:T is almost periodic at y }

is an open set. Proposition 2.2.4. Given T E A ' ( f ) and assume that suppT is compact. a) T is almost periodic o T is almost periodic at each point of r. b) IfT is almost periodic at each y

E

f \ {0}then T is almost periodic.

P r o o f . a) ( 2)We'll prove (2.2.13)

if T is almost periodic and fl=

cp E A ( f ) then Tq is almost periodic.

2.2 Beurling's spectrum

11 1

Because o f the remarks made in Section 2.2.8 we see from (2.2.12) that A

T4-4

=

2

Y ESUPPAP T

&9(Y,X)

is a uniformly convergent series; and so Tq is almost periodic. (e) By our hypothesis o f almost periodicity at each point and the fact that suppT is compact, we can find q l , .. ., qnE A(T) such that Tq, is almost periodic for each i = 1,. .., n and

n

2 q, > 0 on a neighborhood o f supp T. 1 Thus (Z(2Pi)) n SUPPT = 0. By Proposition 1.1.5b there is cp E A(T) for which supp T.

2 qcp, = 1 on a neighborhood of

Using (2.2.13) we see that T is almost periodic since Tq, is almost periodic and b. i) We'll first prove that if T is almost periodic at each y (2.2.14)

VX E G,

- (A,X)

E

r \ (0) then

&

is almost periodic. Take cp E A(f) equal to 1 on a neighborhood o f suppT; and so T = Tq. Also, fix x E G. We have TA - 0 - 9 X) TA = TA(dl) - (1,X) ~(l)),

noting that $(A) = q ( A ) - ( l , x )qW) vanishes at A = 0. Since points are S-sets, there is a sequence {$,,:n = 1,. . .} c j ( ( 0 ) ) such that lim \I$ - $"llA = 0. n+m Thus (2.2.15)

limII(T- (.,x)T) - T$nllAr =O.

n+m

Since Tis almost periodic at each y E T \ {O}and since $,, vanishes on a neighborhoodof 0, we use (2.2.13) and the sufficient condition for almost periodicity in part a) to conclude that T$" is almost periodic. Therefore, from (2.2.15 ) and the comment after Theorem 2.2.3 we have that TA(A,x)TAis almost periodic. ii) Assuming that T is almost periodic at each y E r \ (0) we now use part i) to prove the almost periodicity of F = @. Without loss o f generality assume that @ is real-valued, and set

M

= sup { @(x):x E G)

and

m

= inf { @(x):x E

G}.

112

2 Tauberian theorems

Take E > 0. Choose a, b E G such that (2.2.16)

@(-a) > M

&

-4

&

@(-b) < m + - . 4

and

Because of part i) we set x = a - b in (2.2.14) and obtain @ - ?b - , @ E AP(G); thus (2.2.17)

?a

@ - ' I b @ = Ta (@

- 7b-(3 @) E AP(G).

From (2.2.17) there is a finite set F c G such that (2.2.18)

v

E O(T. @

- Tb @)

3y

E

F

11

for which

- Ty(Ta @ - T b

@)II,

< &/2.

In particular, for each fixed x E G we set Y = T , + g ( ~ , , @ - ?b @); and for this Y there is y E F for which we compute, from (2.2.18), that Y(x

+ a) = @(-a) - @(-b)

and (2.2.19)

I(@(-u)

- @(-b)) - (@(x- y ) - @(x + u - b - y))l < 42.

Thus, given E > 0 we have found a finite set F G G such that for each x E G there are y, z E F for which

(2.2.20)

@(x - y ) > M - E

@(x- z ) < m

and

+

E;

in fact, (2.2.20) follows, for the case @(x - y ) > M - E , from (2.2.16), (2.2.19), and the computation @(x- y ) > @(-a) - @(-b)

&

+ @(x+ a - b - y ) --2 &

In order to prove that @ E AP(G) we take q > 0 and shall find a finite set K G G such that (2.2.21)

v

Y E O(@) 3 u E K

for which

IY - T"

@t i r n

For convenience let q = 28 ( E as above). It is straightforward to check that there is a finite set K c G for which V t EG 3 uEK

(2.2.22)

IITt(@

- 7 , @)

such that

- 7"(@ - 7 y @) [ I r n

Vy

EF(Fas

above)

< E.

(2.2.22) is obviously true for each single y E F since @ - T, @ E AP(G); the point is that it is also true uniformly as y ranges through the finite set F.

2.2 Beurling’s spectrum

113

We now prove that K is the set required to obtain (2.2.21). In order to do this we take t , x E G and use the u E G of (2.2.22) and M to calculate that (2.2.23)

T~ @(x)- T,

@(x)< 2.5;

an analogous argument using m works to check the inequality 72@(x) - 7, @(y)> -2.5. From (2.2.20) we pick y E F such that @(x - u - y ) > M - E . Then we employ (2.2.22) to compute 72 @(X) - 7, @ ( X )

< E -k 71T y @ ( X ) - TUTY @(X) < 2.5 - hf -k 72 T y @(X)

< 2E;

this is (2.2.23). q.e.d. Part ii) of the proof of Proposition 2.2.4b is the generalization to G of Bohr’s theorem: @ E L”(R) is everywhere differentiable and @’ E A P ( R ) then @ E A P ( R ) . An elegant classical treatment of Proposition 2.2.4, including this result on differentiability is found in [Katznelson, 5, pp. 166-1681. The proof of Proposition 2.2.4 is due to [Loomis, 21.

if

2.2.10 Scattered sets, almost periodicity, and strongly synthesizable pseudo-measures. E E r is perfect if it is a closed set without isolated points. E c r is scattered if it is a closed set which does not contain any non-empty perfect subsets. We’ll discuss scattered sets in Paragraph 2.5. For now note that in 5l E is scattered if and only if it is countable. Theorem 2.2.4. Let

T=

T = @ E Cb(C)be uniformly continuous. Ifsupp T is scattered then

2

ady,

YESUPPT

where convergence is in the 1 Ila.-topology(i.e., T is almostperiodic). Proof. i) Because of Proposition 1.4.7, there is a directed system {T,} G A’(T) such that lipIIT- TallA.= 0 and suppT, is compact. If we prove that each T, is almost periodic we can therefore conclude that T is almost periodic. Thus we assume that supp T is a compact scattered set and shall prove that T is almost periodic. ii) Since { y E r : T is almost periodic at y } is an open set and since T is almost periodic at each y 4 suppT we see that the set X c r of points at which T is not almost periodic is a closed subset of supp T. We now observe that X contains no isolated points (a5 a subset of r). In fact, if y E Xis isolated we can choose cp E A ( T ) equal to 1 on a neighborhood of y such that Tcp = 0 on an open neighborhood of X \ (y}. Thus Tcp is almost periodic at each

114

2 Tauberian theorems

point of r \ {y}, and so we conclude that Tq is almost periodic because of Proposition 2.2.4b; this contradicts the fact that y E X . Since suppTis scattered and X contains no isolated points we see that X = 8. Therefore T is almost periodic at each point of I'. Using the compactness of supp T we q.e.d. apply Proposition 2.2.4a to obtain the almost periodicity of T. We knew from the trivial part of Proposition 1.4.7 that if supp T is compact then T is uniformly continuous. It is an elementary property of almost periodic functions that if @ E A P(G) then @ is uniformly continuous. The proof of Theorem 2.2.4 tells us further that

Corollary 2.2.4. If T E A'(r)and supp T is a compact scattered set then T is almost periodic. 2.2.11 An historical remark. Theorem 2.2.4 is due to [Loomis, 21, but the idea of the proof, once we have Proposition 2.2.4, goes back to D i t k i n and is used to obtain general Tauberian theorems, e.g. Section 2.5.2. [Veech, I] has given an alternative proof of Theorem 2.2.4 employing a characterization of AP(C) in terms of the more general notion of almost automorphic functions (such functions were introduced by Bochner). For the case G = R the result was proved by [Levitan, 11 in 1937 and [Beurling, 51 in 1947 in the context of integral equations. For example, L e v i t a n ' s theorem is: let S E A'(R) be almost periodic and assume that rp E A(I1) is continuously differentiable; if the set of limit points of Zq consists of isolatedpoints then every bounded uniformly continuous solution T of

Tq = S is almost periodic. Beurling removed the hypothesis of differentiability and used characterizations of supp Tin terms of Abelian summability in his proof (e.g. Remark 2 in Section 2.1.6 and Exercise 2.1.4). [Reiter, 1, p. 423; 21 made contributions to this problem for the case of G prior to L o om i s ' work.

Exercises 2.2 2.2.1 Properties of the j9 topology a) For each @ E Cb(G)define the family cb(G)of operators Co(G) Y H @Y (cf. Proposition 1.1.6). A strong (resp. weak) operator neighborhood of (i.e., 6 = L o ) has the form < &, j = l,...,n} 4:Co(G)

-+

6 E cb(c>

{ J ? a @ : ~ ~ @ ~ J ~ ~ m

(resp. {La: I < pJ, OYJ > I < E, j = 1,. . - , n } ) where e, { YJ: j = 1,. ..,n} E Co(G), and {p, : j = 1,. . .,n} E M(C) vary. We identify

2.2 kurling's spectrum

1I5

Cb(G)and cb(G)in the obvious way, and thus canonically define the strong and weak operator topologies on Cb(G); these topologies were introduced by v o n N e u m a n n in 1929. Prove that the fl topology is the strong operator topology on Cb(G);and that the canonical locally compact topology on r is the induced fl topology on ((7, . ) : y E r}. The fl topology has been studied for cb(x) for any Hausdorff space X [ H o f f m a n J s r g e n s e n , 11. b) Prove that M(G) is the dual of Cb(G)taken with the fl topology [Buck, 2, pp. 991001 (cf. [Muraz, 11). Of course, this result is true for any topology between a(Cb(G),M(G)) and the Mackey topology on Cb(G),e.g. [Horva th, 1, pp. 203-2061. [Buck, 2, p. 1001 asks if the Mackey topology is equivalent to the topology; contributionstothisproblemarefoundin [Collins, I ; C o n w a y , 1; R u b e l a n d R y f f , 11.

A survey of recent work on the fl topology is found in the Zentralblatt fur Mathematik 244 (1973) 46027,46028 ;and there is interesting new work by S e n t i 1les and Wheeler. Note that if {pn:n = 1,. . ., p,,2 0,II pnII V @ E Cb(R),

!i%

c M ( R ) satisfies the condition that

< pn, 4 > = < P , @ >,

then p E M(R) is non-negative and (lp((l= 1 (convergence on C,(R) would allow the possibility that p = 0, e.g. p,,= d,,). c) Let X C Cb(G)be a convex orbit which is fl closed. Prove that X is closed in the weak * topology a(Lm(C),L1(G))induced on Cb(G).Thus, i f f = @ E Cb(G),Va is the fl closed variety generated by @, and supp,T= (7 E r:h

E

V a l 1 3

then (E2.2.1)

SUPP

T = SUPP, T

(cf. the observation in [Godement, 2, top of p. 1311). (Hint: The proof proceeds in the same manner as the proof of the Kreifi-Smul'yan theorem and uses part b). Since part b) is not true for the u topology the same proof does not work for supp, T, where F = @ E Cb(G)is uniformly continuous. Also, with regard to Section 2.2.7 we can ask about extending the /? topology to Lm(G) and preserving (E2.2.1).

2.2.2 Properties of the u topology a) Prove Theorem 2.2.1 for the case of G, not R.

b) We writefE CD(Rn)c Cm(Rn),1 < p < a,iffand all of its derivatives are elements of LD(Rn). The following convergence criterion gives rise to a metrizable locally convex topology on CD(Rn):&-+ 0 as k + 03 if for each s

116

2 Tauberian theorems

D,(R"), 1 < p 6

00,

is the dual of C,(R"),

1 -+

P

- = 1 ; and if we define f~ C,,o(R") 4

G

C,(R") as V

S,

D'f

E

CO(R"),

then D,(R") is the dual of C,,,(R"). Prove that (E2.2.2)

Dp(R")G Dr(R")E D,(R") E Dt(R")

if p 6 r (e.g. [Schwa r t z, 5, pp. 199-203 and pp. 237-2431) and that L1(R") E D,(R")

and

L"(R")

s D,(R").

c) Because of (E2.2.2) and the remarks on the Fourier transform in Exercise 1.3.6, we see that for each @ E Dm(R") there is T E D,(R") whose Fourier transform is @. Prove the following form of Theorem 2.2.1 (taking Proposition 2.2.2 into account): given T = @ E D,(R"); y E s u p p T o there is a sequence { f , : n= 1 , . ..} c C,d(R") such that f, * @(x)converges to elxyin the T, topology. d) Prove the following form of Wiener's Tauberian theorem [Benedetto, 11: given f E L1(R) with a non-vanishing Fourier transform and @ E Cm(R), and assume that limf* @(x)= rjf(y)dy;

x-m

R

then, for each g E D,(R), limg * @ ( x ) = r ( g , 1).

x-m

2.2.3 Bernstein's inequality for A'(5I) We give this exercise because of the use of Proposition 1.4.7 in the proof of Theorem 2.2.4. a) Prove that if T E D,(R) then its Fourier transform is the restriction to R of an entire function. b) @ :C + C is a function of exponential type A if V

E

> 0 3 K,

such that

V z E C, I@(z)l < K,exp [(A

+ E)IZ~].

Prove that if T ED,(R) and s u p p T c [-A,A] then L ( T ) ( z )is of exponential type A (the Laplace transform L was defined in Exercise 1.3.6). (Hint: Taking E > 0 it is possible to write T = cpcR) + t+b, where cp and t+b are continuous functions supported by [-A, A E ] and qcn) represents distributional differentiation). c) Prove Bernstein's inequality : if T E A'@) and supp T E [-A, A] then

+

2.2 Beurling's spectrum

117

(cf. [Benedetto, 6 , Section A.lO; D o n o g h u e , 1, pp. 227-229; Meyer, 5, pp. 1491511). GArding and H o r m a n d e r have noted that it is easy to prove Wiener's Tauberian theorem using Bernstein's inequality, e.g. [Donoghue, 1 , pp. 230-2311. d) Prove that if T E A'(FI) is supported by [-All], ~ ~ ? ' / / m = A ~ ~and T / ~l ?'A /!m=, , PO,)for some y , then F(x) = Ksin(Ax - y). 2.2.4 The harmonic spectrum and supp, T a) @ E L:(R) if V

CJ

< 0, ll@(x)leuk'dx < a. -W

For such an element @ and for CJ + iy E C, m

@(x)ealx'+ l y x dx

H,(o, y ) = -m

is an harmonic function in the half-plane @ E L:(R), if V

E

> 0, lim a-0-

7.

IH,(a,

CJ

< 0. y E sp, @, the harmonic spectrum of

A)[ dl > 0.

Y-.

Prove that sp, @ is closed for each @ EL:(R); and that there is @ EL:(R) \ (0 )such that sp, @ = 0 (cf. [Vretblad, 11). [Beurling, 31 has determined a satisfactory spectral analysis for a certain subspace ofL:(R) as well as having studied a large class of related spectral analysis problems. A propos this material, Hille and T a m a r k i n have contributed to the ideal theory of the algebra of Laplace-Stieltjes transforms absolutely convergent in a fixed half-plane, e.g. [Hille and Phillips, I ] for references. I and priorities we now comment on some work of b) With regard to the #topology [Povzner, 11 (1947). Let P(R) be the set of even, positive, continuous functionsp which increase to infinity at infinity and which satisfy the condition that p ( x + y ) < M ( y ) p ( x ) for x,y E R. For eachp E P(R), 1 @lip = 1 @/p11, defines a norm on C,(R) and V,(p) will be the 1 IIP-closed variety generated by @ E Cb(R).For F = @ E Cb(R),define supppT= (7 E FI: (7, * )E VG(P)) and supp,~=

n

suppp~.

PEP(U)

If T = @ E Cb(R)\{O} prove that supp,T= supppT= 0 for some p E P ( R ) . P o v z n e r announces (there are essentially no proofs in the paper) that supp, T = SUPP T

118

2 Tauberian theorems

for F ECb(R)(cf. (E2.2.1)). A related “L1”theory is given in Section 1.4.3. [Beurling, 21 (1945) proved that

TfO

=>

supp,T#0

if Fis uniformly continuous, but he does not verify (2.2.6); on the other hand he states that (2.2.6) is easily proved in a footnote in [Beurling, 61 (1949).

2.2.5 The uniform closure of B(G) From Theorem 2.2.3 we know that the uniform closure of M,(T)” is AP(G) and, of course, the uniform closure of L’(T)” is C,(G). An internal characterization of the uniform closure of the set of Fourier-Stieltjes transforms of continuous singular measures is not so easy. In this exercise, B(G) will denote the uniform closure of B(G). a) Prove that @ E C,(G) is an element of AP(G) if and only if for each 1 11 ,-bounded directed system {p,} E M(G), for which lip 0, = 0 pointwise, we can conclude that l i p s @dp,=O c

[Edwards, 4, p. 254; Ramirez, 21. b) Let G be a-compact. Prove that @ E Cb(G) is an element of B(c) if and only if for each I/ II,-bounded sequence {pn:n= 1 , . . .} G M(G), for whichdiyfi,, = 0 pointwise, we can conclude that lim

n+m

s

@dpn=O

c

(cf. [Edwards, 4, Theorem 1.71). This result was first proved by B e u r l i n g and Hewi t t (unpublished). The extension to arbitrary G as well as characterizations of B(G‘) in terms of the /3 and weak operator topologies are due to [Ramirez, 11 (cf. Section 3.2.17). Surveys on the properties of M(G)are found in [Dun k l and Ramirez, 1; Hewitt, 2; Rudin, 51. c) Prove that C,(G) E B(G) c WAP(G) and that each element of WAP(G)is uniformly continuous. [Rudin, 31, e.g. [ D u n k l and Ramirez, I], proved that the inclusion R(G)s WAP(G) is proper. The theory of weakly almost periodic functions was developed by E be r 1 e i n via his research in ergodic theory; in the process he was quite aware that synthesizable pseudo-measures have transforms with generalized almost periodic properties [Eberlein, I , Section 7-81. [Burckel, 13 is an exposition on WAP(G) and [Argabright, 11 shows that the invariant means on WAP(G)provide a simple way to compute the point values of pseudo-measures T for which T E WAP(G). 2.2.6 Almost periodic functions Prove Theorem 2.2.3.

119

2.3 Classical Tauberian theorems

2.3 Classical Tauberian theorems 2.3.1 Abel’s and Tauber’s theorems m

m

Proposition 2.3.1 (Abel). u f ( x ) = xa,x“ conoergeson [0,1 ) and 2 a, = S thenf(1-) 0 0 S.

=

Proof. It is obviously sufficient to prove that 2a,X converges uniformly on [0,1]. Assume each a, E R so that by hypothesis V e >0 3 N

such that V k > m 2 N,

- & < a , + , +...+ak< e.

Now, by partial summation (A be1 ’s technique),

<&(

i

(xk-xk+l)+xn+l

m+l

)

=EX“+’<&.

q.e.d. The first Tauberian theorem (by T a u b e r in 1897) came as a response to the problem of finding some sort of converse to Proposition 2.3.1 besides the trivial converse one obtains by assuming each a, 2 0 (this “a, 2 0” case was observed by P r i n gs h e i m in 1900 although A b el himself had practically done the calculation). T a u be r proved : v f ( x ) = 2 a,xn conoerges on [0,l), f ( x - ) = S, and a, = o(l/n),n --f m, then 2 a, = S. The boundedness condition “an= o( l/n)” is the “Tauberian condition” necessary to effect the converse. Thus, the hypothesis “@ E L“(G)” is the Tauberian condition in Theorem 2.1.5 [Hardy, 1, p. 2851 (cf. [Rudin, 6 , p. 2091).

2.3.2 A technical lemma m Proposition 2.3.2. Dejine f ( x )= 2 a, x“ and 0

(2.3.1)

@(x)=

[‘I:2

a,

forx2O

for x = 0. I f x = e-c then m

m

(2.3.2)

V x E (0,I),

f ( x ) = (1 - x )

2 n-0

@(t)e-“ dt,

@ ( n ) Y= e 0

where E E (0, a),in the sense that v a n y of the quantities in (2.3.2) exists then the other two also exist and all are equal.

120

2 Tauberian theorems

Proof. a) Fix x c 1 and r E (x, 1). Assume thatf(y) converges on [0,1). Thus, there is a constant K such that for each k, lu,l < Kr'. Consequently,nlirn ( @ ( n ) x " l = 0 since x/r -= 1 and +m

m

Therefore (1 - x) 2 @(n)x" converges tof(x) since n-0

n

n-1

2 ukX&= 2

I-I

2

@(k)(xk-xk+')+@(n)x"=(l-x)

k -0

k -0

@(k)xk+@(x)x".

k -0

m

A similar argument works to prove that f(x) converges on [0,1) if (1 - y ) 2 @(n)y" n-0

converges on [0,1). dt = (1 - x)x", where x = e+. Then if E

1

@(t)e-"dt exists,

0

q.e.d. 2.3.3 Frobenius' theorem: C d r o summable implies Abel summable. Observe that

1

$SJ= n+l I-0

J

$(l--&)uk,

wheresI= &2 -0 ak

k -0

(cf. (2.1.31)and (2.1.32)).We write m

2 an=S,

(Cgl),

n-0

if

m

(cf. (E1.3.5) for the case x = 0); and in this case we say that 2 anis Cesdro summabIe to 0

S. Similarly, in light of Proposition 2.3.1,we write

2.3 Classical Tauberian theorems

121

m

and say that

2 a,, is Abel summable to S, if 0

m

lim

x-b 1-

2 aRx"= S. R-0

We give the following improvement of Proposition 2.3.1. Proposition 2.3.3. (Frobenius) If2 a, = S, (C, I), then

can = S, ( A ) .

Proof. Let E = 1/R in (2.3.2). With the notation of Proposition 2.3.2, we set 1 SR =R

m

@(r)e-'IRdt

and

s(x) = f@(t)dr, 0

0

and prove that if (2.3.3)

lim s(R)/R = S R-bm

then (2.3.4)

lim SR = S.

R-bm

By hypothesis, there is K > 0 such that for all R > 0, Is(R)I < KR. Because of this boundedness and since

we compute (2.3.5)

'I

S --

R-R

s(t)e-'IRd(r/R) =

0

R 0

Now, Is(rR)e-'/RI < Kt e-' EL'(O,00). Consequently, we use (2.3.3) and the dominated convergence theorem on (2.3.5) to compute (2.3.4). q.e.d.

2.3.4 Littlewood's Tauberian theorem. The first deep Tauberian theorem is due to [Littlewood, I ] (1910). We shall deduce it from Wiener's Tauberian theorem in the form of Theorem 2.1.5. [Wiener, 7, pp. 105-1061 first showed that iff(x1 = 2a,x" converges on (0,l) andf(1-)

= S then

'i

S=$F~,

@(y)dy ("@"

as in(2.3.1)); he used

his general Tauberian theorem to establish this fact. We present this part of his proof, up to but not including the application of his Tauberian theorem. The second part of

122

2 Tauberian theorems

his proof is Hardy's Tauberian theorem (1909) which states that if a, = O(l/n), n +. co, a n d 1 a, = S, (C, l), then 2 a, = S, e.g. [Hardy, 1, Theorem 631. At this point our proof follows the more direct route observed by [Glasser, 11. On the other hand it is no surprise that Wiener proceeded as he did. In fact, H a r d y had noticed that, in view of Proposition 2.3.3, the easiest way of finding non-convergent series which are Abel summable is to consider Cesiiro summable series (that are not convergent); this, coupled with his (Hardy's) Tauberian theorem, led him to the problem which Littlewood solved by means of the following result. Clearly, Theorem 2.3.1 includes Hardy's Tauberian theorem as well as Tauber's.

Theorem 2.3.1. (Littlewood) Iff(x) (2.3.6)

a,,=O(l/n),

n -+

=

2a,x" conuerges on [O,l),f(l-)

= S, and

m,

then 2a, = S. P r o o f . a) We use (2.3.6) to calculate

+

$7). N+1

Clearly, by expanding e-,IN as a power series,

(2.3.8)

5 0-1

1 - e-n/N

n

r-1

1 n -+-+N 2!N'

" 1

n2 3!N3

J-1

Since {e-"lN/n:n = N + 1,. ..} is a decreasing sequence we can apply the integral test to m l obtain a bound for 2 - e-"IN independent of N. Nfln Thus, from (2.3.7) and (2.3.8),

(2.3.9)

I:

Setting x = e-'lN we apply our hypothesis,f(l-) M such that

(2.3.10)

V x E [0, w),

m

2 a,, - 2 a,,e-,,IN

3 J > 0 such that V N > 1,

0

= S, to

(2.3.9) and obtain a constant

Ip.16 M .

b) Define @(x) as in (2.3.1). Then from Proposition 2.3.2, V e E (0, a),

f(e-') = e

1

@(r)e-"dt;

0

123

2.3 Classical Tauberian theorems

and so if we let E = e-y we have

I

I Q

OD

f ( x ) = e-y

V y E R,

@(t)e-rc-ydt=

0

@(ez)e-e('-y)e(z-y)dz,

-m

where x = e-'. Thus, by Exercise 1 . I . 1 a.i), (2.3.1 1)

S = lim

f

@(e')g(z- y ) dz = lim Y * I-@), Y-tQ

y - r m -m

where g(y) = r ( l vanish. c) We see that

+ iy)

(2.3.12)

g,(x)dx= 1

lim a+m

and Y(z) = @(e'). Recall that the gamma function doesn't

R

for

(e.g. Exercise 1.1.1 a$; and note that Y E Lm(R)-y (2 (2.3.13)

lirn Y * g&)

10). Finally we show that

pointwise a.e.

= Y(z),

n-r m

To do this, write

I W.Z)

- 'y

* gn-(z)l

G

J

IW.Z)~ Ig-(y) - gn-(y)Idy

R

+ j Ig.-Cv)l I W) - 'yo, - Z)ldY. R

The first integral tends to 0 by (2.3.12) and the fact that r(1) = 1. The second integral tends to 0 for almost all z since Y is continuous a.e. and since modulo some trivialities, we only have to worry about small neighborhoods of the origin. Thus, in the terminology of Theorem 2.1.5, L(x) = Y ( x ) and L = 1 ; and obviously (2.3.1 l), (2.3.12), and (2.3.13) are the hypotheses required to apply Theorem 2.1.5. Consequently, '+OD lim Y(z) = S (recalling again that gJ0) = r(1) = 1); and so by the definition of Y,

2 a, = S. q.e.d.

Remark 1. Theorem 2.3.1 is sharp in the sense that if 6, tends to infinity then there is a non-convergent series which is Abel summable and which satisfies the condition, Inaal < bn.

124

2 Tauberian theorems

2. [ K a r a m a t a , 11 (1930) gave a direct proof of Theorem 2.3.1 which depended on

Weierstrass' approximation theorem and polynomials with variable, exe-cx(cf. Remark 2 after Theorem 2.1.4). As in Wiener * s proof (that we referenced above) of L i t t 1ewood's theorem, K a r a m a t a was able to complete his proof by an application of H a r d y 's Tauberian theorem. [W ieland t, 11(1952) was able to carry out K a r a m a t a's approximation technique internally without using H a r d y ' s result. [Delange, 11 and [Korevaar, 11 have also given an elementary proof of Littlewood's theorem; and [Ingham, 11 has made the interesting suggestion that Littlewood's original and technically difficult repeated differentiation procedure and K a r a ma t a's approximation procedure do indeed have a non-trivial relation. 2.3.5 Slowly oscillating functions and Pitt's pointwise conclusion to Wiener's Tauberian theorem. In 1913, [Landau, 11 and H a r d y and L i t t l e w o o d generalized Theorem 2.3.1 to read : $f(x) = a,,x" conoerges on [0,l),f( 1-) = S, and

(2.3.14)

3 Ksuch that V n,

nu,, 2 -K,

then 2a,, = S , e.g. [Wiener, 7, pp. 108-1111. Actually, L a n d a u ' s condition was even weaker than (2.3.14) and perhaps led to R. Schmidt's (1925) notion of a slowly oscillating function. Q :R + C is slowly oscillating if V e > 0 3 Nand 3 S > 0 such that

Ix-yl<S

V 1x1 > Nand V lyl > N,

l@(x)-@(y)l<e.

If Q E C,(R) is uniformly continuous then Q is slowly oscillating, and there are discontinuous slowly oscillating functions. If (2.3.6) is satisfied and Q is defined by (2.3.1) then 9 is slowly oscillating. In fact, if y > x and we write y = rx where r > 1 then

and the right-hand side is estimated by K(1ogrx - logx) = Klogr when x is large. Also observe that iff(x) = 2a,# then the conditionJ(1-) = 0, is written as

I ,

m

(2.3.15)

lim y+m

@(xy)e-"dx=O

because of (2.3.2) (where Q is defined in (2.3.1)). With this background we state R. Schmidt's (1925) generalization of Theorem 2.3.1, e.g. [Wiener, 6, pp. 36-39]: let Q E L"(R) be slowly oscillating and assume that (2.3.15) h o l h ; then (2.3.16)

lim Q(x) = 0. .r+m

2.3 Classical Tauberian theorems

125

It was natural to try to generalize S c h m i d t ' s result by replacing the function g ( x ) = e-x in (2.3.15)by an arbitrary element f E L'(0, a).This attempt fails. In fact, if m

cp(y)=

jx'Yf(x)dX=O 0

and we define @(x)= x i y ,then lim y-m

f

f(x)(yx)"dx = 0,

whereas lim @(x)# 0. Note that if x = eY then x+m

J

x "f(4d x = 8 (y),

0

where g(y) =f(eY)ey. We are now in a position to realize, because of Theorem 1.1.3 and Theorem 2.1.5,that we really couldn't expect this generalization to hold. What is true is the following Tauberian theorem which is a corollary to Theorem 1.1.3.The "pointwise" conclusion is due to [Pit t, 11. Theorem 2.3.2. Let 0 E L"(R) be slowly oscillating and assume that $= cp E A ( A ) never vanishes. If

(2.3.17)

lirn f * @(x)= r

x - +m

1f(t)dt R

then

(2.3.18)

lirn @ ( x )= r. x-tm

Proof. Given E > 0, and choose N and 6 as in the definition of a slowly oscillating function. We see that V

X,

@(x)- g

* @(x) = -6lj:.- a n

[ @(x)- @(x- Y ) ]dY,

1 where g = 3 ~ r - a i z . a / z i . Thus,

v x E [-2N, 2N]",

I@ ( x )- g * @(x)l < 8.

Consequently, since xlim g * @ ( x )= r [ g = r, by Theorem 1.1.3 (or Theorem 2.1.5), +m we conclude with (2.3.18). q.e.d.

126

2 Tauberian theorems

Since the above proof requires no special properties of R, and because our definition of a slowly oscillating function can obviously be made on G, we see that the statement of Theorem 2.3.2 is valid when “R” and “FI” are replaced by “G” and “r”. The conclusion to Theorem 2.3.2 is what one would expect in a generalization of Theorem 2.3.1. The hypothesis that @ is slowly oscillating is the Tauberian condition; and (2.3.17), as a convolution, represents an hypothesis concerning information in the average, and, so, corresponds to the hypothesis, f(1-) = S, in Theorem 2.3.1. Recall Exercise 2.1.3b which is the Abelian theorem to which Theorem 2.3.2 addresses itself. 2.3.6 A remark on Tauberian theorems. Wiener’s Tauberian theorem, Theorem 2.1.5, relates summability methods instead of relating a summability method such as (2.3.17) with pointwise convergence such as (2.3.1 8). Thus Theorem 2.1.4 (or 2.1.5) has the form : if a function @hasa mean of a certain type at infinity and if the growth of the function is properly restricted then the function has a large class of means at infinity (cf. Section 2.1.10). There is a large literature on Tauberian theorems and we list the following classics: [Bochner and C h a n d r a s e k h a r a n , 1, Section 17 of Chapter 1 and Chapter 6; Hardy, 1, Chapters 6,7, and 12; Pitt, 2; Widder, 1, Chapter 5 ; Wiener, 6; 7, Chapters 2 and 31.

2.3.7 C(s) and the fundamental theorem of arithmetic. We shall devote the remainder of Paragraph 2.3 to the study of the distribution of primes; our approach shall be onesidedly Tauberian (sic). Since analytic number theory is so widely advertised we shall show more restraint than usual with historical and motivational notes for this topic.

Proposition 2.3.4 is an analytical statement of the fundamental theorem of arithmetic. Let P denote the set (2, 3, 5 , . ,., 2127-1,. ,., 21279-1,. ..} of primes. Proposition 2.3.4. (Euler) For each s = u + iy E C, u > 1,

n-1

Proof. Fix s. Setting

and expanding the product, we obtain

R€S,

where S, consists of those positive integers which have no prime factor greater than x.

2.3 Classical Tauberian theorems

127

m

Thus, if S = n-1

1 /ns,

where R, consists of those positive integers which have at least one prime factor greater than x. Since n > x if n E Rx we have

and so

5 I)-1

1

n

-= ns

(1 +p-s+p-zs+...).

PEP

Consequently, we obtain (2.3.19) by properties of the geometric series. q.e.d. 2.3.8 Number theoretic functions. Define

v x 2 0,x(x) = card{p E P:p Q XI, V nEZ+,

A(n)=

logp for n = pkfor some p E P and positive integer k, for n # pk for any p E P and positive integer k, 0

and

Proposition 2.3.5.

(2.3.20)

*(4

lim -< lim x+m

X

x ( x ) log x

x+m

Proof. i) We first prove that

To do this, write

X

- x(x) logx d

Q lim x+m

X

- *(x)

lim-. x+m

X

128

2 Tauberian theorems

Now, for a givenp Q x, logp is added to itself k times wherep
[:;].

< x < p”

Taking the log of “p” Q x < p ” + l ” we obtain k < logx < k + 1, i.e. k = logp gives (2.3.21). ii) To prove the first inequality of (2.3.20) we have from (2.3.21) that $(x) 6

2

+

l.

This

log x = x(x) log x.

PCX

Dividing by x yields the result. iii) If x > e, define y = x/(logx)’ (and so x > y). We compute

Therefore, x(x)logx X

ylogx

<-

3

1

<-+log x

+

$(x)logx - 1 -xlogy logx

+ Jl(x)logx x

1 logx - 2loglogx

1 *(x) x 1 - (2 log log x)/log x ’

and the last inequality in (2.3.20) follows. q.e.d. 2.3.9 The prime number theorem. The prime number theorem as erts th t X

(PW

n(x)= log x +

o(G),

x

+

*.

Because of Proposition 2.3.5, (PNT) will follow from Theorem 2.3.3 (prime number theorem) (2.3.22) $(x)=x+ o(x), x + m.

In order to prove (2.3.22) we need Proposition 2.3.6, Proposition 2.3.7, and Proposition 2.3.8. We refer to [Donoghue, 1, p. 2431 for another and interesting proof of Proposition 2.3.6b. Also, we balance the devious trick in the proof of Proposition 2.3.6b by referring to [Titchmarsh, 1, p. 391 for an exposition of Hadamard’s basic idea. 23.10 Fundamental properties of ( ( 8 ) . Proposition 2.3.6. a) (Riemann) [(s) is analyticfor Q > 0 except for apole of order 1 and residue 1 at s = 1.

2.3 Classical Tauberian theorems

b) (Hadamard andde la VallCe-Poussin) V y,

C(1

+ iy) # 0.

Proof. a) Clearly, if D > 1,

The expression (2.3.23)converges for D > 0, s # 1. If oo > 0 is fixed, then V a P a,, and V

X E [1,00),

x - [XI

1

-<-; IXS+11

xao

and so the integral in (2.3.23)converges uniformly, and this yields the analyticity. The pole-residue information is obvious from (2.3.23). b. i) By the Taylor series expansion of log( 1

( is)2 :+.

log 1 - -

=-

n-1

Thus, from (2.3.19),

and so

By (2.3.24)

+ z) we have

129

130

2 Tauberian theorems

Hark! 3 1 1 - + msl+ -cos21= - (1 + C0sI)Z 2 0. 4 4 2

Thus, logf(s) B 0, and so 4

(2.3.25)

1

1.

~ ( a - l ) a[ - 1 ( ~ )IC(a+2iy)l ~ ~ ~ )-,a> ' a-1 ~ ~

iii) We assume ((1 + iy,) = 0 and obtain a contradiction by proving that the left-hand side of (2.3.25) converges to a finite limit as a tends to 1. Note that yo # 0 because of part a). From (2.3.23) we compute lim [(a - l)C(a) l3 = 1

0+l+

and

lim I(a

a+1+

+ 2iy0)l = l((l + 2iyo)l.

Also

since yo # 0. q.e.d. 2.3.11 Lambert series. One would not expect a result about prime numbers to be proved completely on an E - 6 trip. It is time to count. A Lambert series has the form

m

Proposition 2.3.7 a) If 2 any"conoergesfor IyI < 1 then 1

b) V x > 0, e-nx

m

2

- 1) = 1 -e-nx

(&I)

n-1

where

d(n) =

2 1. Jln

m

2 (logn - d(n)) e-nx n-1

2.3 Classical Tauberian theorems

13 1

c) (Dirichlet)

where y is Euler's constant.

Proof. a) Formally, by cross multiplying,

-Y"

.

-y"+yZ"+yJ"+...

1 -y" m

+

Thus, (2.3.26) is 2 an(y" y2"+. ..); and by regrouping we obtain (2.3.27). n-1

These formal operations are legitimate if 2 any" converges for lyl < 1 ; in fact, this hypothesis obviously entails the absolute convergence of (2.3.26) which, in turn, allows for the rearrangement. b) If a, (2.3.28)

=

1 for each n, part a) immediately yields (for y

V x > 0, n-1

I - e-""

-

= e-")

2 d(n)e-"". .. - .

n

Now let a,,,= A(m). Fix a positive integer n = pk, where the product is the unique factorization of n. If mln and A(m) # 0 then m = p k ; consequently, by the definition of

A, (2.3.29)

2A(m) = logn. mln

b) follows when we combine (2.3.29) with (2.3.27) and (2.3.28). c) This is Exercise 2.3.4a. d) Because of b) we use partial summation on 2 (logn - d(n))e-"". Thus, using Stirling's formula, n! n n e - " G , n + (e.g. [Widder, 2, pp. 139-1401), and c) we compute

-

m

m

(2.3.30)

2

(logn - d(n))e-"" = (I - e-")

2

e-""(-2yn

m

Set y = e-x and calculate

m

m

0

1

ny" = 2(n + 1)y" - 1 - 2y"

n -1

= y/( 1 - Y ) ~ .

Thus, (2.3.31)

-2y(l - e-.)

+ O(n1I2)).

n-1

n = l

ED

2 n-1

Y =-2Y ne-"" = -2y 1-y ex-1

132

2 Tauberian theorems

Since lim(ex - I)/x = 1 we shall replace (ex - 1) by x in our final calculation. X-ba

In Exercise 2.3.4b we prove that

5)

1/2

(2.3.32)

3 K such that V y

E

[0,I),

(1 - y ) (log

Q

nl/'Yn G K .

Combining (2.3.31) and (2.3.32) with the right-hand side of (2.3.30), we obtain d). q.e.d. 2.3.12 Theorems of Chebyshev and Mertens. Define

v x 2 1,

H(x)=

2-

A(n) - 1 n

and

Proposition 2.3.8 a) (Chebyshev) $(x) = O(x), x b) (Mertens) QIi e L Q ( R ) . c)f * @&) = -2y+ O(e-('/')'), x +. 03, where

--f

oc).

d e-x exp (-e-3

f (XI = dx 1 - exp(-e-")' Proof. a) This is Exercise 2.3.4~. b.i) We first compute how many times i(p,m) a prime p divides m! (i.e., pf(P.m)lm!but pf(m ~) .+ 1 tm 9. In the set {l,. .., m} there are exactly [m/p] integers which are divisible byp, viz.

In the set {I,. ..,&} there are exactly [m/p2]integers which are divisible by p2, viz. (2.3.34)

(

p2, 2p2,.

[];

..,

p 2 ).

The set in (2.3.34) is a subset of that in (2.3.33). Let S,,be the subset of (1,. ..,m} each of whose elements is divisible by p" but not by p" l . Then card S,, is exactly +

2.3 Classical Tauberian theorems

and sop divides S,, exactly n ([;]-[+])times* Clearly, {I,. . .,m} = U S,, and {S,,:n b 1) is a disjoint collection. nal

Thus

a finite sum. ii) The unique factorization of m! is

n

m! =

pt(p*").

B E P .p=Z m

We compute (2.3.36)

2

logm! =

I [I]+

Setting - = (2.3.37)

m

2

E,,,

[F] 5 []:

2

r(p, m) logp =

pEP.psm

pe

P. u%

logp =

n-1

m

0 < E,, < 1, we have from (2.3.36) that m

$I=2 E,,A(n) + logm!. n = l

n-1

Applying Stirling's formula to the right-hand side of (2.3.37) yields

n = l

n

m

E,,A(n)+logm+0(1), n-1

From the definition of I(I(x),

and so by part a), (2.3.38)

$*)=logm+ n

m

0(1),

-+

m.

On the other hand, it is a result from calculus that (2.3.39)

2 :=

logm+0(1),

m

--f

m.

m

-+

m.

A(n).

133

134

2 Tauberian theorems

Subtracting (2.3.39) from (2.3.38) entails the desired boundedness. c) We first use Proposition 2.3.7d to compute

= -2y+ O(U"2),

u

-+

Of.

Let t = eY and u = e-x in (2.3.40). This yields part c). q.e.d. 2.3.13 Wiener's Tauberian theorem and the proof of the prime number theorem.

Proof. (cf. Theorem 2.3.3) i) Takefas in Proposition 2.3.8~. We evaluate e-x

(2.3.41)

Jf(x) dx = lim -m

-e

-=

e-xe-c-x

- lim

1 -e-c-x

x-,+m

x+-m

1 -e-c-x

+ye-'-' + e-2xe-c-* = lim

+-x

x+m

= lim (1

e-c-x

x+m

ii) Take E > 0. Integrating by parts we compute

so that by the change of variable y = e-* we obtain

s m

f(-y

y(lYta)d Y + is) = (iy + E) 1 -e-y e-y

0

m

= (iy

+ e)

+

yiy+e(e-Y e-2Y

+...) dy

0

= (iy

m

+ E)

e-nYylY+edy n-1

= (iy

+

" E) n-1

= (iy

+

1 nl~+.+~ fe-yyiy+edy 0

+ e + iy)t;(l + E + iy).

+ e-.)

= 1.

2.3 Classical Tauberian theorems

135

iii) We let E tend to 0. There is no problem about switching limits because of the smoothness off: Thus, (2.3.42)

f(-y)

= iyT( I

+ iy) [( 1 + iy).

We note that f never vanishes, the factor “iy” cancels the pole of [(l + iy) at y = 0 (this sort of thing would have to happen since f €L1(R)), and [(l + iy) never vanishes (Proposition 2.3.6). Hence, f never vanishes. iv) Also, m

lim f * Q H ( x )= -27 x-tm

f(y)dy. -m

v) Because of parts iii) and iv), we can apply Wiener’s Tauberian theorem. We could apply it in the form of Theorem 2.1.4 since (2.1.33) is obviously satisfied in our case. For technical convenience we apply it in the form of Theorem 2.3.2. We must check that QH is slowly oscillating. This is easy. Take S < 1 so that if 0 < y - x < 6 then QH(y) - aH(x)= 0 or

in the case that n E [ x , y ] . By Theorem 2.3.2, (2.3.43)

lim H(x) = x -tm

S T

- -2y.

vi) Clearly,

112

and, generally, (2.3.45)

ud(ff(u)

Since lim(H(u) + 27) = 0, the mean, u+m

+

+ 27) = r(H(r)+ 2y) -

0

(H(u) 2y)du. 0

+

( H ( u ) 2y)du, also tends to 0. Thus, (2.3.45)

136

2 Tauberian theorems

becomes

(2.3.46)

lim

r+rn

1

-t ( $ ( t ) - [t]) = 0

with the substitution of (2.3.44). We are done. q.e.d. We have paid our homage to Lambert series. Our proof of (2.3.22) showed that convergence of the “Lambert mean” implied the convergence of the Ceshro mean,

2.3.14 Perspective

Remark 1. As we’ve seen, Wiener’s proof of (PNT) required knowledge of the behavior of 5 only on the line c = 1. The first proofs of (PNT) required that C have no zeros in a region to the lefi of the line c = 1. Our proof generally follows the guidelines set in [Levinson, 3; Widder, 1 , Chapter V, Section 16; Wiener, 7, pp. 112 ff.]. As we’ve indicated in Paragraph 2.1, Wiener was led to his method of proof because of H a r d y and Littlewood’s result that (PNT) is equivalent to a Tauberian theorem concerning Lambert series, and because of their failure to give an independent proof of this latter result and thus obtain a proof of (PNT). Wiener first published his proof in [Wiener, 41, although it is smoother going in [Wiener, 71. 2. One of the most efficient means of proving (PNT) (and also the prime number theorem for primes in an arithmetic progression) is by means of I k e h a r a ’ s complex variable Tauberian theorem [ C h a n d r a s e k h a r a n , 1; I k e h a r a , 1; Widder, 1; Wiener, 71. A deep extension of I k e h a r a ’ s work in complex variable Tauberian theorems is due to [A gmon, 13. Essentially, with the added analytic structure, acomplex variable Tauberian theorem will describe the asymptotic growth of the coefficients a, (in (2.3.1), say), whereas the Tauberian theorems that we have proved only give k

information on the asymptotic growth of the partial sums, 2a,. 0

2.3.15 The prime number theorem and the Riemann hypothesis. The Riemann hypothesis, (RH), asserts that if r(s) = 0, for c > 0, then c = 1/2. It is not known whether or not (RH) is true. (RH) can be considered as a strong form of (PNT) because of the following well-known result, e.g. [Blanchard, 1, pp. 89-93]. Theorem 2.3.4. (RH) is valid ofor each E > 0

(2.3.47)

$(x) = x

(cf. (2.3.22)).

+ O(X”(’’~)),

x

-+

~0

2.3 Classical Tauberian theorems

137

Actually, the original H a d a m a r d and d e l a Vallte-Poussin (PNT) is equivalent to (2.3.48)

V m > 0,

$(x) = x

+ O(x/log'"x),

x

+

[Bombieri, 11. There are even more refined estimates than (2.3.48) and we mention it only as a prototype in which to compare (PNT) with (RH). 2.3.16 Tauberian theorems and the Riemann hypothesis. [Wiener, 61 was aware that (RH) is valid if and only if (2.3.49)

V c E (+, l),

V,.

= L'(R),

where

The prognoses in [Levinson, 1, p. 840; Wiener, 6, pp. 49-50; 7, pp. 136-1371 on the usefulness ofTauberian theorems to prove the validity of (RH) are negative, although nonetheless there have been several interesting results related to (2.3.49) that have been proved since Wiener's work; in particular, we give Theorem 2.3.5 and Theorem 2.3.6 below. These "prognoses" are associated with the numerical Tauberian problem of determining how fast one summability method converges in the conclusion of a Tauberian theorem if in the hypotheses (of the Tauberian theorem) we know how fast the given summability method converges. This problem is also called the Tauberian remainder problem and there is an extensive literature associated with it; we refer to [Beurling, 1, pp. 364-365; Ganelius, 2; K o r e v a a r , 1, Section 7 and Section 8; Lyttkens, 1 ; 21 (e.g. Exercise 2.3.5) and move on.

2.3.17 Salem's Tauberian characterization of the Riemann hypothesis. Theorem 2.3.5. ([Salem, 31) Dejine

Thefollowing are equivalent : a) (RH) is valid; b) v 0 E (+,1>, Vz, = L W ; c) V CT E (3,l) and V @ E L"(R), f,*@=O

-

@=O.

Proof. In light of Theorem 1.1.3 and Theorem 1.3.lc, it is sufficient to prove that for 0 E (i,I), V 7 E FI, JC(7)= T ( s ) ( ~ 2'-")C(s) (2.3.50) (recalling that r never vanishes).

138

2 Tauberian theorems

By the definition of r,

(the result being true if the positive integer n is replaced by any z E C for which Re z > 0) (cf. Exercise 1.4.44. Direct multiplication yields

(2.3.51)

V u > 1,

(1

- 2'-'))r(s)

.=

5 -.

(-:sa+

a-I

Now for each u > 0, {l/n' :n = 1 , . ..} decreases to 0 and so the alternating series 2(-1)"+ '/nu converges for u > 0. Consequently, by the fundamental convergence property of Dirichlet series, the right-hand side of (2.3.51) converges for u > 0. Therefore, because of Proposition 2.3.6a, m

(2.3.52)

VU>O,

r(s)(1-21-a))r(s)=1(-l)n+1( / ~ ' - ~ e - ' ' ~ d x ) . m 1

Now, x 4 - e-x E Ll(0,a)if u > 0 and 1 1 1 1 ()
V s = u + iy,u > 0,

r(s)(l - 2'-'))r(s) =

1 0

s-1

-dx = fu(y), 1 +ex

by the substitution x = P ; this is (2.3.50). q.e.d.

2.3.18 Beurling's functions and a spectral analysis problem. Analogous, although more complicated, results have been proved by [Beurling, 10; Levinson, 11. Because of our concern with spectral analysis problems we shall now comment on Beurling's result (cf. Remark 3 after Exercise 2.1.6). Let S be the multiplicative semigroup {x E R :0 < x < I}. If @ E L2(S)then V;, 1 ,< r Q 2, will denote the L'(S)-closed variety generated by {@(xy):yE S}.As is well known, each (continuous and normalized) character on S has the form ~ ( x=) Y , z E C. It is not difficult to check that a character x E L2(S)if and only if Re z > - 4 (cf. [Paley and Wiener, 1, Chapter 11, e.g. p. 301). The spectral analysis question of whether sets having the form V; contain characters is very complex and generally has a negative answer [Beurling, 7; Nyman, 1; Vretblad, 11. [Beurling, 91 proved the following

2.3 Classical Tauberian theorems

139

spectral analysis result with the restrictive condition (2.3.53): If @ E t 2 ( S )has the property

(2.3.53)

/

V x E S,

, @ ( t ) Idt > 0

0

n

then there is a character x E L*(S) n

V i which is not necessarily in V;.

14rc2

In order to give Beurling's (RH) theorem it is necessary to define a certain class

a G L"(S).Define a:(O,co) -+

x

H

[0,1) x-

[XI.

Clearly, VyyESandVxxE(O,co),

1c

,

6, has a countable set { 1, Y- :n= 1,. ..} of jump discontinuities. n Define

(2.3.54) V x

E (0,a), b(x) =

2 ayb,(x),

where card F < a, F G S, and uy E C ;

YEF

and assume that (2.3.55) 2 u,y=O. YEF

Consequently, if x > max{y:y E F}, then b(x) = 0. Therefore, we can finally define 93 as the vector space of functions b having the form (2.3.54) and satisfying (2.3.55). 2.3.19 Beurling's Tauberian characterization of the Riemann hypothesis. With these preliminaries, we are in a position to state Beurling's theorem.

Theorem 2.3.6 ([Beurling, lo]). (RH) is vulido

a=L2(S).

It is not too difficult to prove that (RH) is valid if =L2(S);we refer to [Beurling, 10; D o n o g h u e, 1, pp. 252-2341 for a proof. We outline the proof of the other and more difficult direction to show the use of (2.3.53).

140

2 Tauberian theorems

O u t l i n e o f P r o o f . WemustprovethatifBsLz(S)then [(ao+iy,)=Oforsome 0 0 E (3,1), Yo E fl. By the Hahn-Banach theorem there is 9 E Lz(S)\ (0) such that V ~ L2(S), E

I f(x) @(x)dx

= 0.

S

It is then easy to check that (2.3.56)

V b E Af and V y

E

S,

b(x) @(xy)dx = 0.

y S

Define V& 1 < r < 2, as above. Since W (2.3.57)

V b E Af and V Y

E

E L"(S), we

see that for any r E [1,2],

J' b(x) Y(x)dx = 0,

V;,

because of (2.3.56). We now prove that 9 satisfies the integral inequality (2.3.53). Assume that 9 = 0 a.e. on (O,y),y > 0. Choose z in the interval (y, min{l,2y}) and set b = zb, - yb,. Thus b E a,b = 0 if x > z, and b(x) = y when x E (y,z). Therefore 7,

0=

J b(x) @(x)dx = y J' @(x) dx, S

Y

and so 9 = 0 a.e. in the interval ( y , min { 1,2y}). We repeat the argument a finite number of times and see that @ = 0 a.e. on S,a contradiction. Consequently, we can apply Beurling's result, (2.3.531, which, when coupled with f2.3.57), yields that (2.3.58)

3 z = (ao- 1) + iy,,

1 Re(ao - 1) > - -, 2

such that V b E B,

1'

b(x)x z dx = 0.

S

1 Define b = 6 , - - by, and, by direct calculation, we obtain from (2.3.58) (for z = so - 1) Y

that VyYE(O,l),

[(SO) (l-y'o-1)=0.

SO

2.3 Classical Tauberian theorems

141

I

If so = 1 then b(x)dx= 0, which is not true. Consequently, we choose y E (0,l) such S

that yso- # 1, and so c(so) = 0. Hence, (RH) is not valid since uo > 112. '6

q.e.d."

Exercises 2.3 2.3.1 Tauber's theorem

Prove T a u b e r ' s Tauberian theorem directly. (Hint: Note that for each E > 0 there is N such that if n 2 N we have la,nl < 43,

and m

then for n >, N and the appropriate choice of x,

2.3.2 Hardy's Tauberian theorem Prove H a r d y ' s Tauberian theorem directly, e.g. [Edwards, 5, I, p. 861.

+

2.3.3 [(l iy) # 0 Use (2.3.22) and not the razzle-dazzle of Proposition 2.3.6b to prove that ((1 for all y E F1. (Hint:Assume [(l yo) has a zero of order m. Then

+

lim s+l+lYg

U s ) (s - (1 + iy,)) ((s)

+ iy) # 0

= m.

Compute

(cf. (2.3.23)), and obtain a contradiction by estimating I['(s)(a - l)/[(s)l). 2.3.4 Number theoretic estimates

a) Prove Proposition 2.3.7~.This is a well-known elementary and ingenious number theoretic estimate. We refer to [ C h a n d r a s e k h a r a n , 1, pp. 53-54].

142

2 Tauberian theorems

b) Prove (2.3.32). Note that [Widder, 1, p. 2311 proves Proposition 2.3.7d in a similar way but with a final appeal to an Abelian theorem. c) Prove Proposition 2.3.8a. (Hint:In light of the proof of Proposition 2.3.5b it is x(x) logx sufficient to find a bound for .This is done in [Chandrasekharan, 1, pp. X

67-68]). 2.3.5 A Tauberian remainder theorem Let f E L1(R) have the property that L(f)-’is analytic in a region R = {s = + iy : IyI < a, a > O} (“L” is the Laplace transform operator), and let @ E Lm(R).Assume that 3 K,, k > 0 such that V s E R, IL(f)(s)-’I< Kl eklsl

(cf. Exercise 2.2.3b) and that 3 x,, E R such that V x > x, 3 K2> 0 and @(x) + K2x is non-decreasing.

If there is a >

i r which f * @(x) = O(e-“%),x --f co, prove that

e.g. [Ganelius, 11.

2.4 Wiener’s inversion of Fourier series 2.4.1 Remarks on Wiener’s inversion of Fourier series. We shall now give W i e n e r ’ s original proof of his theorem on the inversion of Fourier series, viz. Proposition 1.1.5b. Wiener proved Proposition 1.1.5b in [Wiener, 6, p. 14; 7, p. 911. Theorem 2.4.1 b, below, played a major role in this proof; and the property of “local membership” which is used in Theorem 2.4.1 is a valuable concept in harmonic analysis. The approximate identity results, Proposition 1.1.4 and Theorem 1.2.1, are needed in parts b) and c), respectively, of Theorem 2.4.1; and so it is important to note that the proof of Theorem 1.2.1 does not require Proposition 1.1.5b. We also mention W iener’s extension, Proposition 1.2.5a, of Proposition 1.1.5b which was essential in the proof of the generalization of Wiener’s Tauberian theorem that we gave in Theorem 1.2.3. Proposition 1.2.5a tells us that ifZ$ G int Zrp, where $,rp E A ( ~ ) , then rp/$ E A(T), where cp(y)/Jl(y) is dejined to be 0 if&) = 0. ([Wiener, 7, p. 921 and [Mandelbrojt, 1, p. 501 have given a strange acknowledgement of this result to Denjoy and Luzin; in fact, the papers by Denj oy and Luzin that they cite deal with the Denjoy-Luzin theorem, e.g. [Kahane and Salem, 4, Chapitre VII], and [Wiener, 7, pp. 91-92] actually gives a proof of the Denjoy-Luzin theorem after his proof of Proposition 1.2.5a).

2.4 Wiener's inversion of Fourier series

143

Recall from Paragraph 2.1.1 1 that Wiener's original proof of his Tauberian theorem, Theorem 1.1.3, in 1928, did not use Proposition 1.1.5b and Proposition 1.2.5a, which are intrinsic properties of A(r),but did assume added integrability conditions such as (2.1.33). 2.4.2 Local membership. Let I s A(f) be an ideal. A function rp: f locally at y E r (resp. at infinity) if (2.4.1)

3 $,

E I and

v 1 E N,, (resp. 3 $,

--f

C belongs to I

3 N,,an open neighborhood of y, such that

$4) = rp(4 EI

and 3 K c f ,compact, such that

V ~ E K " , $m(d)=V(l));

in this case we write cp E Ilo,(y) (resp. rp E Il,,(w)). Thus, if G is discrete and I C A(f) is an ideal then

v cp E 40,

rp E I , o c ( w ) .

Theorem 2.4.1. Let I c A(T) be an ideal and let rp : f --f C be a function. a) Zfcp has compact support then cp E I,o,(w). b) Zfcp E I,,,(y)for each y E f and ifcp E Zloc(m) then cp E I. c) Given cp E A(r).Zfq E Iloc(y)for each y E f and if1 is closed then rp E I.

Proof. a) Take K = suppcp and $, = 0 in (2.4.1). b) Since cp E I,,,(w) there is a compact set L and an element $ E I such that cp = on L". Setting 8 = cp - $ we see that K = supp8 is compact, that 0 E Z l O c ( ~ ) by , part a), and that 8 E Z,,, ( y ) for each y E r. Using this information we'll show that 0 E I; and, thus, cp E I. For each y E Kchoose I), E Iand an open neighborhood N , of y for which $, = 8 on N,. By the compactness, let { N l , ...,N,}E { N , :y E K} cover K and, for e a c h j = 1,. . .,n,let K,

c N, be a compact neighborhood such that K G UK,.

If N, = N y we write I), = $,.

I-1

Because of Proposition 1.1.4 we choose {cp, : j K, and q, = 0 on N ; . Since I is an ideal, cp, $, E I.

= 1,.

. .,n ) c A(T) for which rpj = 1 on

We now note that (2.4.2)

V y e r a n d V j = 1,...,n,

~~,(Y)$,((Y)='PJ(Y)~(Y).

This follows easily by checking the various possibilities.

144

2 Tauberian theorems

Define (2.4.3)

8, = e[i

- (1 -

(1 - p2)...(1 - (~31,

and observe that 8, E Z because of (2.4.2) (the “1’s’’ cancel when we compute the right-hand side of (2.4.3)). Finally we show that 8 = B1. In fact, if y # K then 8(y) = 0 and so &(y) = 0 by (2.4.3); and if y E K then 1 - cp,(y) = 0 for somej, and so the value of the expression inside the bracket (of (2.4.3)) is 1. c) Choose the directed system {cp,} c A,(r)of Theorem 1.2.1. for each y E r and for each a. Then it is clear that cppa E Zy(,) Because of a) and b), we have cpcp, E 1; and so cp E I by Theorem 1.2.1. q.e.d. There are generalizations of Theorem 2.4.1, e.g. [Katznelson, 5, Chapter VIII, Section 5.2; Loomis, 1, Section 25E] and Exercise 2.4.5a. 2.4.3 Wiener’s proof to invert Fourier series. Wiener used the following example to prove Proposition 1.1.5b. Example 2.4.1. Given!= cp E A(T) andassume that

T

1 We shall show that - E A(T). We combine the series expansion, 1/(1 + JI) = 1 - $ cp

J12 - 4b3 + ..., and (2.4.4) to compute for $(y) I

=

+

(4 elnTthat 2’ff(0)

1

Thus, by (2.4.4) again,

This example is obviously valid when T is replaced by any compact r. We can now give Wiener’s proof of Proposition 1.1.5b. It should be noted that we use Theorem 1.2.2a, a fact which Wiener proved [Wiener, 7, pp. 88-90], and that the proof of Theorem 1.223 did not in any way employ Proposition 1.1.5b. Wiener realized the importance of Theorem 1.2.2a in [Wiener, 6, pp. 12- 15J ;also, compare the proof of Theorem 1.2.2a contained in [Zygmund, 1, pp. 141-1431.

2.4 Wiener’s inversion of Fourier series

145

Theorem 2.4.2. Let T be compact and take cp E A(T). a) If (p(yo) # 0 then there is I(/ E A(T) such that cp = I) on a neighborhood of yo and 1I* E 40. b) I f c p never vanishes then 1/cp E A(T). Proof. a) Let yo = 0 and define (2.4.5)

8,(Y) = d o )

+

*U(Y)(cp(Y)

- cp(O)),

where {1+9~}c A,(T) is the directed system of Theorem 1.2.2. Choose q = lcp(0)1/3;and apply Theorem 1.2.2a to find a such that l$,(cp For this a set I(/ = 8,.

- cp(0))liA < q.

Clearly, since the modulus of the “0th Fourier coefficient”, I /$=(y)(cp(y) - cp(O))dyl, r

is bounded by ll$u(cp - cp(0))/lA,we have

On the other hand it is immediate from (2.4.5) that (2.4.7)

ll*llA

< I 4 v l + tl.

Combining (2.4.6)and (2.4.7) we obtain

From the definition of $, y5 = cp on a neighborhood of 0; and because of (2.4.8) and 1 Example 2.4.1, - E A(T).

*

b) For each y E T choose t,by E A(T) such that $y = cp on a neighborhood of y and 1 1 1 - E A(T). Thus, - E A(T),,,(y) for each y E T and so - E A(T) by Theorem 2.4.la,b. *Y

cp

cp

q.e.d. 2.4.4 Wiener-Levy theorem. In an important, and in some sense elementary, paper, [Ltvy, 11 proved the following generalization of Theorem 2.4.2b for the case T = T. Theorem 2.4.3 is the Wiener-Lkvy theorem. Theorem 2.4.3. Let N E C be an open neighborhood and let F : N function. Take cp E A(T). a) I f K E T is compact and cp(K) E N then F o cp E A(T).

+ C be an analytic

146

2 Tauberian theorems

b) Ifr is compact and ~ ( c fN ) then F o cp E A ( r ) . c) IfF(0) = 0 and cp(r) c N then F o cp E A(r). Proof. i) We shall prove that for each 1E K there is $ A a neighborhoodof A, i.e., F o cp E A ( f ) , J A ) . Let z I = ~(1). By hypothesis,

E A(T) such

that F o cp = $ A in

m

(2.4.9)

F(z) = F(z,)

+2

-

C ~ ( Z zA)”

n-1

in some neighborhood of zA. By H a d a m a r d ’ s formula for the radius of convergence of an analytic function we have that 3 r E R and 3 K > 0

such that

V n 2 0, JcnIQ Ken‘.

+

Take E > 0 such that r log&< 0. Choose the de la VallCe-Poussin kernel {$a} about 1,as in Theorem 1.2.2, so that for the appropriate a, (2.4.10)

II(cp

-cp(4)$all,4

< 6.

Thus, from (2.4.10), $A

= F(zA)

$a

+ n2- 1

cn

[(cp

- cp(l))

and because of (2.4.9) and the fact that $a = 1 in a neighborhood of 1,we have that = F o cp in a neighborhood of 1. ii) Part a) follows from part i) and Theorem 2.4.la, b. iii) Parts b) and c) are now easy. q.e.d.

2.4.5 Other proofs of the Wiener-IKvy theorem. C a l d e r 6 n has given a slick classical proof of Theorem 2.4.3 for the case o f f = T which is independent of Theorem 2.4.1 [Zygmund, 2, I, pp. 245-2461. A soft proof of Theorem 2.4.2b, which tries to skirt around classical considerations and maximal ideal spaces is found in [Eisen and Gindler, 11; this proof should be compared with the proof of Proposition 1.1.6. There is also the proof by [Carleman, 1, pp. 67 ff.] which dates back to 1935 (cf. M a n d e l brojt’s introductory remarks in [Akutowicz, 11). 2.4.6 Generalizations of the Wiener-Lkvy theorem. Remark. In 1939, [Gelfand, I] proved (among other things) the analogue of Theorem 2.4.2b for a commutative Banach algebra with unit (cf., the relevant introductory remarks in [Cohen 41). Then, in [Gelfand, 21 (1941), he proved the analogue of the

2.4 Wiener’s inversion of Fourier series

147

Wiener-Lkvy theorem for commutative Banach algebras with unit a t the end of what is a fundamental exposition on Banach algebra theory (an English translation of this paper appears as the appendix in [Mackey, 11). Segal [Katznelson, 5, Chapter VIII, Section 3; Mirkil, 2, pp. 13-14; Segal, 31 extended G e l f a n d ’ s Wiener-LCvy result to commutative Banach algebras without unit. The Wiener-LCvy theorem has since been generalized to locally analytic operations on commutative Banach algebras ; this latter situation includes analytic functions of several complex variables and was devised to deal with “the square root” and “the logarithm” operations (cf. Exercise 2.4.4~).The main work in this program is due to S h i l o v (1953 and 1960) and [Arens and C a l d e r b n , 13; and expositions are found in [Gelfand, R a i k o v , and Shilov, 1, Section 13 and Chapter IX; H o f f m a n , 21. 2.4.7 The maximal ideal space of M ( G ) and the extension of the Wiener-Levy theorem to B(T). With Theorem 2.4.3 and Section 2.4.6, with its list of generalizations (of Theorem 2.4.3), it is natural to ask if the analogue of Theorem 2.4.3 is true for B(T) 1 instead of A(T). In particular, since - E A(T) if (D E A(T) never vanishes and r is comcp

1

pact, is it true that - E B(T) if cp E B(T) satisfies the condition inf {Icp(y)l

:y E

cp

and

r}> 0

r is non-compact ?

In general the question is answered in the negative and this is due to the complicated structure of the maximal ideal space, M(G)“‘, of M(G), when G is non-discrete; there are general characterizations of M(G)”’dating from S r e i d e r (1948-1951), e.g. [ D u n k l and Ramirez, 1, Chapter 1; Hewitt, 2, Chapter 3, Section 3-Section 51, but a good deal of mystery still surrounds such aspects of M(G). [Pitt and Wiener, 11 gave the I first example of a function cp E B(R) for which inf {I(~(y)l : y E A} > 0 and - 4 B(R); cp

there was some question as to the details of their proof, but this is settled in [Pitt, 2, pp. 104 ff.]. The problem of finding such examples was pursued by S r e i d e r (1950), [Hewitt, I], and [Williamson, I ] ; and is closely related to the fact that M(G), G non-discrete, is not self-adjoint, e.g. [Katznelson, 5, Chapter VIII, Section 9; Rudin, 5, Section 5.31. 2.4.8 The Wiener-Pitt theorem. There are some positive results of the Wiener-LCvy type for B(T). Essentially, the analogue of the Wiener-Ltvy result for B(T) is true for each p E M(G) that has a trivial continuous singular part, e.g. [Segal, 3, pp. 99-1001. Classical contributions to this problem are due to [Beurling, 1, pp. 356-358; C a m e r o n and Wiener, 1; P i t t and Wiener, I ] ; and the following result was proved in [Pitt and Wiener, 13 with the hypothesis that inf {Ij2(y)l : y E r}> 0.

+ +

Theorem 2.4.4. Given p = pa ps p‘, E M(G), where pa is the absolutely continuous part, ps is the singular continuous part, and p’,is the discontinuous part. Assume that j2

148

2 Tauberian theorems

1

Then = E B ( r ) .

P

Proof. i) Since M(G) is a commutative Banach algebra with unit, we know that p is invertible if and only if

P‘ E M G ) ” ,

(p’,p> f 0.

In ii) we consider the case that ( p ’ , f ) # 0 for some YEL1(G)E M(G), and in iii) we suppose ( p ’ , f > = 0 for each f E L’(G);consequently, in this latter case, p> =(p’, px>

+ (p’,pd).

We observe, from elementary considerations, that l(P’,Ps>l 6 IIPslI1

and m

(p‘, pd) =

2 a,(x,xI), I-

3‘

1 m

m

1

1

where x : G + C is a character, pd = 2aJ6,,, and 2 laJl c co. ii) Assume (p’,f) # 0 for somef E L’(G). We shall find y V v E MG),

E

r such that

(p’,v> = ~XY),

and thus (p’, p) # 0. By the L1-theory, there is y E r for which ( p ’ , g ) = i ( y ) for each g E L1(G). Since L’(G) is an ideal in M(G), v * f E L1(G), and so

f(r) (P’,v> = ( P c ’ , f >(P’,v > = W,f* v > = (f* V

) W =

PO)XY).

iii) From part ii) and the discussion in part i) it is sufficient to prove that ( p ’ , p , ) + ($,pd) # 0, and this follows, again from the discussion in part i), once we verify that /(p’,pd)l 3 inf{lfld(7)l : y E

r).

Kronecker’s theorem tells us that for each E > 0 and each n there is A E r such that V j = 1,.

..,n,

I(x,

XI)

- (1,XI) 1 < ~ / ( 42 la&

(We shall prove Kronecker’s theorem in Section 3.2.12 after recording some necessary facts about the Bohr compactification in Section 3.1.16 which will allow for a very quick proof.)

2.4 Wiener's inversion of Fourier series

Choosing n for which

149

2 la, I < 814, we obtain

n +1

1

m

(p', pd) -

2

a j ( k x l ) (< & *

J-1

Since this inequality is valid for all E > 0 (where A varies with E ) , we obtain the desired inequality. q.e.d. 2.4.9 The homomorphism problem. Note that iff = cp E A @ ) then (pa&) = cp(q+ b) is of the form cp o u E A@), where a,b E FI, a # 0, and u(y) = ay + b ; in fact, 2 = qab where g(x) =

eiXb/"f(f). Besides proving Theorem 2.4.3, [Levy, llalso posed the la1 following problem (for the case rl = r, = T) : find all functions u :r2-+ rl such that

(2.4.1 1)

v cp E A(rl),

cp o

E A(r,).

Clearly, (2.4.11) defines an algebra homomorphism, and we refer to L 15 v y * s problem as the homomorphism problem. This problem has been solved in the following works : f o r r , = T z =FIby[BeurlingandHelson, 1](1953);forrl=r, = T b y [ L e i b e n s o n , I ] (1954) with an assist by [Kahane, 1; 21 (1955); for rl = r 2= Z by [Rudin, 11 (1956); for the general case, after further contributions by H e l s o n and R u d i n , by [Cohen, 2; 3](1960). Expositionsofthe homomorphismproblemarefoundin[Cohen, 5 ; Kahane, 7; Katznelson, 5, Chapter VIII, Section4; R u d i n , 5, Chapters 3 and 41. Naturally, there is an homomorphism problem for quotient algebras, A(,!?), and spectral synthesis can play a role in this case, e.g. [Kahane, 13, Chapitre 1x1. With quotient algebras in mind we give the following result in a fairly general setting.

Proposition 2.4.1. Let X I , i = 1,2, be complex semi-simple commutative Banach algebras x ) (x',y j}, with maximal ideal spaces X p = {x' E X,' \ (0) : V x,y E X i , ( x ' , x y ) = i= 1,2. a) Let k: X , -+ X , be an algebra homomorphism. There is a function u : X y Xp U(0) c X i such that i f x ; E XY then ( X I ,

(2.4.1 2)

V

XIE

XI,

(u(x;),x;) = (xi, A ( X I ) )

(i.e., the right-hand side of(2.4.12) defines a). b) Let X1 = X , = X and let R :X --f X be an algebra automorphism. Then u, defined by (2.4.12), is an homeomorphism X" --+ X"', where X" has the induced u(X',X ) topology. c) Let XI = X , = X and let u :X" + X"' be an homeomorphism. u is defined in terms of an endomorphism A :X + X o VXEX,

x o a ~ X ,

where x o u : X" + C is definedas x o u(x') = (u(x'),x).

150

2 Tauberian theorems

The proof of Proposition 2.4.1 is left as Exercise 2.4.5b. Remark 1. If ct: T --f T is permitted in the sense that (2.4.1 1) is valid, then A ( q ) = cp o u defines an endomorphism A : A ( T ) --+ A(T), and so X is continuous (e.g. the remarks at the end of Exercise 2.4.5b). In particular,

(2.4.13)

V n,

((einallA G 1 R 11 < co.

It turns out that the boundedness of (2.4.13) is the condition required on any function ct :T --+ T to ensure that (2.4.14)

a(?) = ay

+ b,

aEZ

[Beurling and Helson, 1; Leibenson, 11. This solves the homomorphism problem. Thus, in terms of Proposition 2.4.1, if X :A(T) -+A(T) is an algebra automorphism then there is an homeomorphism ct:T -+T of the form (2.4.14) such that R(cp) = cp o ct for each cp E A(T). This result can actually be strengthened by assuming only that A:A(T) -+ A(T) is an algebra endomorphism, e.g. [K atznelson, 5 , pp. 220-2211. A(T) and A’(T)norm boundedness of “exponentials” (as in (2.4.13)) play an important role in spectral synthesis and we shall return to this matter in Chapter 3. 2. Co hen’s solution to the homomorphism problem is based on the following result: let 4 :A ( T , ) .+ A(TJ be an algebra homomorphism, let ct be defined as in (2.4.12), assume that rl and r2are discrete, and set S = {(y2, d y d ) : y 2 E f

z

and a(y2) E rd E r2x

fl

;

then /I= xs,where j i is an idempotent measure. Thus, a solution of the homomorphism problem is more or less reduced to the problem of characterizing idempotent measures, cf. Section 1.2.5. C o hen used his technique for idempotent measures to prove that there is a constant K such that for any set { n l , ..,nk} E Z,

[Davenport, 11 modified one of Cohen’s combinatorial lemmata and was able to replace the exponent, 1/8, by 1/4 and to take K 2 1/8 for large k ; P i c h o r i d e s has gone a step further and been able to replace 1/4 by 1/2. In his famous list of research problems (actually there were three such lists), L i t t l e w o o d conjectured that the right-hand side of (2.4.15) can be replaced by Klogk; if the conjecture is true, an upper bound K has been computed by K a r a n i k o l o v . 2.4.10 Beurling’s criterion for A(T). Getting back to the spirit of Theorem 2.4.3 we shall now prove a theorem, which stems from [Beurling, 61, from which it is possible to find conditions on E A(T) so that if F(z) = IzI then F o cp = IcpI E A(T). Note that

2.4 Wiener’s inversion of Fourier series

151

and

As such, we define

and

wherej= cp E L2(T).We say A (resp. B or C ) isfinite for thesequence {d,: n = 1 , .. .} c R+ if A(cp) < m (resp. B(cp) < m or C(cp) < m) wheref= cp, f ( n ) = d l n land , f ( 0 )=0.

Theorem 2.4.5. a)Iff’= cp E L2(T)and B(cp)< m then cp E A(T). (b) I f {dn:n= 1 , . ..} G R+ is a decreasing sequence and 2 d, < m then B and C are finite for {d,:n = 1, ...}.

c) I f { d n : n= 1,. ..} E R+ and 2d, fi< m then B and C arejinite for {dn:n= 1 , . ..}. d)

uf=cp

1

E L2(T) and

+

B(cp) C(cp) < oc, then A(cp) < co.

Proof. a) First note, by drawing the triangular arrays, that formally

Also, formally,

152

2 Tauberian theorems

Combining (2.4.20) and (2.4.21) yields

n-1 k-n

and, using Holder's inequality, the right-hand side of (2.4.22) is bounded by

b) We shall use the Cauchy "2" -criterion" for convergence: i f { c n :n = 1,. . .} decreases to 0 then 2cn converges ifand only if2 2"c2n converges, e.g. [Bary, 2, I, pp. 5-61, i) There is a constant a > 0 such that

by an integral estimate of 2"+'-1

I

x-jI2dx.

2-

Since {dk:k = 1,. ..} is decreasing, the right-hand side of (2.4.23) is

by an integral estimate of 2J+'-1

x2dx. 2J

The right-hand side of (2.4.24) is bounded by

2.4 Wiener's inversion of Fourier series

153

m

a, if each a, 2 0 and because

m

Now,

2 2-"'12 is estimated by an integral estimate to be 2-'12, m-

and so (2.4.25) becomes

I

22" d2-.

m-1

Thus the left-hand side of (2.4.23) is bounded. ii) To show that B is finite for {dn:n= 1,. ..} it is sufficient to prove that

n-0

\k-2"

1

Since {dk:k = 1,. ..} is decreasing and 2J+ - 2J = 2', the left-hand side of (2.4.26) is

and, as in the analogous calculation in part i), the right-hand side of (2.4.27) is bounded by

Thus, we apply the Cauchy "T-criterion" to obtain (2.4.26). m

c) lf

2 dnfi< co, then, by using integral estimates in the obvious way, we see that 1

d) From the definition of A ( q ) and Parseval's formula it is sufficient to estimate

+

$ n-2

111 ( n - ~ ) - l

(

/f(k)12) k-n+l

1

n-1

1-312dA< C ( q ) + E ( q ) < 03. q.e.d.

154

2 Tauberian theorems

The simple proof of Theorem 2.4.5a is due to SteEkin; there are more complicated proofs. Actually, f o r f = cp, Leindler has shown that A ( q ) < 00 if and only if B(cp) < co; and [Sunouchi, 11 has used Jensen’s inequality to show that B(cp) < 00 if and only if C(cp)< co. On this matter we also mention the contributions of K o n y u s h k o v (1958) and Boas (1960). Using (2.4.16), Theorem 2.4.5a,b,d, and the direction, A(cp)< m

=r

B(cp)< m,

of Lei n d 1e r ’s theorem, we have : given 3= cp E A( T) ; i f there is a decreasing sequence {d,:n = 1,. ..} G R+for which 2 d, < co and i f

h then

’0,

If(&

n>l

4,

IcpI E A(T).

Contraction conditions such as (2.4.16) have also been developed by Beurling into a very important tool in spectral synthesis. In fact, not only does A ( p ) < 00 imply that cp E A(T), but, as we’ll prove in Section 3.2.4, it implies that cp is synthesizable. 2.4.1 1 Katznelson’s converse to the Wiener-LCvy theorem. As we note in Exercise 2.4.4b, F(z) = IzI does not have the property that F(A(T))G A(T). In 1958 [Kahane, 31 proved that there are infinitely differentiable functions Fsuch that F o cp 4 A(T) for some cp E A(T) (cf. Theorem 2.4.3). Later in 1958, [Katznelson, 11 made the breakthrough that establishes the converse to Theorem 2.4.3 ; the general statement of this result as developed in [Helson and K a h a n e , 1; H e l s o n , K a h a n e , Katznelson, and Rudin, 1; Katznelson, 11 and exposited in [Katznelson, 5, Chapter VIII, Section 8; Rudin, 5, Chapter 61 essentially says that if a function F satisfies the condition, F(A(T)) E A ( f ) , then Fis analytic (an exact statement is given in Exercise 2.4.2). Just as the Wiener (-Ltvy) theorem is fundamental in establishing a solution to the spectral analysis problem, so the K a t z n e 1s o n theorem is intimately related to the solution of spectral synthesis problems. [Varopoulos, 61 has given a tensor algebra proof of K a t z n e l s o n ’ s theorem. The Katznelson conjecture [K a t z nelson, 41 associated with this theorem is discussed in Exercise 2.4.2. There is a natural analogue of K a t z n e l s o n ’ s theorem when A ( f ) is replaced by B(r) [Graham, 1; Rudin, 5, pp. 135 ff.]; also see [Katznelson, 21. 2.4.12 Tensor products. We shall now give an application of Theorem 2.4.1 to tensor products; the introduction of tensor products allows us to give the intrinsic definition of convolution that we promised after Proposition 1.3.1. As we shall later see, tensor products play a surprising and interesting role in harmonic analysis.

2.4 Wiener’s inversion of Fourier series

155

Let Xl, X2 be complex Hausdorff locally convex topological vector spaces, let Y and Z be complex vector spaces, and let Z* be the algebraic dual ofZ. We denote the space of bilinear maps,

x,x x,

Y,

-+

by By. The tensorproduct of X , and X , is a vector space X18 X , and a map u E 9YxIBx2 such that for any vector space Y and any map E gy there is a unique linear map L : X 1@ X ,

Y

+

for which the following diagram commutes-

xl@x2 7 y It is a standard algebraic fact that XI @ X , exists and is uniquely defined for given X1, X 2 . For each ( 3 , t ) E X , x X , we designate t;

a(s, t ) = s

and it is easy to check that (2.4.29)

u=2 s

V u E XI 8 X,,

8 t, a finite sum,

C3 t =s1 @ t + s , 8 t ,

(s1 + 5 * )

s @ t l + s 8 t , = s 8 (tl

+ tz),

and s

@ ( cIt ) = c(s @

r) = (cs) 8 t ,

where s,s, E X1, t,t, E X2,and c E C. The representation in (2.4.29) is not necessarily unique; on the other hand it can be shown that each u E X1 X2 has a unique representation (2.4.30)

u=

2

$1

8 hi,

IS1

E

where I is an index set for an Hamel basis {hi :i E I } of X2 and where si finitely many i, e.g. [Benedetto, 2, pp. 128-1301.

=0

except for

An important property of algebraic tensor products is that (XI

8 X,)*

= 9Yc.

This “equation” is (one of) the raison d’Ctre for tensor products since, because of it, vector spaces of multilinear maps can be considered as vector spaces of intrinsically linear maps. More generally, if YYis the vector space of linear maps XI @ X2 --f Y

156

2 Tauberian theorems

then 4ey=

Py.

2.4.13 An intrinsic definition of suppT. Note that the dual group of G x G is f x f .

Proposition 2.4.2. a) There is a natural injection A(f) @A(f)

-+

A(fxr).

b) A ( f ) @ A ( f ) = A ( f x f). Proof. a) If f , g E L ' ( G ) then fg EL'(G x G) and E(1,y)= cp(l)$(y), where f = cp and g = $ ~ A ( f ) .Thus A ( f ) @ A ( f ) c A ( f x r) and so we have a linear map a : A ( f ) @ A ( f ) -+ A ( f x f). To prove that this map is injective first note that if (0, :i E I } is an Hamel basis of A ( f ) then (0, €3 OJ:(i,j)E I x I}is an Hamel basis for A ( f ) €3 A ( f ) . Let @ = 2cp @ $ E A ( f ) @ A ( f ) , and, using (2.4.30), expand each cp (i.e., each SJ as a finite linear combination of elements from (0, :i E I } . We then regroup this sum and obtain the unique representation

9 = 2 C J k 0,@ 8 k 9

a finite sum,

where {d,} and { e k } are two linearly independent sets. Now if a( 9)= 0 we have

v A,? r, 2 C J k eJ(A)8d?) = 0. Thus, for each fixed A E f , v k, 2 C J ej(1) ~ =0 I

by the linear independence; and by a second application of the linear independence, we see that CJk = 0 for each j and k. Consequently, @ = 0 and so a is injective. b) Write I =A ( f ) €3 A ( f ) G A(r x r).Clearly Zis a closed ideal by Proposition 1.2.6. Let 9 E A(T x f),and take (1,y) E f x f . Choose q~ E A(f) (resp. $ E A ( f ) ) which is 1 in a neighborhood of i.(resp. y). Then, cp$ E A(r)€3 A ( f ) , and so @cp$ E I since I is an ideal. Also, @cp$= @ in a neighborhood of ( i , y ) . Consequently, by Theorem 2.4.lc, @ E I. q.e.d. The problem solved in Proposition 2.4.2b has an analogue in distribution theory where it is necessary to prove that C:(Rn) @ C,"(R") = C;(R" x R"); this latter result is proved by means of a lemma due to H. C a r t a n [ H o w At h, 1, pp. 368-369; Sc h w a r t z, 51.

2.4 Wiener’s inversion of Fourier series

Proposition 2.4.3. Given S, T E A’(r). a) There is a unique element S @ T E A‘(T x

v cp,

*

E

Am,

b) For any @ E A(T x

157

r)such that

( S 8 T,cp*) = (S, cp> (T, *>.

r),

( s 8 T, @)

= (sY,

y))) = (TA,

(TA,

(sY,

y))).

Both parts of Proposition 2.4.3 are proved by verifying the result on A(T) @ A ( f ) , and then extending to A(T x r)via Proposition 2.4.2b and an easy calculation. S @ T is the tensor product of S and T; and it is now immediate that (2.4.31)

V S, T EA’(r) and V cp E A(T),

(S * T,cp) = (S 8 T, (P),

where @(A,?) = cp(A + y). (2.4.31) is the promised intrinsic (in A’(r)) definition of convolution. The proof of the following is left as Exercise 2.4.6.

Proposition 2.4.4. Given S, T E A’(T) and cp E A(T). a) suppS @ T = (suppS) x (suppT). b) Assume that for each compact set K c r, (supps x SuppT) n {(i,y)

E

r x r :A + y E K)c r x r

is compact. Then supp S + supp T is closed and (2.4.32)

suppS * T E suppS

+ suppT.

c) IfsuppS is compact then (2.4.32) is valid. d) suppS * cp c suppS + supp cp.

Exercises 2.4 2.4.1 Local membership and a property of A,(T)

a) Let E G r be closed and let cp E k(E). Prove that

v Y E r \ (azcp n a ~ ) ,

cp E A E ) , ~ ~ ( Y ) .

b) We noted in Proposition 1.2.4a that if I c A(T) is an ideal then I f l A,(T) is norm dense in I. Prove that A,(T) = fl{ I : I c A(T) is a norm dense ideal in A(T)}. (Hint: Use Proposition 1.1.5b).

158

2 Tauberian theorems

2.4.2 The conjecture of dichotomy Let E c T be a closed set. A function F: I + C, where I s R is an open interval, operates inA(E)ifF o cp E A(E)whenever cp E A ( E ) has itsrangecontainedinI. K a t z n e l s o n ’ s theorem mentioned after Theorem 2.4.5 tells us that if E contains an interval and F : I + C operates in A(E) then F is analytic in a neighborhood of I ; the same result holds if E has the property that for each n E contains an arithmetic progression having nterms[KahaneandKatznelson, l ; K a t z n e l s o n , 3 ] . [Kahaneand K a t z n e l s o n , 21 have also proved that if E is a set of strict multiplicity (in particular, if mE > 0) and Foperates in A(E) then Fis analytic; more generally, they obtained the same result for closed sets E with the property that V e > 0 3 /A E M(E) such that a) Let E c T be closed and infinite, and let I G R be an open interval. Prove that if F : I + C operates in A(E) then F is continuous. b) Let E c T be an Helson set. Prove that if F : I --f C is a continuous function then F operates in A(,?). Helson sets and sets of strict multiplicity are mutually exclusive classes, e.g. [Benedetto, 6 , Chapter 71. We say that a closed set E c T is a set of analyticity if F is analytic whenever F : I + C operates in A(E). The Katznelson conjecture, or conjecture of dichotomy, mentioned after Theorem 2.4.5, is that any closed set E c T is either an Helson set or a set of analyticity; an interesting recent result on the Katznelson conjecture is found in [Katznelson and McGehee, 21. A criterion for E to be a set of analyticity, proved by means of a combinatoric argument, is given in [Salinger and V a r o p o u l o s , 13; and a probabilistic contribution to the Katznelson conjecture, whose presentation is a shockastic process, is given in [Katznelson and Malliavin, 1; 21 (cf. [ K a t z n e l s o n and Malliavin, 31). K o r n e r ’ s example of a pseudo-function supported by an Helson set provides motivation to pursue the techniques in [ K a h a n e and K a t z n e l s o n , 21. 2.4.3 Norm estimatesfor Ilcp Assume that]= cp E A ( T ) doesn’t vanish. a) Set m = inf {Icp(y)l : y E T}. Provide an estimate for IIl/cpllmin terms of and m,where N i s chosen so that

2 1f(n)l < m/4, e.g. [Cohen, 41. Inl>N

b) Can you provide an estimate for 11 1/&

in terms of 11 cp([A ?

2.4.4 F o cp # A(T)for cp E A(T) a) Prove that cp o cp 4 A(T), where cp E A(T) is defined by for y = I[

(linear

otherwise.

or y E (-7~01,

2 InbN

Inf(n)I

2.4 Wiener's inversion of Fourier series

159

This example is due to [Marcinkiewicz, 21. Thus, cp" E A(T) if cp E A(T), whereas it is not necessarily true that cp o . .. o cp E A ( T ) if cp E A(T). Also, if cp is a uniformly convergent Fourier series then cp2 is not necessarily uniformly convergent. (Hint: If f = cp then

= O(l/lnlog2nl), In1

+- m.

By using an argument with conjugate functions it is not difficult to prove that if $ E A(T) then

exists and is finite; whereas it is easy to check that this limit is infinite for $ = cp o cp. Cf. the example in [Bary, 2, 11, pp. 194-1961which uses Shilov's criterion for the absolute convergence of Fourier series [Bary, 2,II, pp. 181-1831). Marcinkiewicz's main theorem associated with this example is: l e t f = cp E A(T) be real-valued and assume that 2 If(n)ls < m for some s E [0,1]; if V z E (a, b) and V n,

IF'")(z)I < nnls,

where (infcp(y), supcp(y)) 5 (a,b),then F o cp E A(T) (cf. [Kahane, 111 which is interesting in this regard for several other related problems). b) Define

and

where K > 0 is chosen large enough so that cp=cpt - (P2

is non-negative on [0,2n]. Check that cpl, cpz E A(T) (and so cp E A(T)), and prove that IcpI 4 A(T), e.g. [Kahane, 1; 2, pp. 255-2591. c) Given cp E A(T) and write cp(y) = r(y)(cosr, Prove that fi E A ( T ) if and only if

+ isinrJ,

where r(y) 2 0 and rv E R.

+

V y E FI, 3 n, E Z such that ry = 4nYn ry+2n

(cf. the proof of the Wiener-LCvy theorem in [Zygmund, 2, I, pp. 245-2461).

160 2 Tauberian theorems

2.4.5 Algebra homomorphisms a) Let X be a commutative semi-simple regular Banach algebra with unit and let Y E X be a subalgebra which separates the points of the maximal ideal space of X.Prove that X = Y if each element of Xis locally in Y, e.g. [Rainwater, I]. b) Prove Proposition 2.4.1. With regard to Proposition 2.4.la it is well-known and easy to prove (by the closed graph theorem) that every algebra homomorphism A :XI X2 of Banach algebras XIand X2is continuous as long as XIis semi-simple. The analogous problem for an algebra homomorphism A:X + C, where X is a Frtchet algebra, is unsolved; a spectral synthesis technique is used with regard to this problem in [Benedetto, 71. --f

2.4.6 Tensor products and support Prove Proposition 2.4.4. (Hint: Draw a planar picture of the sets for parts b) and c). Part d) is much easier).

2.4.7 Potential theory Apotential kernel K : R + R is an even positive function which is convex on (0, a)E FI. K, is the potential kernel which in a neighborhood of 0 E FI equals l/lyl”, for a > 0, or log( l/lyl), for a = 0, and which is infinitely differentiablefor each y # 0. The a-potential of T E D(FI) is U; = T * K,, a E [0,a).A compact set E c FI has zero a-capacity if V T E D(FI) \ (0) for which suppT c E,

U; 4 L“(FI)

[Hedberg, 11; the usual definition of zero a-capacity replaces “D(FI)” by “M(FI)”. If a = 0 the a-capacity is called the logarithmic capacity. a) Let E E FI be a compact set having zero logarithmic capacity and assume that cp takes constant values on the countable collection of open intervals contiguous to E. If

v Y E A,

cp E 4 f l ) , O C ( Y ) ,

prove that cp is a constant function. (Hint: Let T = cp‘, the distributional derivative of cp. By the local membership hypothesis there is rl/ E A(FI) and a bounded open interval Z z E such that cp = rl/ on 1, rl/ = 0 on I-, and II/ is infinitely differentiable on I - . Define the distributional derivative S = rl/‘ of rl/; and note that T=S+O, and

where 8 E C:(FI). By taking Fourier transforms we see that U,”E L“(FI). Consequently, T = 0 because of our hypothesis, and so we can conclude that rp is a constant). This result is due to [Hedberg, I]; and he also showed that the result is false for each K“,a > 0.

2.4 Wiener's inversion of Fourier series

b) Let E E Fi be the perfect symmetric set characterized by the condition that y and only if y = 2 EJrJ,E, = 0,1, e.g. (E1.1.3). Prove that if

'J

E

161

E if

+1

then E has zero logarithmic capacity. The conclusion of part a) was first proved for sets Ewith this lirn property in [ K a h a n e and K a t z n e l s o n , 31. 2.4.8 Inverses in M ( G ) , multipliers, and the Tauberian theorem This exercise should be compared with Theorem 2.4.4. a) Prove that

(E2.4.1)

1(

M(G),

Ilpdll.4.

6

~

~

~

~

~

A

8

,

where pd is the discontinuous part of p . (Hint: Use Wiener's characterization of continuous measures to prove that for E > 0 and y E T there is a sequence {y,,,: m = 1,. . .} G T such that i f j < n then

<&I2.

I(p-pd)'(y+Yn-Y,)I

A subsequence of {ym :m Ifld(Y)

- PI(?

= 1,.

. .} exists for which

+ 7. - 7J)I < &/2

f o r j large and n > j . Thus, for suchj and n, IPd

(7) - p(7

+ Y n - ?,)I

< &.I*

(E2.4.1) is due to [Eberlein, 31 who proved that projections WAP(T) + A P ( T ) of norm 1 exist (noting, of course, that B(T) E WAP(T));the above proof is due to [Glicksbergand Wik, 11. b) Recall that L'(G) is a closed ideal in M(G). Prove the following generalization of Wiener's Tauberian theorem: given p E M(G);jl is never 0 if and only if (E2.4.2)

-

P A ( f ) = A(T).

c) Given p E M(G). Prove that (E2.4.3)

p-l E

M(G) if and only if

P A ( f ) = A(T).

(Hint: If p-l E M(G) then (E2.4.3) is evident. The converse requires the following wellknown multiplier theorem: if A E L(L'(G),L'(G)) has the property that

(E2.4.4)

V f E L'(G) and V x E G, A(.r,f)

= .r&J),

then A(f) = f * p for some p E M(G) (cf. Exercise 3.1.3). If (E2.4.3) holds then the map A(T) + A(T), cp H Ccp, is a continuous bijection; consequently, the inverse map A is

162

2 Tauberian theorems

continuous by the open mapping theorem. R satisfies (E2.4.4) and so there is an element v E M(G) such that

v cp E A ( 0 , cp = P@). Thus,ifj??=PA(T)andPnevervanishes,f o r p E M(G),thenp-’ E M(G).[Beurling, 11 initially posed the problem of finding when P A ( f ) is closed in A ( f ) . A recent contribution is due to [Glicksberg, 11: assume that jl E B(T)\{O} and that r is connected; PA(r)is closed in A ( T ) ifand only ifp-’ E M(G). With regard to(E2.4.3), [DieudonnC, 1](1960) hasshown thatifTisnotcompact then

v cp E

cpA(0 # A m

(cf. Proposition 1.1.6b and the Cohen factorization theorem).

2.5 The Tauberian theorem in spectral synthesis 2.5.1 Historical commentary on S-sets and Tauberian theorems. We shall now trace the development from Wiener’s Tauberian theorem to a generalization (Theorem 2.5.1 and Theorem 2.5.2) which is most important in spectral synthesis. As we shall see, it evolved in the 1940’s that such a generalization was possible in light of the fact, which Wiener proved in [Wiener, 7, Lemma 613,p. 881 (1933) and which we discussed in Paragraph 1.2, that one-point sets have very good spectral synthesis properties. For perspective, note that Wiener’s Tauberian theorem, or, at a more basic level, Proposition 1.1.5b, can be viewed as a theorem in both spectral analysis and synthesis; the spectral analysis result was given in Section 1.4.1 and the spectral synthesis result is that 0 is an S-set (i.e., choose cp ~ j ( 0which ) never vanishes and conclude that)-) = k(0) = A(FI) by Theorem 1.1.3). [Carleman, 1, p. 781 (1935) showed that finite sets are S-sets by proving that if cardspy@< m, for @ E L“(R), then @(x) = 2 ayeiyx,a finite sum. As we indicated earlier, this result was essentially given by Wiener. C a r l e m a n ’ s technique involved the function-theoretic “Carleman transform”, e.g. [Katznelson, 5 , Chapter VI, Section 81. Next came the remarkable paper by [D i t k i n, I] (1939). D i t ki n proved, in our terminology, that ifI 5 A(FI) is a closed idea1,ZI = E c FI, and cardaE < w,,, then I = k(E), i.e., E is an S-set. His technique is based on the D(q,I)lemma: if I s A(R) is a closed ideal, cp E A @ ) , ZI G Zrp, and D(cp9 I ) = {Y E r:cp $ l,.ac(Y)}

(for f = FI in this case), then D(q,I ) is perfect. The proof of the D(cp,l)lemma depends on the likes of Theorem 1.2.2a which was then available for f = FI, i.e., Wiener’s “Lemma 6,3’’ mentioned above. The proof also uses Theorem 2.4.1. Shortly after

2.5 The Tauberian theorem in spectral synthesis

163

Ditkin’s work, and unaware of it, [Segal, 1 ; 3, p. 881 (1940) essentially proved the above D i t k i n S-set theorem for E G FI and card JE < so. His theorem depended on Wiener’~‘‘Lemrna6,~” [Segal, 3, Lemma2.7.11 coupled with hisgeneral results on the “spectral resolution of ideals” [Segal, 3, Theorem 2.21. [Fukamiya, I ] (1942) and [Sega 1,3] also proved Theorem 1.1.3 for G and f ; Fu k a m i ya was aware of [Segal, 11 but unaware of D i t k i n ’ s paper. The importance of proving a result for G and r corresponding to Wiener’s “Lemma 613” was established beyond a doubt in [Shilov, I ] which dates from his thesis in 1941. His major theorem in this regard [Loomis, I , p. 86; Shilov, 1, Theorem 13, p. 1101 was proved for regular semi-simple Banach algebras X and has the following form for X = A ( T ) : assume that one-point sets in r and the point at infinity (off) are C-sets, and let I c A ( f ) be a closed ideal with E = Z I ; ifcp E k ( E ) then D ( q ,I ) isperfect; with this result it is easy to check that i f E = E,U E2 is closed, where El is scattered and E2 is open, then E is an S-set, cf. Exercise 2.5.8. The next stage of development then was to ascertain the spectral synthesis properties of finite sets in an arbitrary f , The fact thatfinite sets are synthesizable for any f was established in 1949 by [Kaplansky, 11 and Riss, e.g. Exercise 1.3.7. K a p l a n s k y seemed unaware of the above S h i l ov paper (which appeared in 1947) but was aware of Ditkin’s and Segal’s work. His technique used structure theory and Segal’s methods; and as such he was able to prove also the above Ditkin-Segal S-set theorem for a large family of LCAG f . In 1951, [He I son, 1 ] proved thatfinite sets are S-sets in any r without using structure theory; and then, using a modification of an unpublished Beurling result for r = FI (from Beurling’s Harvard lectures in 1949), Helson proved that U T E A’(r)and JsuppTis scattered then T E A x f ) . Independent of Helson, [Reiter, I ] (1952) proved that i f I E A(T) is a closed ideal, cp E A(T), ZI E Zcp, and JZq f l aZI is denumerable, then cp E I. The proof of this result works if “denumerable” is replaced by “scattered”; and, as R e i t e r points out, H e l s o n ’ s proof actually gives the full generality of Reiter’s statement. R e i t e r was aware of Ditkin’s and Segal’s work but was unaware of Shilov’s work; his methods, although abstract, are influenced by [Agmon and M a n d e l b r o j t , I ] (1950) which utilized the C a r l e m a n transform. Ag m o n and M a n d e 1 b r o j t ’ s theorem is the same as Rei ter’s but is proved for the case that r = FI ;as such, the A g m o n-M a n d e l b r o j t result also follows by Shilov’s theorem and Wiener’s “Lemma 613”. The use of the C a r l e m a n transform as well as some corollaries special f o r r = FI still make [Agmon and M a n d e l b r o j t , 11 a valuable contribution. The other major contributions during this period are due to [Beurling, 21 (1945) (cf. Paragraph 2.2), [Godement, I ; 21 (1946), and [Mackey, 11 (1951 lectures). Later, [Cotlar, 11 (1954) provided a technical clarification.

2.5.2 Ditkin’s lemma, C-sets, and the major generalization of Wiener’s Tauberian theorem. We shall now prove the most appropriate general form of the Tauberian

164

2 Tauberian theorems

theorem for spectral synthesis, viz. Theorem 2.5.2. The idea of proof stems from Wiener's Theorem 2.4.1, D i t k i n ' s D(cp,I) lemma, Shilov's work, and the fact (which we'll prove) that scattered sets are C-sets; the result is due to [Warner, 11. We begin with D i t k i n ' s D(cp,I)lemma.

Proposition 2.5.1. Let I c A(T) be an ideal and take cp E A(T).

4 0-7 $ 2 1 then cp E J,oc(?)* b) YZ is closed and ZI c Zcp then D(q, I ) is a perfect set. Proof. a) Take y $27. Let $ E I be 1 at y and choose 0 E A,(T) such that llOllA < E c 1 and (2.5.1)

O=I(I(y)-I(I=

1 +-I/!

in some neighborhood V of y; this can be accomplished via Exercise 2.5.lb, which, as we prove there, is an easy consequence of Theorem 1.2.2a. Since Ijq 2 OnllA ,< IIqllA2 IlOili < llqllA2 E" < m, we compute, by means of the 0

0

geometric series, that cp E A(T).

1-8

40$

Since $ E I we have -E I ; and because of (2.5.1) 1-0 -= cp*

1-e

cp

on V.

b) For each y $ D(q, I ) we can choose an open neighborhood N , s D(q, I)" of y because of (2.4.1). Thus, D(v,Z)* is open and so D(cp,l) is closed. Let y E D(q, I ) be an isolated point; we shall obtain a contradiction and thus conclude that D(q, I ) is perfect. Because of part a), y E ZI, and therefore cp(y) = 0 by hypothesis. Since y is an isolated point of D(pJ we can find a compact neighborhood U of y such that (2.5.2)

w \{Y))~w,I)=o.

Let $ E A(T) \ (0) vanish outside of U. Since one-point sets are C-sets, there is a sequence { q n : n= 1,. ..) ~ j ( { y ) for ) which lim I cp - cpcp"llA = 0. Using (2.5.2) we easily calculate that V 1E

r and Vn,

rpI(Icp,, E I,&).

2.5 The Tauberian theorem in spectral synthesis

We apply Theorem 2.4.1 and conclude that cp$ J(q$qn-(P$IIA

II$IIAIIvPP~-

E

165

I since I is closed and

PIIA.

We can specify $ to be 1 on a neighborhood V c U of y, and so cp E I,oc(y). This is the desired contradiction. q.e.d. Thus, if E c r is closed and cp E k ( E ) theti D(cp,j(E)) G aE. For E c r closed we define D ( U

= U {D(cp,j(E)):cp E k(E)}.

Thus, if E is an S-set then D(k,j) = 0. The proof of Theorem 2.5.lc, especially part a i ) , shows that the conuerse is also true. D(k.j) is closed if r is metric.

Example 2.5.1 a) Let n 2 2 be an integer and let Z ( n ) be the cyclic subgroup of T A consisting of the n elements y = 2nj/n,j = 0,. . ., n - 1. Z ( n ) = Z(n). Define the complere direct sum rn.1 =

2

zi(n),

ier

where I is an index set and for each i E I, Z,(n)= Z(n). Clearly, rner is a compact, abelian, perfect, and totally disconnected group. If I = Z+then rn,l is metrizable with metric d(A,Y) =

2

df(A1,Yl) 2'

1

f=1

where A = (Al,. . .), y = (yl,. . .) E rn,Iand di is the natural metric on Z,(n).Thus, rn,I is cardI= No, by homeomorphic to the Cantor set when cardI=N,. We designate r2,1, DaY b) Let I = [0, I ] and set

E ={y

E

r2,r:card{i E I : y , = n} < I}.

The group identity 0 E T z , ris the only limit point of E and E is closed. E provides an example of an uncountable scattered set. Scattered sets are countable in separable groups. c) A closed interval contained in R is an uncountable set with a scattered boundary. There are no uncountable closed subsets of R",n 2 2, which have scattered boundaries.

Theorem 2.5.1. Let E s r be closed. a) Assume that for each cp E k ( E ) , aE fl aZcp is scattered. Then E is a C-set. b) Assume that for each cp E k(E), aE fl aZcp is a C-set. Then E is a C-set. c) ZfaE is scattered then E is a C-set.

166

2 Tauberian theorems

Proof. a.i) Assuming that cp Ej(E),,,c(y)we’llprove that

(2.5.3)

(PE c p . P ) l o c ( Y ) ;

it is not necessary that aE be scattered to verify (2.5.3). Take $, EJ’(E)fA,(T) l and N , (as in (2.4.1)) such that

vI

€

NY,

Since E n supp $, on SUPP $ 7 . Thus,

= 0 and

v I EN

Y

9

=4 4 ) .

*y(4

supp $, is compact, there is $ EJ’(E)fl A,(T) for which $ = 1

cp(4

=*Y(4

= W)* Y ( 4

= *(A)

cp(4,

and we have (2.5.3). ii) Let cp E k(E).It is clear that

v Y E \ Wcp n W ,

cp M3,0c(~),

e.g. Exercise 2.4.la.

BY part 9, (2.5.4)

V YE

\ Wcp n W ,

cp E cpj(E),,,(y);

it is not necessary that aE be scattered to verify (2.5.4). -

iii) Let I = cpj(E)where cp E k(E). Then, by Proposition 2.5.1 and the fact that ZI ~ Z c p , we see that D(cp,I) is perfect. Because of (2.5.4), we have q c p , I ) c azq n a

~ ,

and, consequently, D ( p , I) = 0 since aZrp fl aE is scattered. Thus, we apply Theorem 2.4.1, and conclude that cp E

m).

b) Take cp E k(E), set F = 8Zcp flaE, and choose a sequence {cpn:n= 1 , . . .} s j ( F ) f l A,(T) for which

lim IIP - VVnlla = 0,

n+m

since F is a C-set. Note that ( E n supp cp) fl supp cpn = F f l supp cp,, = 0; and, therefore, since cp,, we choose +,, E ~ ( Ewhich ) is equal to 1 on supp cp fl supp cpn. Clearly, cpcp,,

= cpcp.

$,,. Thus

lim IIP - cp(4o.*n)llA

n+m

= 0,

E A,(T),

2.5 The Tauberian theorem in spectral synthesis 167

and, because cp,, $,, Ej ( E ) , we conclude that E is a C-set. c) is clear from part a). q.e.d. In light of Theorem 2.5.1b, c the following is tempting to state and, alas, is true (Exercise 2.5.3a). Proposition 2.5.2. Let E c r be closed. Zf aE is a C-set then E i$ a C-set. Theorem 2 . 5 . 1 is ~ interesting in light of the above historical development and Theorem 2.5.2. Note that if ZZG Zcp then

azz n azcp = zz n SUPP cp. Theorem 2.5.2. a) Let Z G A(r)be a closed ideal and assume that ZZ E Zq, IfaZZ

where cp E A(F).

n iEcp is a C-set then cp E Z.

b) Given T E A’(r)and set E = suppT. (TI cp> = 0.

If cp E k ( E ) and

aE fl aZcp is a C-set then

Proof. a) Let E = Z I so that c p ~ k ( E )By . hypothesis we choose {I(/,,:n= 1, ...} G j(aEfl a@) fl A @ ) such that lim 11 cp - cpI(/,,llA = 0. -OI

(2.5.5)

n

From Exercise 2.4.1 and the fact that IF)c Z we see that V Y E r and Vn,

Thus, cp$,

E

VJIn E l i o c ( ~ ) .

Zand hence cp E Z because of (2.5.5).

b) is clear from a).

q.e.d. Theorem 1.2.3 is an immediate corollary of Theorem 2.5.2. 2.5.3 Singular support. Given T E A’(T). Let U c f consist of all points y E r for which there is an open neighborhood U,of y and p, E M(T) such that T - p, = 0 on U,. U is open and the singular support of T is defined as singsuppT = r \ U.

[H o r m a n d e r, 21 used a similar notion with the measure p, replaced by a C“-function (on R”). [Edwards, 31 proved that if E c R” is a closed set then there is T E A’(R“) for which E = sing supp T; this result is clear when “singsupp” is replaced by “supp”. Clearly, singsuppT E suppT, and [Edwards, 31 gives conditions for equality in terms

168

2 Tauberian theorems

of Helson sets. In light of Theorem 2.5.2 (properly translated) and the definition of the singular support the following comes as no surprise: given T E A'(r),suppT = E, and cp E k ( E ) ;i f c p Ej(singsuppT) then Tq = 0. [Edwards, 4, p. 2731 makes an interesting observation on this point and the /Itopology.

2.5.4 Herz sets. As we mentioned before Theorem 1.4.4, it is not known if every S-set is a C-set; this is the C-set-S-set problem. Let E G T be an S-set and let cp E A(T) have the property that Zcp = E (not, Zcp 2 E). If 1 llcp - cpnlla <-for some cp, E ~ ( E then, ) , by Proposition 1.1.5b, (PI),,= 1 on (int Zcp,,)" n for some I),, E A(T). Thus, qn= cp(cp, $,,) on T, and so we have the C-set property for cp E k(E) when Zcp = E.

This calculation shows two things: first, the essence of the C-set-S-set problem is to examine the zeros of cp E k ( E ) which are not in E ; and, second, Tauberian methods, in this case, Proposition 1.1.5b, give C-set information from S-set data. In fact, for the case Z=/(E) and cp E k(E), Theorem 2.5.lb and Theorem 2.5.2 tell us that the (generalized) Tauberian techniques at our disposal are those which determine C-sets; and that if there are S-sets which are not C-sets then Tauberian techniques will not find them for us. Clearly, if E E r is an S-set then E is a C-set if and only if V c p ~ k ( E ) , 3{cpn:n=1, ...} s k ( E )

suchthat

lim IIcp -

n+m

Since the C-set property is valid for S-sets E and functions $ with the property that Z$ = E, we have the following fact: let E be an S-set, and assume that V cp E k(E),

3 I),0 E A ( f ' ) ,

for which

cp = $0,

where Z$ = E ; then E is a C-set.

In 1956, [Herz, I] proved that the Cantor set, C,is an S-set; it is not known if C is a C-set. We prove what is now a common generalization (viz. Theorem 2.5.3) of Herz's Cantor set result, e.g. [ K a h a n e and Salem, 4, pp. 124-1251. [Herz, 5, pp. 229-2301 has formulated the result for any r and for the o! topology, e.g. Paragraph 2.2. The crux of Herz's proof is contained in the following Proposition 2.5.3. Given cp E A(T). Define the continuous function c p N : [0,2n] -+ C to be cp at the points 2rrklN, k = 0 , ..., N , and to be extended linearly on the remainder of [0,211]. Then cpN E A(T) and limllcp - q& = 0. N+m

2.5 The Tauberian theorem in spectral synthesis

169

Proof. a) If rp E C2(T)then r p i exists on [0,2n) \{2nr:r E Q fl (0,l)) and {cpl;:N= 1 , . ..} converges uniformly to rp' on this set. Integration by parts shows that C2(T)E A(T) (cf. Exercise 2.5.2). Clearly, rpN E A(T), and if gN = cp - cpN we use Holder's inequality and the Plancherel theorem to compute

Since limllq - rpNIILm (T)= 0 and limllrp' - rpNIILm(T) = 0 we conclude now that lim/lV - VNll.4 = O. =~1 ~ . A b) Setting e,(y) = elnVwe now show that ~ ~ ( e , ) N Define the triangle function A S n l N this ; is the functionfl, = rp, of Proposition 1.2.2 and is the analogue on T of the function, A , E A ( R ) , defined in Exercise 1.2.6. Hence, supp AZnlNG [-2n/N, 2n/N] and dZnlN(0) = 1. Thus,

where

Clearly, by the convention (1.3.7), (jjX)" = p * cp where p E M(T) and f = rp E A(T). Now, &(m) = N for infinitely many m E Z,and, for the remainder, j&(m) =

N 2 zk = z(l - zN)/(l - z) = 0, k-1

where z = e,, - ,,(2n/N). Consequently, negative since

fi- 2 0 and so the Fourier coefficientsg(m), of g = p N * AZn,N are non-

' x 2 0" (as we proved in Proposition 1.2.2).

Thus, lI(%)NIIA = 2 g(m>= (en)N(o)

1.

170

2 Tauberian theorems

C)If V ( Y )= 2f(n)einYE A(T) then (PAY) = C f ( n ) (en)N(Y), and SO II(PNIIA

II(Pl\A,

by part b). d) Take cp E A(T) and 8 > 0. Choose I// E C2(T) such that IIq - I/l, < 4 3 and choose Mlarge enough so that

v N 2 M, llI// - $ “ [ / A by part a). Then, because IIcpN

- $NllA

v N 2 M,

= l (cp

< 8/39

- I//)NIIA

< IIcp - $/IA

by part c), we conclude that

IlcP - C O N I ~ A < 8 .

q.e.d. The reason to formulate Proposition 2.5.3 becomes clear in the proof of Theorem 2.5.3a.

A closed set E c T is an Herz set if there is a sequence {N,:n= 1,. ..} of positive integers tending to infinity such that (2.5.6)

V n = 1,...

V k = 1,...,N,,

and

d(E,T)<”

Nn

3

-2 E x kE ,

Nn

where d(E,y) = inf (d(1,y) :1 E E}and V I,y

E T,

d(1,y) = inf(ll- y

+ 2xjl :

j Z}. ~

Theorem 2.5.3 a) Each Herz set E is an S-set. b)

E G T is closed then there is D s T, card D < No, such that F = E U D is closed and

is an S-set.

Proof. a) Given T E A’(E), cp E k(E), and the sequence {Nn:n = 1,. ..}. We’ll prove that ( T, cp> = 0. Set

(“A” was defined in Proposition 2.5.3) and define

2.5 The Tauberian theorem in spectral synthesis

171

The key observation is that

(2.5.71

jTn(Y)~ ( ydpn(y) ) = 0; t

I

in fact, suppT, E y:d(E,y)Q - so that we obtain (2.5.7) from (2.5.6). 25c n A trivial calculation shows that lim Tn= Tin a(A'(T), A(T)),and so, since " p n -+ m",it is reasonable to conjecture that ( T , cp) = 0 in light of (2.5.7). This is where we need Proposition 2.5.3.

The left-hand side of (2.5.7) is

= ( T , VN.);

so that since limllq - qN1l,,= 0 we obtain that ( T , cp) = 0 from (2.5.7). b) Take Nn= n and let D be the subset of X =

= 1,. ..,nand

n = I, ...

by the condition that -2nk -ED 2 d(E,?).". n Then F = E U D is an Herz set.

n q.e.d.

Let E c T be an Herz set. If T E A'(E) then the proof of Theorem 2.5.3 exhibits a sequence { p n : n= 1,. ..} c M(E)for which lim pn = Tin a(A'(T), A(T)) and for which SUP n

II Pnll AT Q I TI1 A *

*

The fact that we can synthesize with a sequence describes the phenomenon of bounded synthesis which we shall discuss in Section 3.2.13. There are generalizations of Herz sets for which bounded synthesis holds and is essentially characterized [Katznelson and K o r n e r , I]. 2.5.5 Pisot numbers and spectral synthesis properties of perfect symmetric sets

Example 2.5.2 a) The Cantor set C E T is an S-set. This follows from Theorem 2.5.3 since C is an Herz set for Nn= 3". b) If E c T is a perfect symmetric set determined by (5, :k = 1,. ..} c ( 0 , t ) where each 1

t k

=

5 E Q and - 4 Z,then E = E, is not an Herz set [Rosenthal, 3, Theorem 41.

r

172

2 Tauberian theorems

Remark 1. In light of Example 2.5.2b it is natural to investigate when E, c T, for an arbitrary E (O,+), is an S-set. An algebraic integer a > 1, each of whose conjugates b # a satisfies the condition I/?\ < 1, is a Pisot number.The main interest of these numbers in harmonic analysis stems from Salem’s incredible result (1943-1955), e.g. [Salem, 41: let E (O,+); E, is a U-set ifand only ifa = l / < is a Pisot number. Prior to Salem’s 1 work, Bary (1937) proved: given E Q n (0,J); E, is a U-set if and only i f a = - E Z

r

r

<

5

(cf. Example 2.5.2b). The uniqueness properties of perfect symmetric sets are dealt with in [Bary, 1; 21. [Meyer, 3; 4; 5; 61 proved that ifa > 2 is a Pisot number then Ee, 5 = l/a, is an S-set. His technique depends on a generalization of the arithmetic used in Theorem 2.5.3; and he synthesizes in the progressions, {2nk/N,,:k= 1,. . ., Nn}, narrow topology, e.g. Paragraph 2.2. It is not known if E, is a non-S-set for some

r

E (033).

2. [Rosenthal, 31 gives a “structural” characterization of Herz sets. 2.5.6 The finite union of S-sets. We know from Theorem 2 . 5 . 1 ~or Exercise 2.5.3b that countable closed sets E G T are C-sets. With regard to Theorem 2.5.3b it is natural to ask if a closed countable set D E T can be added to a non-S-set E E T so that E U D is an S-set. In fact this situation can never arise because of the following result [Warner, 2, Theorem 1-41. Theorem 2.5.4. Let E l , E, 2 f be closed sets and assume that Elf l E2 is a C-set. E = El U E2 is an S-set ifand only ifboth El and E2 are S-sets. W a r n e r points out Reiter’s influence in the formulation and proof ofTheorem 2.5.4. In fact, [Reiter, 6, pp. 557-5581 proved the analogue of Theorem 2.5.4 for the case that El fl E2 = 0. [Calderbn, 1, p. 31 and [Herz, 5, p. 2281 proved one direction of Theorem 2.5.4, viz. $ E l , E2 c r are S-setsfor which El n E2 is a C-set then El U E2 is an S-set. A recent result by [Saeki, 41 (1969) is: let E l , E2 E f be S-sets and let F E ElU E2 be a C-set which contains aEl fl aE2 n aE; then El U E2 is an S-set (cf. Proposition 2.5.4). Generally, it is not known if the union of two S-sets (even for the case r = T) is an S-set (cf. Exercise 2.5.3b). This union problem is obviously closely related to the C-set-S-set problem, In this regard, we say that a closed set E C T is a Saekiset [Saeki, 1, p. 2461 if Q

cp, JI E ME),

cp $ 4 3 .

Clearly, S-sets are Saeki sets. The following is trivial to check: Proposition2.5.4. Assume that every non-S-set in T is a non-Saeki set. If E l , E, G T are S-sets then El U E2 is an S-set.

2.5.7 The intersection of S-sets. Proposition 2.5.5. Let E E FI” be a closed convex set. Then E is an S-set.

2.5 The Tauberian theorem in spectral synthesis

173

Proof. Suppose 0 E FI" is the center of E and let cp E k ( E ) .

Then $,, E j ( E ) and limll$,, - q(la= 0. n+m

q.e.d. Thus B" = {y E A": IyI 6 I} is an S-set and the same argument shows that (intB")" is an S-set. Consequently, the intersection of S-sets is not necessarily an S-set because of Exercise 2.5.5 (where we show that S2G R3 is a non-S-set). [Varopoulos, 51 has shown that this phenomenon about the non-S-set intersection of (perfect) S-sets is valid in the case of any compact r. On the other hand, it is not difficult to check that if r is compact and E G r and F c f are C-sets, then E f l F is a C-set when (int E ) U (int F ) = r. E c R" is a polygon if it is a compact connected set obtained from closed half-spaces by a finite number of intersections and unions. Polygons in FI", n > 1, are not strong Ditkin sets; and a polygon E E R", n > 1, is convex if and only if (intE)" is a strong Ditkin set [Meyer and R o s e n t h a l , 11.

2.5.8 Sets of spectral resolution and arithmetic conditions. In Theorem 2.4.4 we made use of Kronecker's theorem and we shall give another application in Section 3.2.12. Because of the importance of various independent sets in harmonic analysis (e.g. [Benede t t o,6]), Kronecker's theorem has led to the following definition. A compact set E E r is a Kronecker set if V

E

> 0 and V r p C(E), ~ for which IrpI = 1,3 x E G such that

IP(Y) - (7, x)l < & *

Y E E,

A compact set E E r is independent if k

21 nJyJ= 0,

where nJ E Z, Y J E E, and y i # Y J ,

implies n, = . . . = nk = 0. It is easy to check that Kronecker sets are independent sets; and a simple application of the Radon-Nikodym theorem shows further that if E is a Kronecker set then +dP

E

M(E),

IlPllN = IlPlll

(cf. Example 2.5.4). It is interesting to note that every compact Hausdorff space X is homeomorphic to some Kronecker set E contained in some compact r ;in fact, let r = II {T,:rp E C(x>, (cpI = 1, and T, = 1)and let E=f'(X) where for each x E X , f ( x ) = {rp(x): rp E C ( X )and IqI = I}. We say that r is a torsion module (over Z) if VyE

r

3nEZ

\ (0)

such that ny = 0.

174

2 Tauberian theorems

Thus D , is a torsion module and so has no independent subsets. Ifp is a prime number we say that a compact set E c r is p-independent if

2 n ,y , k

1

= 0,

where n, E 2, y, E E, and y1 # y,,

implies n, = 0 (mod p ) for each j . Thus there are points y E D , for which { y } is a pindependent set. Also we say that a closed set E 2 r is a set of spectral resolution if every closed subset F E E is an S-set; and a closed set E c r is a set of strong spectral resolution if A’(E) = M(E). There are sets E of spectral resolution for which A g E ) \ M(E) # 0 [Varopoulos, 31. In light of our remarks about the union of S-sets the following result [Saeki, 41 is interesting: gG c r is an S-set and E2 is a set of strong spectral resolution then El U E2 is an S-set, cf. Exercise 2.5.8. [Varopoulos, 21 (1965) proved that Kronecker sets in T are sets of strong spectral resolution, e.g. Example 2.5.3; and the reason we mention these notions now is that it is not known if all sets of strong spectral resolution, or even Kronecker sets, are C-sets. We conjecture that a counterexample for the C-set-S-set problem can be found among the sets of strong spectral resolution, contrary to the devil-may-care opinion we expressed in [Benedetto, 6, p. 1591. As we noted above, it is also not known if the Cantor set, C, is a C-set; Cis not a set of spectral resolution [Kahane and Katznelson, 21. Sets of spectral resolution were introduced by [Malliavin, 41 (1962) for r = T, and he proved that such sets are U-sets; [Filippi, 11 later extended this result to arbitrary r.

2.5.9 Totally disconnected sets and measure theoretic properties of pseudo-measures. Theorem 2.5.1 and Theorem 2.5.2 show the importance of boundary sets to determine spectral synthesis properties; and the potpourri of definitions, unproved results, and open problems listed after these two theorems indicates a need to take a closer look at “irregular” boundary sets. Detailed expositions of the role of “irregular” sets in spectral synthesis are found in [Benedetto, 6;Hewitt and Ross, 1,II; Kahane, 13; Kahane and Salem, 41. We proceed in the following framework. Let E E r be a compact totally disconnected set and define 9 = { F E E: F is open in E and closed}. It is easy to check that since E is compact then the total disconnectednessof E is equivalent to the fact that 9 is a topological basis for E. 9is also an algebraof sets. Afinite disjoint family {F,: j = 1,. ..,n} c 9 is a finite decomposition of F E 9 if F = F,. For

0 1

F E 9, we let $F E A,(T) be 1 on a neighborhood of F and 0 on a neighborhood of E \ F. Then T E A’(E) is a well-defined finitely additive set function with domain 9 and defined by

(2.5.8)

V F E 9,

T(F)=(T,$F).

2.5 The Tauberian theorem in spectral synthesis

175

A finite decomposition of T E A‘(E) is a finite set {T, : j = 1.. ..n} c A ’ ( f ) and a finite

decomposition {F,:j = 1,. ..,n} E 9 of E such that T =

2 T, and suppT,

G F,

for

1

e a c h j = 1, ..., n. If T E A ’ ( E ) and {F,:j= 1, ...,n} c 9 is a finite decomposition of E then there is a unique finite decomposition of T given by T, = TtjF,,j = 1,. . .,n. Because E is compact we see that T, as a finitely additive set function on F,is also a countably additive set function on 9. It is natural then, especially in light of the notion of strong spectral resolution, to see if we can determine some element p T e M ( E ) for which /lT = T o n 9. Define the semi-norms, 11

/Iu

and 11

[Iv,

on A’(E) as:

V TEA’(,!?), I I T J I , = s u ~ { J T ( F ) J : F E ~ }

and V T EA ’ ( E ) ,

IITllv = sup{ 2 IT(F,)I : { F , } is a finite decomposition of E } .

Clearly, IITJI, < a if and only if llTllv < 03. Define Y = { T EA ’ ( E ) : ~ ~ < T ~a}, ~u

and let 9,be the smallest a-algebra containing 9. By the Caratheodory-Hopf theorem, e.g. [Hewitt and S t r o m b e r g , 1, pp. 141-142; Royden, 1, pp. 253-2591, we obtain the following result since T E A‘(E) is countably additive on 9: Proposition 2.5.6. Let E c f be a compact totally disconnected set and let T E A’(E) be an element of Y . Then there is a unique complex valued countably additive set function vT defined on 9, for which vT = Ton 9. 2.5.10 Measures associated with pseudo-measures. It is conceivable that there are open and thus 9, is properly contained in the Bore1 algebra sets U c E which are not in F,, 9?(E). Clearly, if E has a countable topological basis then 8,= a(,!?). Problems with axioms of countability are dispensed with by Theorem 2.5.5. Define (2.5.9)

AE(f)= {cp

E

A ( f ) : 3 U, 2 E, open in f , such that card cp(V,) < a}.

The space A E ( f )was used by [Katznel son and R u d i n , 11 in their study of semi-simple commutative Banach algebras X for which the Stone-Weierstrass theorem is valid; in particular, they proved that a necessary and suficient condition in order that every sevadjoint separating subalgebra X of A ( f ) , with the property Vy

E

f,

3 40 E Xsuch that ~ ( y #) 0,

be dense in A ( f ) is that r be totally disconnected.

The following result was used by V a r o p o u l o s in his proof that Kronecker sets are sets of strong spectral resolution.

176

2 Taubetian theorems

Theorem 2.5.5. Let E G I' be a compact totally-disconnected set and let T E A'(E) be an element of Y .Then there is a unique complex measure pT E M (suppT) for which pT = T on F (and pT = vT (defined by Proposition 2.5.6) on Fa). Proof. a) AE(r) = C,(T) in the sup norm because of the Stone-Weierstrass theorem and the fact that E is totally disconnected. b) Define the linear functional (2.5.10)

pT:AE(r) cp

+

C,

I-+

For each fixed cp E AE(r) with corresponding U, (as in (2.5.9)), set (2.5.11)

v x E cp(~),

u, = u,n

cp-yx).

Clearly, (2.5.12)

E G U { U , : x E cp(E)}.

It is also easy to see that each U, is open. In fact, let N be an open neighborhood of x such that N n (cp(U,) \ {XI) = 0.

Then U, I Icp-'(N) = U, fl cp-'(x) and, obviously, U, fl cp-'(N) is open. { U,:x E cp(E)}is a finite disjoint family. If we set F, = E n U,, for x E cp(E), then F, is closed by (2.5.1 1) and open in E by the previous observation. Thus {Fx:x E cp(E)}G 9 is a finite decomposition of E. We write T = 2 T,, where x

,(E)

{ T, :x E cp(E)}is the corresponding finite decomposition of T.

c)If cp E AE(r) we have cp = x on U, and so (2.5.13)

PT(V)=

2

X€,(E)

2 XEQ(E)

x(T,$Fx)=

2

x T(Fx).

XEQ(E)

Thus

v cp E A

m , IPT(cp)I G Il~llc(e)IITll". Therefore, by part a), pLTE M(T),and by the definition of pT we have that p T = Ton F. Also by part a), the restriction of AE(r)t o E is sup norm dense in C(E) and so we have supp pT c E. Even more, the above estimate allows us to take supp pT E supp T.

q.e.d. The following is easy to check.

Proposition 2.5.7. Let E be a compact totally disconnected set. a) For all v E M ( E ) , pLy, defined by (2.5.10), is equal to v. b) For all TE A ' W , IITIIY = 11PTll1.

2.5 The Tauberian theorem in spectral synthesis 177

c) IfT E A'(E) is an element of Y then

[IT-

PTIIY = O.

Tlreorem 2.5.6. Let E c r be a compact totally disconnectedset and let T E A'(E). Zfthere is NT > 0 such that for every finite decomposition { Tj :j = 1,. ..,n} of T, (2.5.14) then

1 llTjll.t

(2.5.15) IITIIA, and, in fact,

Q

NT,

IITII.

<

T = PT.

Proof. a) Given E > 0 and let cp E A(T) have the properties that IcpI < 1 and I IITIIA.(T,cp)I < 42. Take I E E and a neighborhood U , of I.By Exercise 2.5.1b there is 8, E A,(T) such that llt9,11A < e/(2NT)and 8, = 0 on U ; , and there is a neighborhood V , E U , of I for which

v Y E v,,

P(Y) - cp(4

= 8,(Y)*

b) Since E is totally disconnected and V , is a neighborhood we can choose FAE f such that I E FAand FAn V ; = 0. Let K, = E \ FAso that since FAand K, are compact subsets of r we can find a compact neighborhood W, s V , of FAwhich is disjoint from K,. Thus a W , n FA= 0 and a W, n K, = 0 and, hence, we have found a compact neighborhood W, E V , (in T)of I for which (2.5.16)

aw,n E = 0.

Since E is compact we choose { W,,, . .., W,"}, whose union covers E and such that (2.5.16) is satisfied for each W,,. c) Because of (2.5.16) we can choose {W,,, ...,W,"} to be a disjoint collection (cf. [Benedetto, 3, Lemma 4.11). d) FAJ= W A J nE E 9,j = 1 ,...,n, and we have ( T , cp)

= (T9

2V $ F A j ) = 1

(T9

cp('j)

$FAj)

+2

(T@FAj,

Consequently,

By hypothesis, then, IITIIA.G IITIJY. Obviously, for any finite decomposition {F, : j = 1,. ..,n} E 9 of E,

2I T ( F , ) I and, thus, llTllY < m.

G 2llT$F~llA'

NTS

'AJ)'

178

2 Tauberian theorems

e) T = p T because of (2.5.15) and Proposition 2.5.7~;in fact,

2/ I T , and SO 117'-

- pTjI(A'

pTIIA,

< IIT-

NT

+ llpTlll,

p ~ l l v= O .

q.e.d. The above result is found in [Benedetto, 3, Theorem 4.laI. (2.5.14) and (2.5.15) are clear if E is a set of strong spectral resolution, and the content of Theorem 2.5.6 is to obtain the "measure inequality" 11 JIA, Q 11 11" for pseudo-measures which satisfy (2.5.14). 2.5.11 Varopoulos' lemma. The following result is due to [Varopoulos, 2, Lemme 2;

9, Chapitre IV]; but has its origin in R a j c h m a n ' s proof that the Cantor set, C,is a U-set ; the technique should be compared with Exercise 2.5.1,

r be compact. For each E > 0, there is 6 ( ~E)(0,E ) such that V E E r where E is closed, V x l , x2 E G, and V c E C for which IcI = 1,

Theorem 2.5.7. Let

SUP I(Y9 x1) - C(Y,X J I < YE€

implies V T EA'(E), c

I'F(x1) - c ~ ( x ~<) I~ l l T l l A * .

Proof. i) Because of Exercise 2.5.la there is V ( E ) > 0 and 0 E A,(T) such that (2.5.17)

Vy

11 - ely[< ~ ( e )

E T,

0(y) = 1 - e"'

=>

and llOlla < E . Considering T as the unit circle in C, (2.5.17) becomes (2.5.18)

V z E C for which IzI

ii) Thus for any fixed x

E

r, Clearly, if O(z) = 2a,z", y

0x.d

1,

(1- zI < V ( E )

=- 0(z) = 1 - Z.

G and d E C for which [dl = 1 we set

ex,d

(y) =

=

(7) = e(d(y, x))*

IzI = 1, then

an P(yi nx) =

2

(an

d")(7,Y ) ;

YEOX

nez

and thus, not only does O x , d

E A ( r ) , but

also

iii) From (2.5.18) we conclude that for each d E C for which Id1 = 1 and for all x E G (2.5.19)

11

-

E

> 0 there is V ( E ) > 0 such that for all

- ~ ( Y , X ) I < ~ ( & ) I -d(y,x)-ex.,(y)=o.

2.5 The Tauberian theorem in spectral synthesis

179

iv) Now, given c, xl, and x2 as in the statement of the theorem, we define P(Y) = F(Y, XI) [1 - C(Y, x2 - x1) - &,c(Y)l,

where x = x2 - xl. Let 8 ( ~=) min

(E,?).

B(E) is independent of xl, x2, c, and E.

q.e.d. 2.5.12 E Kronecker implies A’(E) = M(E).

Example 2.5.3 a) Using Theorem 2.5.7 it is straightforward to prove that if r is compact and E G f is a Kronecker set then (2.5.14) is valid for each T EA’(E). In fact, if {T, : j = 1,. . .,n} is a finite decomposition of T E A‘(E) and E > 0, there are elements xlr.. .,xn E G and complex numbers cl,. ..,cnwith each lc,l = 1 such that

Since E is a Kronecker set we consider the continuous function p(y)

=

2n c,(y, x,) xF, (y)

I-1

on E, where FJ = suppT,, and have that V r >0 3x E G

such that sup { sup Jc,(y, x,) - (7, x)l} < r. 1<j
YeFj

E

Consequently, by Theorem 2.5.7 and for r = 6

(~~,/TI,A,)’

180

2 Tauberian theorems

b) If E is independent and r is non-discrete it is easy to check that mE = 0, e.g. [Benedetto, 6, Section 4.4; Rudin, 5, Section 5.3.61(cf. [Graham, 21 for a generalization). Recall that Kronecker sets are independent. c) If T E A'(E) is in Y ,where E 2 T is closed and m E = 0, there is a routine way (e.g. Section 3.2.11) to prove that T = pT E M(E), where pT was defined in Theorem 2.5.5. d) Combining a) and Theorem 2.5.6, we see that totally disconnected Kronecker sets in compact r are sets of strong spectral resolution. Parts a), b), and c) provide another proof of this result for r = T. The extension to arbitrary r is routine. 2.5.13 A'(E) = M ( E ) .

Remark 1. The idea for Theorem 2.5.7 involves a composition of a character and an element of small norm from A(T), so that an operation by a pseudo-measure T yields a difference of two Fourier coefficients of T and preserves the small norm. The statement of Theorem 2.5.7 estimates the differences between Fourier coefficients of pseudomeasures given a sup-norm relation between the corresponding exponentials. 2. Theorem 2.5.7 has had other applications in [Chauve, 1; Drury, 1; Kahane, 121. We shall mention Kahane's result explicitly. A closed set E c T is a Dirichlet set if

lim sup 11 -elnYl=o, YEE

and it is a strong U-set if

It is clear that strong U-sets are U-sets, and trivial to check that Kronecker sets are Dirichlet sets. Kahane proved that Dirichlet sets are strong U-sets; and, for perspective, especially in light of Example 2.5.3, we recall the above-mentioned theorem by Malliavin that sets of spectral resolution are U-sets. Kahane's proof is a direct application of Theorem 2.5.7. 3. In 1969, [Saeki, 4, Theorem 1 I ] was able to prove that any Kronecker set in any r is a set of strong spectral resolution. In 1971, [Saeki, 51 generalized this result in the

2.5 The Tauberian theorem in spectral synthesis

181

following way. He first showed that i f E c r is a Kronecker set then it satisfies the condition : 3 b > 0 such that V Fl,F2 E E for which FlflF2 = 0 and each F, is closed, 3 p E A(T) with the properties that

and

.=(

1 on a neighborhood of Fl, 0 on a neighborhood of F2.

Then he proved that E E r is a set of strong spectral resolution if and only i f there is b > 0for which condition (Kb)is valid.

Example 2.5.4. In 1953, [Reiter, 21 proved that i f E E r is closed, independent, and countable then E is a set of strong spectral resolution, and, in fact, by the countability, each T E A ‘ ( E ) is a discrete measure. The proof, e.g. [Benedetto, 6, Prop. 5.10; Rudin, 5, Section 5.6.71, depends on Kronecker’s theorem and shows that IlpllA. = 11 pII for each p E M(E).It is not too difficult to construct compact countable independent sets satisfying this norm equality but which are not Kronecker sets, e.g. [Benedetto, 6, Example 5.2.1 We alluded to Reiter’s result at the end of Paragraph 2.2, and it is interesting to compare it now with Theorem 2.2.4 where F(x) = 2a,(y,x) converges Y

uniformly instead of absolutely as in this case. We saw that T EA’(r)can have countable support and a discontinuous Fourier transform (e.g. Exercise 1.3.2) but that if supp T is compact and scattered then T is almost periodic.

Exercises 2.5 2.5.1 A norm estimate for A(r)

a) Given E > 0. Prove that there is 8 E A ( T ) and 6 > 0 such that l1811A < E and 8(y) = 1 - ely if 11 - elyI < 6. (Hint: Define 8 to be 1 - ely if (y1 < a, 1 - e1(2a-y) i f a < y < k, 1 - e-1(2a+y) if -2a < y < -a, and 0 if 2a < IyI < x . Iff= 8, then by Holder’s inequality

(i

(IWI’ + lWy)l’)dy

1

l”).

Compare [V a r o p o u 1o s, 2, Lemme 1] for a different proof of this result. b) We now give a generalization of part a). Given cp E A ( f ) , I E r, E > 0, and U a neighborhood of A. Prove that there is 8 E A,(T) and a neighborhood V E U of 1such

182

2 Tauberian theorems

that 8 = 0 on U",l1811A < E , and liY E

v,

V ( Y )-

cpw = NY).

(Hint: Choose cpA E A(T) for which cpn = cp(1) on a neighborhood Vl of 1.By Theorem 1.2.2a we can choose $ E A,(T) such that $ = 1 in a neighborhood V2 E U of I , $ = 0 on U",and II(cp - cpA) $]IA < E . Set 8 = (cp - pA)$and V = V, f l VJ. Unfortunately, we have no control over the size of V and it may be too small to obtain good norm estimates. c) Given E > 0 and a compact set K c r.Prove that there is 8 E A,(r)such that 0 < 8 < 1, tl= 1 on K, and

llell" < 1 + 6.

(Hint: By Proposition 1.1.4, choose cp E A,(T) such that cp = 1 on K and 0 < cp < 1. By Theorem 1.2.1 take $ E A#), such that ll$llA = 1 and llcp - (PI& < E . Set 8 = $ + cp - cp$. It is trivial to check that 8 has the desired properties). 2.5.2 Lip, (T) c A(T), a > 1/2 Using -integration by parts we see that Cm(T)E A(T) and C;(R") E A(R"). Prove that Cm(T)= A ( T ) and C,"(R")= A(FI") in the 11 /[,-topology (cf. Proposition 1.1.3). (Hint: Take {cp,:rn = 1,. ..} E Cm(T) such that cp,(y)dy = 1, cpm 2 0, and supp cp, c

s

T

[- i, ]: then a direct calculation, using the dominated convergence theorem, shows ;

that lim IIcp - cp * cpmIIA = 0.

V cp E A(T),

m+m

For the R"case, let { K , :rn = 1,...} E R" be an increasing sequence of compact neighborhoods such that R" = UK,. Choose cp, E C?(R") such that cp, > 0 on intK, and rn

supp cpm E K,; and let I E A(R") be the closed ideal generated by {cp, :rn = 1,. ..}. Then I -. A(R") by Theorem 1.2.4). The fact, Cm(T)E A ( T ) , can be extensively refined as we indicated in our remark about Lip, (T) in Exercise 1.3.5. We'll now be more specific. Define

v cp E C(TL

=SUP{(CP (cp(~ y)I

:A,Y E T and d(2,y) < S}

("d" was defined before Theorem 2.5.3). cp E C(T) is an element of Lip,(T),a > 0, if w,(6)= O(S'),

6 -+ Oand6 >O.

S . Bernstein proved that $2u , ( ~ ) / f i < m

then cp E A(T) and, thus, if cp E Lip,(T),

a > 1/2, then cp E A(T) (cf. Exercise 2.1.2e and [Kahane, 13, pp. 13-20; K a h a n e and

Salem, 4, Chapitre XI).

2.5 The Tauberian theorem in spectral synthesis

183

The Hardy-Littlewood function, cp(y)

5

elnlog einY, n

e.g. [Zygmund, 2, I, pp. 197-1991, and the Rudin-Shapiro function, e.g. [ K a h a n e and Salem, 4, pp. 129-130; Katznelson, 5, pp. 33-34], are examples of elements in (Lip1,2(T))\ A(T). As is well known,

if { ~ , : = n 1,. . .} decreases to 0 (the series converges uniformly) and so $ $ A ( T ) if 2 E,/n diverges, cf. Exercise 1.3.2. 1

2.5.3 Properties of C-sets The properties of C-sets have been developed mainly by [Calderbn, 1 ; Herz, 5 , Section 6; Rudin, 5, Section 7.5; Warner, 21; Theorem 2.5.1~is found in [Reiter, 8, Chapter 2, Section 5.3; Warner, 11. a) Prove Proposition 2.5.2. (Hint: Take cp E k ( E ) . To show cp E cp/o.Since cp E k(8E) and aE is a C-set we have that

3{p,,:n= 1, ...} s j ( a E ) n A c ( f )

such that lim llq-cpcpPnllA=O. n+ m

Note that U,, = ((f\ E ) f l supp cp,)" 2 E is open. Choose $,, E A ( f ) such that $,, = 0 on U."and +,, = 1 on E f l supp cp.. It is easy to check that cpn - cp,, $,, = 0 on Unand cpcp,, $,, = 0.Thus, 4OYn = P(Cpn - Vn $ n ) and (Pn - rPn $n EJ'(E)fIA c ( f ) ) . m b) Let { E n : n= 1,. . .} 5 f be a sequence of C-sets and assume that E = U En is closed. 1

Prove that Eis a C-set. (Hint: Take cp E k ( E ) and E > 0. By hypothesis and an inductive procedure we have V n 2 1, 3 pmE j ( E m ) n A c ( f ) rn , = 1,. . .,n,

IIP'P1...pn-1

such that

- ( ~ ' ~ 1 . * . ~ n l l A <

Let Urn2 Em be an open set for which qmvanishes on Urn.Since E f l supp cpl is compact and m

EnsuPPqlSUUm 1

we see that E

n SUPP (PI

N

U urn 1

for some N . Set $ = p l . ..cpN. Note that $ EJ'(E)and IIcp - (

~ $ 1 1 ~< E ) .

184

2 Tauberian theorems

Another important C-set criterion is the fact that closed subgroups of r are C-sets, e.g. [Rudin, 5, Section 7.5.1 We also refer to the treatment of C-sets in [Reiter, 81, where C-sets are called Wiener-Ditkin sets. 2.5.4 A classical generalization of Wiener's Tauberian theorem We give a generalization due to Beurling and [Moh, 11 of the classical Wiener Tauberian theorem. The basic technique goes back to [Beurling, 21. Let h be a nonnegative Lebesgue measurable function on R such that

(E2.5.1)

lim h(x)/x = 0

x+ m

and VY

(E2.5.2) lim x-m

E

X E R,

for whichrn(R\X) =0,

h(x + YhW) = 1. h(x)

Prove the following result. Given h which satisfies (E2.5.1) and (E2.5.2),f~L1(R) for whichfnever vanishes, and @ E L"(R); if

then VgeL'(R),

X-Y

lirn

x-m

I

(Hint: Take r = 0 and define

Assuming that)@-F(x) = 0 and that there is a sequence {x,:n = 1,. ..} G R increasing to infinity such that & I IG(x,)( = q > 0, we shall obtain a contradiction. To this end we define FAX)= F(x,

+ h(x,) x)

and

G,(x) = G(x,

+ h(x,) x).

Using the dominated convergence theorem and (E2.5.2), calculate that V x E R,

lim (f* G,(x) - g * F,(x)) = 0.

n+m

By (E2.5.1) and the hypothesis,?lmF(y) = 0, we see that limg * F,(x) = 0 and so 1-9)

V x E R,

lirn f * G,(x) = 0.

n-m

2.5 The Tauberian theorem in spectral synthesis

185

The definition of G and properties of the weak * topology yield the fact that lim Grim = Y EL=(R) in the a(L"(R), L'(R)) topology. Thusf* Y = 0. Note that Y(0) exists and I Y(0)I k q. Consequently, once Y is shown to be continuous in a neighborhood of the origin we obtain the desired contradiction because of Theorem 1.3.1~.Actually M o h effects this part of the proof, as Beurli ng did in 1945, by calculating that Y is uniformly continuous and appealing to Theorem 2.2.1, cf. part c) of the proof of Theorem 2.2.1). 2.5.5 Schwartr's example

+3 ( where m, is Haar measure on R";we don't make use of the specific value of this constant. a) Let n > 1 and set B" = {y E R":IyI d 1). It is well-known that m,B" = nn12/r 1 -

Let cp: S"-' +. C be continuous, where S"-' is considered as a subspace of A". Define $ : R " + C asfollows: $(O)=O; $(ry)=cp(y)ifO 1 and y E S"- l . Clearly, $ E L"(R") and $ = 0 on B"" ;in particular, the integral, 1

c

exists for each cp E C(S"-l). Because of (E2.5.3) we readily see that p E M(S"-'), p 2 0, and (p, 1) = 1. If B, E R" is an open ball of radius E E (0,l) and BEn S"-' # 0, prove that p(B, f l Sn- l ) > 0. b) Let cp E k(S"-l)f l C,"(R"),n > 1, have the property that acp/ayl 2 0 on Sn-l and acp/ayl > 0 for some point of S"-'. Prove that ap/ayl, defined by

is in D(R"),and that

c) Recall that C:(R")

c A(R"). Prove that T = ap/ayl E A'(R"), if n 2 3. Once we prove all

this it is clear from part b), the fact that supp -c S"- l, and the definitionof cp, that aY, T E A'(S"-')\A;(S"-'). (Hint: It is sufficient to prove that sup(lxfl(x)l:x~R"} < 03. If R: R" + R" is an orthogonal transformation, verify the change of variable formula (E2.5.4)

V cp E L;(S"-'),

I p - 1

cp o Rdp =

1 s"-l

cpdp.

186

2 Tauberian theorems

Using (E2.5.3), (E2.5.4), and the orthogonal transformation R:F13 ( r l 7 Y 2 r 73)

-+

FI'

k>

(090,Irl),

a direct calculation shows that

B3

By considering the spherical coordinates, y1 = csinacosb, y2 = csinasinb, y3 = ccosa, compute that

This exercise completes the proof of the original counterexample to synthesis due to [Schwartz, 21 (1948) that we referred to in Section 1.4.8. Analogous examples are found in [Dixmier, 21. d) Letfe L'(R") be a radial function and write @(p) =fir) as in Exercise 1.4.4a. Prove that @ E C'(0, w), and that for each p > 0 there is K,,(p) > 0 independent o f f , such that I @'(p)l < K,,(p)JJfII'. (Hint: Compare (E1.4.5)). e) Let YE L1(Rn),f=cp and n 2 3, be a radial function for which xf(x) E L'(R"), and assume that @( 1) = 0 and @'( 1) > 0.Some relatively simple examples of such functions are given in [Reiter, 5, p. 4691. Note that cp E k(S"-l).Prove that Iez 5 I, [Reiter, 5, Theorem I]. Cf. the condition "IQ2 5 Zq" with the results of Paragraph 3.1. f ) Prove that S"' x FI" E FI"+", n 2 3, is a non-S-set. g) The order of T E D,(FI"),ord T, is the smallest value of m for which T is a continuous linear functional on the space Cm(FIn),taken with its natural topology, e.g. [Schwartz, 51. Prove that for each n 2 1 there is T,, E D,(FI")f l A'(FIn) such that IimordT,, = a. (Hint: The technique of part c) can be used). S"-' GFI", n 2 3, is a non-S-set in the following strong way: if E c R" is closed and S"-' \ E Z 0thenE U S"-'isa non-S-ser [Reiter, 5 , Theorem 21 (cf. Theorem 2.5.3). The fact that S"-'c A", n 2 3, is a non-S-set is equivalent to the statement that primary ideals are not necessarily maximal ideals in the Banach algebra of radial functions on FIn[Rei ter, 5, Section 3 ;V a r o p o u l o s, 7; V a r o p o u l o s, 9, Chapter V]. References for examples of non-S-sets which essentially generalize the S"- c A", n 2 3, example are given in [Reiter, 8, p. 371. [Herz, 41 proved that S' E R2is an S-set, and it is not known if S' c F12 is a C-set. Even though S"-' c FI", n 2 3, is not an S-set, F a r o p o u l o s , 71 was able to use the method in [Herz, 41 to prove:

'

VE >0 lIcp-I(/IlA<&.

and

V cp E k ( 9 - l ) 3 I(/ E C,"(Rn)nk(S"-') such that

2.5 The Tauberian theorem in spectral synthesis

187

2.5.6 Maximal primary ideals in Banach algebras Let X be a regular semi-simple commutative Banach algebra. Prove that each closed primary ideal in Xis maximal if and only if for each x E X and for each maximal ideal y for which the Gelfand transform of x at y, 2 (y), vanishes, there is a sequence {x, :n = 1,. . .} c X such that each R, equals R on a neighborhood V, of y and lim IIx,II = 0 n+m

(cf. Theorem 1.2.2 and Exercise 1.2.4). A(T) is such an algebra. 2.5.7 A clafsical equivalentform of Wiener’s Tauberian theorem

Let f

E

C,(R) have the property that

llfll=

2

n=-m

SUP

If(x)I <

x€[n.n+l]

and let W(R)be the class of Fourier transforms of these functions. W ( a )is a Banach algebra where the norm off= cp is defined as 11cp11 = llfll. Prove that Wiener’s Tauberian theorem (in the form of Theorem 1.2.4) is equivalent to the statement: let ZE W(R) be a closed ideal and assume that for each y E tl there is q E Z such that cp(y) # 0; then Z = W(R),e.g. [Wiener, 71. Thecomplete proofis foundin [Edwards,2]andonedirection was given in [Hardy, 1, Section 12.71. There is a generalization in [Bochner and C h a n d r a s e k h a r a n , 1, Chapter VI]. 2.5.8 The union of S-sets a) Given T E A ’ ( r ) and q E A(r)and assume that cp E k(supp T ) and that Tcp is almost periodic. Prove that Tcp = 0. (Hint: By Theorem 2.2.3a it is sufficient to prove that SUPP,, T q = 0.Assume not and let y E supp,, Tq. Then the mean value has absolute (F)) I = r > 0. Take E E (0,r ) . Since cp = 0 on supp T, supp,, T q C value I suppT, and points are C-sets, we can choose $, E j ( { y } ) such that lcp - cp$,I, < e/IIT(IA,.Then, observing that y 4 supp,,Tq$,, compute that r < E).

M(G(x)

b) Let El c r be an S-set and assume that E, E r contains no non-empty perfect subsets. Let El U E, be closed. If T E A’(E, U E,) and cp E k(El U E,) prove that Tq is almost periodic. (Hint:Let {$,:n = 1,. . .} Ej(El) have the property thatj@Jlcp - $, , l A = 0, and, hence, suppT$, E E,; then suppT$, is scattered. By Theorem 2.2.4 and the fact that is uniformly continuous we conclude that T$, is almost periodic. Therefore Tcp is almost periodic sinceJipJTq - T$,,ll,. = 0). c) Parts a) and b) combine to yield: ifEl E r is an S-set, ifE, c r contains no non-empty perject subsets, and if El U E2 is closed, then El U E, is an S-set. C f . Theorem 2.5.lc, Theorem 2.5.4, and the discussion after Theorem 2.5.4. This result has also been proved by Rosenthal(for E,countable)and [Veech, 2, p. 4211;Veech’sproofisinteresting but complicated in that it uses properties of minimal and almost automorphic functions.

Sn

3 Results in spectral synthesis 3.1 Non-synthesizable phenomena 3.1.1 Introductory remarks on non-synthesis. The fact that there are non-synthesizable pseudo-measures, discovered for R3 by [Schwartz, 21 in 1948 and for any nondiscrete I‘ by [Malliavin, 1; 2; 31 in 1959, has warranted a finer study of the structure of pseudo-measures, first as to their synthesizable properties and ultimately to the general problem of relating various aspects of the support of a pseudo-measure to the behavior of its Fourier transform. Thus, questions, as to whether various sets with arithmetic properties, such as Kronecker sets, are synthesizable or not, become meaningful. We refer to the fact that there are non-s-sets in any non-discrete r as Malliavin’s theorem.

A large literature has evolved giving many diverse examples of non-synthesizable phenomena. In Theorem 3.1.1 we shall give the general method developed by M alliavin to determine such non-synthesis. Generally, it is necessary to construct specific elements of A(r)in order to employ Theorem 3.1.1 and we shall refer to-and not provide the details to-an assortment of these constructions. We gave S c h w a r t z’ s example, that S”-’ E A”, n 2 3, is a non-S-set, in Exercise 2.5.5; and we shall give V a r o p o u 1o s’ proof of Malliavin’s theorem starting from this example and employing tensor algebra techniques.

3.1.2 Malliavin’s idea. Malliavin’s original idea [Malliavin, 1, Section 5; K a t z nelson, 5, Section 8.81 is rooted in previously quoted (Paragraph 2.4) work on the symbolic calculus by [Kahane, 31 and [Katznelson, 11. Basically, Malliavin’s procedure yields a result in the “individual symbolic calculus” which finally comes down to extending a linear functional and preserving its support; a different twist on this latter issue is found in [Benedetto, 81. An interesting analysis of the role of these support preserving extensions and of differentiation (e.g. Exercise 2.5.5~and (3.1.1)(3.1.2) below) in the construction of non-S-sets has been given by [Atzmon, 11, cf. [Katznelson and Rudin, 1; Rudin, 5, Section 7.7.11. Without going into detail about the symbolic calculus we now outline Malliavin’s idea in the following way. The problem is to find T E A‘(T) and cp E A(r)such that cp E k (supp T) and (T, cp) # 0. For 6‘ E D(R) we have (3.1-1)

(b’(y),y) = -1,

3.1 Non-synthesizable phenomena

189

and, by a formal Fourier inversion formula, (3.1.2)

I e-IxY(ix)dx.

a’(?) =

R

We can generalize the formulas (3.1 .l)and (3.1.2) by considering functions cp:f

-+

R

instead of the identity function

id:R

-+

R.

Then (3.1.1) and (3.1.2) become (3.1.1)’

@’(P),cp) # 0

and (3.1.2)’

6’(cp)

‘i

=-

e-lxCp(ix) dx,

2A

-m

respectively. We know, of course, that supp6‘ = (0)c A ; considered in terms of the identity function, id,we could say that 6‘(y) has its support contained in the zero-set of the function, id(y) = y. With this latter interpretation, we say that (3.1.3)

S’(cp) has its support contained in Zcp.

Thus, if cp E A(f) can be chosen in such a way that the formal expression, 6’(cp), given in (3.1.2)’ is a pseudo-measure then (3.1.1)’ and (3.1.3) yield a formal solution to the problem. In fact, we now prove that all of these formulas can be justified. 3.1.3 Malliavin’s operational calculus. First of all note that if A(f) c A ‘ ( f ) and the map

r is compact

then

R + A’(f) u

H

(iu)keiu@(y)

is continuous. Thus, if (3.1.4)

[

Ileiu@llA. du < co,

-m

then the elementary results of vector-valued Riemann integration theory yield the fact that the integrals (3.1.5)

T,,,= 1 j(iu)Jeiu@du, 2A -m

i = o ,...k

190

3 Results in spectral synthesis

are elements of A ' ( r ) . We now verify that Tv,j E A'(r)without using vector-valued integration theory; and then are able to give rigorous proofs of (3.1.1)' and (3.1.3). Theorem 3.1.1. Let I' be compact, and assume that rp E A(T) is real-valued and satisfies (3.1.4)for afixed non-negative integer k. Then a) For each JI E A(r)and for each j = 0,.. .,k,

r

-w

exists and T,,,: A(r)+ C is in A'(T). b) For j = 0,...,k, supp T,,, G Zrp. c) Assume

Then,for each j = 1,. ..,k, (3.1.7)

(Tv,,,pJ)#O

and

Tv,JrpJ+l=O.

all but a countable number of terms in each sum is zero. Then

a) follows from (3.1.S) b) Write E -m

so that if

-W

> 0,

3.1 Non-synthesizable phenomena

191

then $:"(cp(Y))

= Te.,.c(Y).

The fact that Tv,,,cE A(T) follows from the Wiener-Ltvy theorem. By Exercise l.l.la,

and so if K c R is compact and 0 q! K then lim $?) = 0,

uniformly on K.

Z+O

Consequently, (3.1.9)

lim T @ , , ,= c 0, C-+O

uniformly on every compact subset of T disjoint from 29. Using the Lebesgue dominated convergence theorem, (3.1.4), and computing as in (3: 1.8), we see that

(3.1.9) and (3.1.10) combine to yield b). c) Using (3.1.6) we verify that (T@,,,cpJ) # 0, for j = 1,. ..,k. (3.1.11)

(T,,,,cpJ) = lim c+o

d Since du

2n

(iu)Je1u@(v)9(Y)Je-c1u1dudy.

= icp(y)eiU@(Y), integration by parts shows that the right-hand side

(3.1.11) is -1im c+02n

S r

=

~elu@(~)(iucp(y))'-l.-clul~j2~~u~ldudy -m

-j(Te,,-

cpJ-')

= (-1)'j!

f",,o(~).

of

192

3 Results in spectral synthesis

Finally, we prove that f o r j = l , ...,k.

(T,,,,cpJ+l)=O,

In fact,

(T.,

1,

d ucp(y)e-L'Y'- elrp(y)dudy du

cpz) = lim 8-80

2x

r

-m

r

-m

q.e.d. 3.1.4 Non-spectral functions and Lipscbitz conditions.

Remark 1. Assume that cp E A ( r ) is a real function that satisfies (3.1.4). We shall now prove that the condition (3.1.6) is no restriction in the statement of Theorem 3.1.1~. In fact, if we set f ( u ) = (eiu9(y),1)

then f e L1(R) is continuous andf(0) # 0, and so!(r) # 0 for some real r. Thus f"I+r,o(0)=

m

1

2~ f f eiurei"@"~)dudy # 0, r

and, clearly, the analogue of (3.1.4) holds for cp + r if it holds for cp. 2. In light of Theorem 3.1.1~and the above remark, the construction of functions rp E A(T) which satisfy (3.1.4) emerges as an important part of the problem to determine non-S-sets; we shall refer to such functions cp as non-spectral functions. Malliavin's original non-spectral function cp did not satisfy any Lipschitz condition of positive order. Using probabilistic methods, [Kahane, 4; 61 (1959-60) was able to ameliorate the situation by providing non-spectral cp E Lip,(T), for each a < 1/4 (cf. [Kaufman, 11). If E E T is closed we write

(3.1.12)

VI>O,

E,=(y~T:d(y,E)
193

3.1 Non-synthesizable phenomena

(“d” was defined before Theorem 2.5.3); and note that if cp E Lip,(T), a > 0, then (3.1.13)

SUP{IcP(Y)I:YEE,}=0(1”),1

+

0

Y

(the converse is not true!). In 1967, by a refinement of his earlier method [Kahane, 61, [K a h a n e, 10; 13, pp. 64-68] proved :for each ct < 1/2 there is a non-spectral cp which satisfies (3.1.13); this result is important in light of the Beurling-Pollard type result: each cp E A(T) which satisfies (3.1.13) for c1 = 1/2 is synthesizable (e.g. Section 3.2.5). The first explicit construction of a non-spectral element cp E Lip, (T), ct > 0, is due to [ K a h a n e and Katznelson, 21 (1963). Another, more or less combinatoric construction, is due to [Richards, 11 (1967). The simplest example of a non-spectral function is due to [Katznelson, 5, Chapter VIII, Section 7.61. K a t z n e l s o n works on f = D, and, as such, is able to give V a r o p o u l o s ’ proof of Malliavin’s theorem without using Schwartz’s example. His method extends easily to f = 1.

3.1.5 Principal ideals and non-synthesis. M a 11i a v i n verified (3.1.4) for compact the following strong way : 3 cp E A(f), real-valued, and 3 6 > 0 such that V s > 1, (M) = O(e-dIuI”’ I4 O3 I/eluvl!A, 1 7

r in

+

(cf. [Benedetto, 9, Theorem 2.11). Condition (M) obviously implies (3.1.4) for each k . Condition (M), Remark 1 and Theorem 3.1.1 combine to yield the following strengthening of Malliavin’s theorem for compact f (e.g. [Rudin, 41).

Theorem 3.1.2. Let f be compact and let cp E A(f) satisfy condition (M). There is an interval (a,b),possibly of infinite length, such that for each r E (a,b) {I(@+ r)n :n = 1,. ..} is afamily consisting of distinct closed principal ideals; and, in particular,for each r E (a,b), Z(cp + r ) is a non-S-set. A standard argument can be used to obtain the same conclusion for any non-discrete f (e.g. Theorem 3.1.7). As indicated at the beginning of Paragraph 3.1 we shall not prove the likes of (M).

Remark. We use the notation of Theorem 3.1.2. Note that Z(cp + r l ) fl Z(cp + r2)= 0 for rl # rz and so mZ(cp + r ) = 0 for some r E (a,b). For such an r let tj = (cp + r)2 so that if g = $ then g E L1(G); consequently, g E LP(G)for all p 2 1. It is not difficult to adapt the above techniques to be able to choose T E A’(Ztj) for which ( T , $) # 0 and such that E Lq(G)for a given q > 2. Thus, by the Hahn-Banach theorem, we have a proof of Segal’s theorem mentioned in Remark 1 after Exercise 2.1.6. 3.1.6 Malliavin’s theorem and a property of A(f). Condition (M) can essentially not be strengthened : Proposition 3.1.1. Let r be compact. There is no real-valued element cp E A(f)for which 6 > 0 can be found such that (3.1.14)

(le*uvII~. = O(e-’lul), IuI

+

03.

194

3 Results in spectral synthesis

Proof. Let eiucp(y) = 2 a(x,u)(y,x) E A(T) and for each q E [1,03] let u, be the D(G) xeG

norm of the sequence {a(x,u ) :x E G}. Since IleiupIILztr)= 1 we have u2 = 1 by the Plancherel theorem. By the spectral radius formula, l i m ~ ~ ( e i @=)IleiV’llrn ” ~ ~ ~=~ n1, and so limu:” = 1. n-rm

u+m

Let 6 > 0. Then lim(u,/edu)l~u = e-6 and hence u+m

u-+m

U

Since lim(l/u) = 0 we see that limlog u+m

(3.1.15)

Y

lim Y

-+ m

+rn

= -m, and, consequently,

U1 =O.

eau

Clearly, u:

< UlUm,

so that, because u2 = 1, we have 1

- 6 u,. u1

Therefore, edU/ul< urnedu which, in light of (3.1.19, tells us that lim(Ilei”ell,f/e-d”)= m. u+m

q.e.d. 3.1.7 Helson’s theorem on non-S-sets and the structure of ideals. In 1952, [Helson, 21 proved that if J, K c A(T) are closed ideals for which (3.1.16)

E=ZJ=ZK

and J s K ,

then there is a closed ideal I S A(T) for which (3.1.17)

Jrf Irf K.

Consequently, once Malliavin’s theorem was proved a weak form of Theorem 3.1.2 was immediately available; the strength of Theorem 3.1.2 is that the ideals can be chosen as closed principal ideals. Helson’s proof depended on a result by Godement on unitary representations of abelian groups. Katznelson (1966) (unpublished) strengthened (3.1.17) and we shall prove Katznelson’s result in Theorem 3.1.3a; his basic tool is Ditkin’s D(cp,Z) lemma. [Saeki, 11 (1968) also proved Helson’s theorem by using Ditkin’s lemma (cf. Reiter, 7;Stegeman, 11).

3.1 Non-synthesizable phenomena

195

3.1.8 Katznelson’s proof of Helson’s theorem. meorem 3.1.3 a) Let J, K G A(T) be closed ideals which satisfy (3.1.16) (and therefore r is non-discrete). There is a continuum, {I,}, of distinct closed ideals of A ( T ) such that Va,

JsI,SK,

and there are pairs { I l ,I,}

s {I,} such that

I, n z2 = J .

b) Denumerable decreasing and increasing sequences can be chosenfrom {I,}. c) Ifeach point of ZJ has a countable open basis in the relative topology of E then there is a decreasing sequence {I,,:n = 1 , . ..} E {I,} such that

n I,, = J . Proof. a.i) Define D(K,J) to be the set of points y E r such that for each neighborhood N7of y there is cp E K such that V $ E J, 3 3, E NT for which

$(A)

# cp(3,).

Note that if K = k ( E ) and J = j m then D(k,j) c D(K,J), where D(k,j) was defined after Proposition 2.5.1. The proof of Proposition 2.5.lb shows that D(K,J) is perfect. Since J g K we see that D ( K , J ) # 0 because of Theorem 2.4.1. Let P s D(K,J) be a compact perfect non-empty set which is the closure of its interior taken relative to D(K,J). We shall associate a closed ideal Ip, with the required properties, to each such perfect set, P. ii) Take P as in part i) and set Q = D(K,J) \ P. Define

I P = { P E K : ~Q,~ Ec ~ ~ J r o c ( ~ ) } . Clearly Ip G A(r)is an ideal and J E Ip E K . We shall now prove that I p is closed. Take $ E 1, and choose any y E Q. Let 8, E A,(T) be 1 on a neighborhood of y and assume that it vanishes on a neighborhood of P.

LetlimII$,,-~11,=Ofor{$,:n= 1, ...} zip. If 3, E Q then $,07 E J,,,(3,) since $, E Ip. If 1 4 D(K,J) then, since $,,8,E K, we have $,O,

E

Jloc(L).

196

3 Results in spectra1 synthesis

Consequently, by the way we have defined 8,, we see that Ic/, 8, E J because Ic/,,BY E Jloc(A) for each A E r. Thus, Ic/& E J and so I(/ E Jloc(y).Therefore Ic/ E Zp since y was an arbitrary element of Q . iii) We show that I , \ J # 0. Choose 1 E r in the interior of P relative to the topology on D(K,J), and let V , E D ( K , J ) be an open neighborhood of 1 (in the relative topology of D(K,J)) such that V , n Q = 0. Take an open set N , c r for which N, f l D(K,J) = V,; in particular, N , n Q = 0 since V, r l Q = 0. Let B, c N , be a compact neighborhood of A, and choose 0 E A,(T) such that 8(B,) = 1 and supp8 c N,. Thus 8(Q) = 0. For A and B, choose cp E K as in the definition of D(K,J ) . Then cp8 is not equal to any element of J on B,. On the other hand, cp8 E J,oc(y)for each y E Q ; and so cp0 E I p . iv) Note that D(Ip,J) E D(K,J). We shall prove (3.1.18)

p = D(IP,J).

from which it follows immediately that Ip 5 K. The proof of part iii) tells us that P E D(Zp,J).To see this first let I be in the interior ofP, relative to the topology on D(K,J), and let N , be a neighborhood of 1.As in part iii) we can take N , so that NA flQ = 0. Then choose B,, 8, and cp as in part iii). Thus qobl E Ip, and, by applying the definition of D(Zp,J),we see that P c D(Ip,J). To prove that D(Ip,J)E P we take I E Q f l D ( I p , J ) and obtain a contradiction. Choose N,, a compact neighborhood of A, disjoint from P. By the definition of D(I,,J) there is cp E Ip so that cp is not equal to any element of J on N,. Let 8 E A,(T) be 1 on N , and assume (supp8) f l P = 0. Then 8cp E I p and 8cp E JloC(y)for each y E P. Thus, by Theorem 2.4.1, 8cp E J ; and this contradicts the fact that cp is not equal to any element of J on N,. v) If PI, P2are disjoint compact perfect non-empty subsets of D ( K , J )then I p , f l Z, = J. b) is now obvious, and c) is routine. q.e.d.

A constructive procedure has been given in [Osipov, 11 to determine a continuum, {Ia}, of distinct closed ideals where ZZ, =S2 E R3.

3.1 Non-synthesizable phenomena

197

3.1.9 The projective tensor product. In 1965, in an important series of papers (e.g. [Varopoulos, 4]), V a r o p o u l o s proved Malliavin’s theorem using S c h w a r t z ’ s example and projective tensor products of spaces of continuous functions. Expositions of V a r o p o u l o s ’ work are due to [Herz, 7; 81 and [Kahane, 81; and the basic reference for the connection between harmonic analysis and tensor algebras is [Va r o p o u I o s, 81. The relation of tensor products to Hilbert spaces was discovered by G r o t h e n d i e c k in what he called the “fundamental theorem on the metric theory of tensor products”, e.g. [Lindenstrauss and Pelczynski, I]. Analogously, V a r o p o u l o s ’ results show the relation of tensor products to harmonic analysis.

Let El and E, be compact Hausdorff spaces. The projective tensor product of C(E,) and C(E,), denoted by V(El,E,)

or

C(E1) 6 W,)

(we write V ( E ) instead of V(E,E)), is the set of elements @ E C(E, x E,) which have the form

where

Because of (3.1.20), V(E,,E,) can be normed by

where the infimum is taken over all representations (3.1.19) of @; as such V ( E 1 , E 2is) a commutative Banach algebra (under pointwise multiplication) with unit. 3.1.10 Grothendieck’s characterization of the projective tensor product. We shall now sketch some properties of tensor products. These will not be necessary to read Var o p o ulos’ applications of tensor algebra methods to harmonic analysis in Sections 3.1.12-3.1.15; but are only presented for some added perspective.

Let X , , X , be complex Hausdorff locally convex topological vector spaces with tensor product, X , 6 X,, as defined at the end of Paragraph 2.4. Also, let { p i: i E I } and {q, :j E J } , where I and J are index sets, be families of semi-norms on X , and X,, respectively, which define their respective topologies. Preserving the notation, u : X , x X , + X , 6 A’,, of Paragraph 2.4 it is standard to prove (e.g. [Benedetto, 6, pp. 231-2331) that: there is afinest locally convex Hausdorfl topology, T,, on X, €3 X, such that u is. continuous; and the semi-norms which characterize T, are of the form (3.1.21)

v u 6 x16 x2,Pi 6 9

, = inf ~ {2pt(sn)q,(tn):u = 2 s, 8 1, n

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3 Results in spectral synthesis

i E Z,j E J. ( X , Q9 X,, T,) is the projective topological tensor product. If X , and X , are normed spaces then ( X , @I X,, T,) is normed by (3.1.21) (cf. (3.1.20)). Since (XI Q9 X,, T,) is a Hausdorf'f locally convex space it is also a uniform space, and so we can discuss its completion, X, 6% X,. We give Pietsch's proof of the following theorem due to Grothendieck (cf. the definition of V(E,, E2)).

Theorem 3.1.4. Let X, and X, be metrizable locally convex vector spaces. Then each u E X , 6 X2 has the representation m

(3.1.22)

2 C,S,

u=

Q9 t,,

C, E C, S,

E

Xi, tn E x,,

n-1

where lims,, = 0 in X,, limt, = 0 in X,, (3.1.22) is in the topology of XI 6% X,.

2 1c.l

< a,and the convergence of the sum in

Proof. Assume without loss of generality that the sequences, { p , : i = l,.. .} and {q, :j = 1,. ..}, of semi-norms describing the topologies of X , and X2 are increasing; and also assume that {pk €3 qk :k = 1,. ..} is increasing. Let rk be the continuous extension ofpk Q9 qk to X, 6% X,. Take u E XI 6% X,and choose a sequence {w,,,:m = 1,. . .} contained in X , @ X , for which limw, = u in X , &I X,. Clearly, we can choose a sub-sequence {uk:k = 1 , ,..} E {w,,,: m = 1,. ..} E Xl @ XZ such that rk(u - uk) < k-, 2-(k+l). V k, Set (3.1.23)

UI

=

2

. n-1

Cnsn

€9 m t

to be any representation of u,, and define Vk,

Uk=Uk+l-Uk.

Thus, for each k, (3.1.24)

rk(uk)G rk(u--k) +rk(U-Uk+l) ~ r k ( ~ - u k ) + r k + 1 ( U - u k + l )

3.1 Non-synthesizable phenomena

199

Generally, then 'k+1

vk

2

=

cnsn

€3

tn*

n-nk+l

Now,because of (3.1.21) and (3.1.24), the above representation of

can be chosen so

that "k+l

1

Jcn(d T k

n-nt+l

and V n E (nk,nk + 11.

Pk(sn)

< 1lk

and qk(tn) 6 1/k.

m

This completes the proof since u = u1

+ 20,. k -1

q.e.d.

3.1.11 Further properties of topological tensor products. Without getting involved in too much detail we now define the other most popular topology on X , @I X , for purposes of comparison.

Let 42, be a neighborhood basis at 0 E X , of the HausdorfT locally convex space X, and define

u; = ( R E X i :v r E uj, I(R,r>l d 1).

tl V, E %J,

Consider the tensor product X i €3 X i with correspondingmap, a':X; x Xi -+ X i 8 Xi; and let T, be the topology in Xl @ X , of uniform convergence on all sets having the form a'( Vf,Up)c Xi €3 Xi where VJE qJ.T, is aptly called the topology of equicontinuous convergence on XI €3 X,.( X , €3 X,, T,)is a HausdorfT locally convex space and we designate its completion by X , 6 X,. We have T, E 5 in XI €3 X , so that, generally, XI &I X , c X , &I X,.An HausdorfT locally convex space X, is nuclear if X , &I X , = Xl 6 X , for each HausdorfT locally convex space X,. An infinite dimensional Banach space is never nuclear. Proposition 3.1.2. Let XIbe a complex vector space and let X, be a complex vector space of C-valued functions defined on a set E. Then there is a vector space isomorphism of Xl €3 X , onto a complex vector space, F(E, Xl), of functions, E

-+

Xl,

which take their values in finite dimensional subspaces of X,.

Proof. The correspondence for the required isomorphism is given by XI @ X2

3

F(K,Xl)

2 sn @ 1,

I-+

2 Snrn

200

3 Results in spectral synthesis

It is straightforward to check the necessary details. q.e.d. As a corollary we have : let X be an Hausdorff locally convex space and let E be a compact space; then X 8 C(E)is a space of continuousfunctions on E taking values in X . Now, let {p[:i E I} be a family of semi-norms characterizing the topology on an Hausdorff locally convex space X , and let C(E, X ) be the space of continuous functions on the compact Hausdorff space E taking values in X . As such it is natural to topologize C(E,X) with the family {pi:i E Z} of semi-norms defined by (3.1.25)

V

Q, E

C(E, X ) and V i E Z,

pi(@)= supp, (@(y)). YEE

We know that X 8 C(E)E C(E,X ) and it turns out that the induced topology of (3.1.25) on X @ C(E)is T,. Even more is true:

x €3 C(E) = C(E,x) and

X is complete then

(3.1.26)

X

6 C(E)= C(E, XI,

algebraically and topologically. We now go one step further.

Proposition 3.1.3. Let El and E2 be compact Hausdorfspaces and take C(E,),i = 1, 2, and C(El x E2)with the sup norm topology. Then C(E1 x

E2) =

C(Ei) @ C(Ez),

algebraically and topologically.

Proof. In light of (3.1.26), which we haven’t proved, it is sufficient to show that C(EI x E J We define the map

= C(Ez,C(Ei)).

C(EZ,C(El))

-+

C(E1 x E2)

Q , W 6 where

v 71 E El ,

%Jl, 72)

= [@(72)1(71).

It is straightforward to check the necessary details. q.e.d.

3.1 Non-synthesizable phenomena

201

In light of these remarks, we see that if El and E2 are compact Hausdorff spaces then

where the injection is a norm-decreasing linear map. Of course, this brings us back to (3.1.19), (3.1.20), and the norm 11 I( @.

As we mentioned in Section 3.1.9, V ( E l , E 2 )is a commutative Banach algebra with unit, and it is straightforward to check that it is also semi-simple, regular, self-adjoint, and with maximal ideal space,

3.1.12 Varopoulos' idea. Let T be infinite and compact. The initial relations between

harmonic analysis and tensor products that were discovered by V a r o p o u 1o s can be described by the two statements: (V.1)

A ( f ) is a closed subalgebra of V(f)

and

for certain sets El, E2 c T.By means of (V.1) information about A(T) can be translated into information about V(T). In particular, the fact that S2 E R3 is a non-S-set for A ( W ) (Exercise 2.5.5) or, equivalently, for A(T3) will tell us that there are non-S-sets for V(T3) (where the notion of an "S-set for V(T3)" is defined in a natural way). By means of (V.2) information about V ( E l ,E,) can be translated into information about A(El + E,). In particular, the fact, which we expand on in a moment, that there are non-S-sets for V(D,), will tell us that there are non-S-sets for the Banach algebra A(El + E2),where, among other conditions, we demand that Ej be homeomorphic to D,; it is then easy to see that El + E2 contains non-S-sets for A(T). As we mentioned earlier, it is then not too difficult to prove that there are non-S-sets in any non-discrete LCAG, given the fact that there are non-S-sets in every infinite compact f. There is one interesting hitch in this scheme of tensor-talking S c h w a r t z ' s example into Malliavin's theorem. Basically the above plan has to include a method of transferring the non-S-set for V(T3) into the non-S-set for V(D,). The question is whether the solution to this latter problem is any easier than transferring S c h w a r t z ' s non-S-set E G T3directly into a non-S-set for A(T), without first climbing down into V(T3) and then climbing up from V(D,). It turns out, by a general theorem of Reiter, Herz, and D e Leeuw, which we discuss below, that if there is a continuous homomorphic injection a:T3

--f

f,

202

3 Results in spectral synthesis

then a(E) E r is a non-s-set. Of course, if we use the above-mentioned (in Section 3.1.5) example of a non-S-set for A(D,), due to Katznelson, then we only need (V.1) and (V. 2) to obtain Malliavin's theorem. The issue of transferring non-S-sets from one group to another still maintains its importance. 3.1.13 Imbeddings of group algebras into tensor algebras. Let r be infinite and compact. Clearly, the following linear maps are well-defined [Herz, 7; 81:

M :c(r)

+

c(rx r) Mcp=@,

cp I+

where

@(A, Y ) = cp(1

+Y),

P:c(rx r)

-+

and

c(r)

@ H- P @ = q ,

where cp(Y) =

J N Y - A,4dl.

r

Theorem 3.1.5. Let r be infinite and compact. a) M A ( r ) G V(r). b) The map (3.1.27)

M:(W),Il l l ~ ) + (J'(r),II lld

is an (injective)isometry, and so M A ( r ) is a closed subalgebra of( V(r),)I )I@). c) A(T) = {cp E C(r):McpE V (r)}(where the inclusion, M A ( Q E V(r),ispart a)).

Proof. a) Given cp(y)

=

2a,(y,x) E A(f).

XCG

Then @(A, Y ) = M d A , Y ) =

z:

XCG

ax@, x ) (7, -4.

b.i) First note that P o M: C(r)+ C ( f )is the identity since

Consequently, M is injective.

3.1 Non-synthesizablephenomena

203

ii) We now prove that P ( V ( r ) )c A(T), and that llPll Q 1 where IlPll is the canonical norm for the map

P : ( W ) ,It It@)

+. (AV),II

IlA).

Let @ = cp €4 $, where cp, $ E C(T).Thus =

j cp(Y - 4Jl(4dY

E

W)

r

since L z ( f ) * L2(r)= A(T) (a bona fide application of Cor. 1.1.2!). Also (3.1.28)

I I P @I I A

= Ilcp

* $11”

Q

IIcplls11$112 Q IIcpIlm 1 $1 ,

and so

IIWIAQ ll@l!@.

(3.1.29)

Because of (3.1.28), (3.1.29), the definition of 11 have P V ( r ) E A(T) and llPll Q 1.

/la,

and the fact that P is linear we

iii) We now prove that IlMll Q 1 where llMll is the canonical norm for (3.1.27). Take x E G and compute ~ ~ ~ ( * ,<x1 )=~II(*,X)liA ~ @ = (~,X)(Y,X). since (M( X))(~,Y) Consequently, a ,

v cp E W ) ,

tlMcpIl@ Q IIcpllA,

c) Take cp E C(T)for which Mcp E V(T).

By part b.ii), P o M(cp) E A(T), and by part b.i), P o M(cp) = cp. Thus, {cp E

c(1-1: M P E qr))E A ( r )

and we have c). q.e.d.

204

3 Results in spectral synthesis

A closed set F c I' x r is an S-set for V ( r )if for each @ E V(T)which vanishes on F there is a sequence (0, : n = 1,. ..} E V ( r )each of whose elements vanishes on a neighborhood of F such that (3.1.30)

limJI@- @,,Il@=O. n

Proposition 3.1.4. Let A(T)). Then

F = {(A,?)

r be infinite and compact, and let E G r be a non-S-set (for

E

rxPA+

y

E E} C

rxr

is a non-S-set for V(r). Proof. Choose q E k ( E ) \ j v ) a n d set @ = Mq. Suppose @,, E V(T)vanishes on a neighborhood of F and (3.1.30) is valid. ) Then, by Theorem 3.1.5, P@,,E ~ ( Eand IIq - Pan IIA = IIP(Mv - @ n )

IIA

II@ - @n IIa.

This yields the desired contradiction. q.e.d. Herz proved the converse of Proposition 3.1.4, e.g. [Varopoulos, 8, pp. 98-99]: E E r is an S-set for A(T) then F is an S-set for V(r).

if

3.1.14 Imbeddings of tensor algebras into restrictions of group algebras. Let r be infinite and compact. An initial, crucial, and simple-to-prove observation by Varopo ulos is the fact that i f E , , E2 s r are Helson sets then A(E1 x Ez)= V(Ei, E2);

this led to conditions of the form (V.2). If El, E2G r are closed we generalize the map M of Section 3.1.13 by defining (3.1.31)

M:C(Ei +E2)

-+

C(E1 x &)

where

thus, M is the canonical homomorphism induced by the continuous map El x Ez

+-

El + E 2

(A,?) w- 1+ Y .

3.1 Non-synthesizablephenomena

205

As in parts a) and b.iii) of Theorem 3.1.5 it is easy to check that MA(El + E2)c V(El,E2) and (3.1.32)

+

M : A ( E , EJ

V(E1,EZ)

-+

is continuous with llMll < 1. We now strengthen (3.1.32) to obtain the likes of (V.2). Theorem 3.1.6. Let r be infinite and compact. Assume that r contains a Kronecker set which is homeomorphic to D,. Then there are closed sets El,E2 c r such that

(3.1.33)

V(D,)

= A(E1

+ E2),

isometrically and algebraically.

P r o o f . i) If E is the given Kronecker set we let El and E2 be disjoint non-empty relatively open subsets of E whose union is E. Since E is independent the canonical maps D, x D,

-+

El x E2

-+

E l + E2

are bijections; and so the canonical homomorphism, M , given by (3.1.31), actually identifies C(El + E2) and C(D, x Dm). Thus, the map (3.1.32) is an injective continuous linear map, norm bounded by 1 ; and we shall now prove that it is surjective (the proof of surjectivity is completed in part v)). ii) Take CP E V ( D , ) and let cp E C(El + EJ have the property that Mcp = @. We write (3.1.34)

@=

2q n 8

$n,

V

E El,

y

E

Vn E C(E1)and

$n

E

C(EJ9

so that

For each

E E

E2,

~ ( lY )+= C c P n ( l )

$n(Y)*

(0,l) choose a representation (3.1.34) so that

Since E is a Kronecker set we can use an argument involving the Tietze extension theorem to prove that each E,, j = 1,2, is a Kronecker set, and then we are able to prove that

v n,

IIVnllc(E,) = I I q n l l A ( E I )

and IWnlIc(E2)

= Il$nllA(E,).

(Actually, if we only use the more evident fact that each E, is an Helson set (e.g. [Benedetto, 6, pp. 120-1211 or the appropriate remark in Section 2.5.8), the proof of the entire theorem goes through except that instead of proving norm equality in (3.1.33) we prove norm equivalence.)

206

3 Results in spectral synthesis

Thus,

Using (3.1.35) we compute that

where cx,y= 2a,(n)b,(n) and n

(3.1.36)

c Icx.yl 6 ll@ll@+

8.

iii) For each x , y E G define cpx,v

E C(E) as

Since E is a Kronecker set and Iqx,yl= 1 we see that V x,y

EG

SUP I P X , Y ( Y ) Y E E

and V q > 0 3 u(q,x, y) = u E G such that

- (Y,41 < tl.

3.1 Non-synthesizable phenomena

iv) Take q = ~ / ( 2 2

207

I C ~ , in~ ~part ) iii) and define

X.YEG

2

=

CX,Y(440,X,YN

E 40,

X.YEG

so that t,be(A Note that (3.1.37)

+ y ) = Yy,(A,y)E M(A(El + E,)). -

Q

E,

because of (3.1.36) and part iii), and that (3.1.38)

Il$*ll”

Q

ll@Il@+ E

because of (3.1.36). v) We now use (3.1.37) and (3.1.38) in the following way. Given E > 0. Choose d1 E A(El + E,) such that for 0, = Me,

II@ - Q,lI@

E

Q

2

and

114I I A ( € l +

E

€2)

Q

11@Il@+ 2’

Then, setting O1 = 0 - O1 E V(D,), choose 0, E A(&

11% - @ Z I l @

Q

z;E

llez IIA(El + € 2 )

Q

11% II@+ 2.’

and E

m

Proceeding in this way, we set cp = 2 8,. 1

Note that

+ E,) such that for e2= Md,,

208

3 Results in spectral synthesis

and so cp E A(E, + E,) since

It is also clear that Mq

=@

since

and thus we obtain the surjectivity that we promised in part i). The norm equality follows from (3.1.39). q.e.d.

3.1.15 Remarks on Varopoulos’ method. By means of some technical modifications, Theorem 3.1.6 can be strengthened to the formTheorem 3.1.6‘. Let r be injinite and compact. Then there are closedsets El, E2 s rsuch that V(Dm)= A(Ei

+ E2),

topologically and algebraically.

Instead of providing the details required to prove Theorem 3.1.6’ we shall make some relevant remarks and move on.

Remark 1. In the mid-l950’s, using a construction due to Carleson, R u d i n was the first to construct perfect non-empty Kronecker sets in T.It then became apparent that some groups I‘ did not contain any Kronecker sets homeomorphic to D,, and a related notion, that of a Kp-set,was devised. Just as Kronecker sets are independent, so Kp-sets are p-independent. Also, for any non-discrete r there is a Kronecker or Kp-set,p prime, contained in f and homeomorphic to D , [Rudin, 5, Chapter 51 (cf. the remark in Section 2.5.8 on Kronecker set homeomorphs to compact spaces). This fact is precisely what is needed to go from Theorem 3.1.6 to Theorem 3.1.6‘. The construction of Kronecker sets as well as K a u f m a n ’ s existential proof of perfect Kronecker sets are dealt with in several places, e.g. [Benedetto, 6; H e w i t t and Ross, 1, 11; K a h a n e , 13; Katznelson, 51. 2. Because of Theorem 3.1.6, a non-S-set for V(D,) is transferred into a non-S-set for A(El + E2), and, consequently, El E2 is not a set of spectral resolution (for A(T)). Thus in order to complete the proof of Malliavin’s theorem for compact f, using Theorem 3.1.5 and Theorem 3.1.6, we need only prove that a non-S-set for V(T3) is transferred into a non-S-set for V(D,); we do this in Exercise 3.1.5. The extension of Malliavin’s theorem to the non-compact non-discrete case is proved by means of Theorem 3.1.7.

+

3.1 Non-synthesizable phenomena

209

3. In light of the fact that we can find non-S-sets in El + E, when E = El U E2 is a Kronecker set, etc., it is interesting to generalize Theorem 3.1.6 to a result with the following form: A(El + E,) = V(D,) if El and E, are disjoint sets homeomorphic to D, and if E, U E2 is an Helson set. Theorems such as this have been proved recently by [Kaijser, I] (cf. [Saeki, 61). Actually, V a r o p o u l o s has proved that ifr is compact and if El and E, are closed uncountable metrizable subsets of r then El + E2 is not a set of spectral resolution. 4. V a r o p o u l o s ’ proof of Malliavin’s theorem leaves the interesting question of determining which non-S-sets for A(T), say, are “tensor images” of Schwartz’s example. 5. Using these methods it can also be shown (for El countable) that El

+ E, is a perfect

set of spectral resolution as well as being a set of analyticity.

3.1.16 Mappings of restriction algebras. Let r be a non-discrete LCAG. If G with the discrete topology is denoted by Gdthen f$= gr is the Bohr compactification of r. As such (e.g. [Rudin, 5, Chapter l]), gr is a compact group and there is a continuous homomorphic injection, b : r -+ br, such that 6i== gr; the natural transpose map, Gd -+ G, is, of course, a continuous bijection. If El E r and E2 E r are disjoint compact sets homeomorphic to D, and if El U E, is a Kronecker or K,-set, then it is clear that bEl and bE, are disjoint compact subsets of br such that each is homeomorphic to 0,and for which b(El U E2) = bEl U bE, is a Kronecker or K,,-set. To avoid notational confusion we write A(E, X ) = A ( X ) / k ( E ) where X is a LCAG, E E X is closed, and k ( E ) = {cp E A ( X ) :cp = 0 on E}. Because of Theorem 3.1.6 (or Theorem 3.1.6‘) we can prove that there are non-S-sets for A(b(E, + E2),bT); thus, once we establish Theorem 3.1.7b, we shall have proved that there are non-S-sets for A(E, + E,, r),and, hence, that El + E2 E r is not a set of spectral resolution (for A(T)).This completes the proof of Malliavin’s theorem for any non-discrete r. A form of Theorem 3.1.7a was first used by Wiener [Wiener, 7, p. 801 for the case of

FI and 1.

r be a LCAG and let E c r be compact. then there are sequences {x,,:n = 1,. ..} c G and {a,,:n = a) If cp E A(E) and r > I( cpII

Theorem 3.1.7. Let

1,. ..} c C such that

and

210

3 Results in spectral synthesis

b) The map R:A(bE,/?r)

--f

cp I-t

A(E,r) cpob

is a bijective isometry and algebraic homomorphism.

Proof. a.i) Given {bn:n= 1,. ..} E C and { y n : n= 1 , . ..} G G for which 2 16, I c prove that (3.1.40)

03.

We

IEbnYnIIA(€)<2 lbnl.

-

Take e > 0 and let p = 2 bn6,,. Choosef E L'(G) such thatf= 1 on E and 11f 11' c I Then f * p E A(r)and== C bnynon E. The norm inequality, (3.1.40),follows since

+ e.

+ 6) 2 IbnI

< (1

II~BIIA~~)

implies that

11 2 bnYn

IIA(€)

< ( 1 + &) 2 Ibn I

for every > 0.

ii) By the Hahn-Banach theorem, { 2 b n y n : 2 1 b n l < andy,EGforn=l, l ...)

is A(E)-norm dense in the unit ball of A(E). iii)Choosee > OsuchthatIIcpIIAcn+e c r.Take$,=~bb,,,y,,,forwhichIIcp - $l I IA( E ) < n

E

-and21bl,,,l
and p, - is given then *J

=

2 b,,n Y J , n

is chosen so that Icp,-

- $J(IA(E)

e

d y ) = 2 b,, n(?,YJ,n ) on E and J.n

e

< -and 2 Jb,,, I c F~ . 2' n

2 lbJ, I < r*

J."

n

(There are more complicated proofs of this fact.) b.i) Let $ E A(E, r).By part a) we have Vy

E

E,

$(Y)= L: an(?,Xn), Xn E.G and 2 Ian I < OJ.

3.1 Non-synthesizable phenomena

If I

E bE

21 1

we define q ( A ) = 2 an(IrXn)r

and so 4 is surjective. Once we show that A is well-defined it follows that it is an algebraic .,omomorF ism and an injection. ii) We prove (3.1.41)

v cp E A(bE,Br),

11 d 1 A w . r ) Q I / ( P I I A ( b E . B r ) ;

this yields the continuity of A, and, consequently, the fact that it is well-defined.

q.e.d.

212

3 Results in spectral synthesis

Remark, In Theorem 3.1.7a we proved that if 2 lbnl < 03, E E FI is compact, and {y,:n = 1, ...} c R, then 2 b,y, E A(E). [Katznelson and M c G e h e e , 21 have constructed a denumerable compact set E c FI with one limit point yo such that 2b,y, E C(E) \ A(E) for some choice of {b, E C : 2 lbnl < m} and { y , : n = 1, ...} c PR; their construction uses the fact that there are non-synthesis sets in FI as well as Herz's procedure to prove that the Cantor set, C c T, is an S-set (e.g. Paragraph 2.5). Using this set, E, [Graham, 31 has proved that ( E \ {YO})' \ ( E \ {YO}) c b(FId ) is a non-S-set in !(Ad), where "-/?" indicates closure in B(&) (cf. [Drury, 13). Note that FI c j(FI,) E PFI. [De Leeuw and Herz, 1, Theorem 11 (1963) have strengthened Theorem 3.1.7b in the following way.

Theorem 3.1.8. Let rl and r2 be LCAG and let a : rl --f T z be an injective continuous homomorphism. If El is compact and aEl = Ez then the canonical transpose, (3.1.43)

A :A(E,, r,)

-,A(E, rl),

is a well-defined bijective isometric isomorphism.

Proof. We give a fairly thorough sketch of the proof. The details are to be provided in Exercise 3.1.6. i) Since a is a continuous homomorphism it is not difficult to prove that A(B(r,)) c B(T,), e.g. [Rudin, 5 , p. 791. ii) If E E r is compact then B(E) = A ( E ) ; this is proved using the fact that for each E > 0 there is cp E A(T) for which cp = 1 on E and IIqII,, < 1 + E . iii) Since a is a continuous homomorphism, (3.1.43) is a well-defined norm decreasing map. To prove iii), take cp E A(&, r,) and use i) and ii) in succession to obtain the inclusion, A(A(E2,r,))c A(&, rl). The norm inequality follows by the estimates used to obtain the inclusion. iv) It is easy to check that A is injective. v) Starting from the hypothesis that a is a continuous homomorphism we can define the canonical transpose (of A ) A':Ai(E,) -+ A;(E,). We now prove in a straightforward way that A ' is an isometry since a is injective. vi) The fact that A is a surjective isometry is a consequence of the following general theorem about Banach spaces (e.g. [Rudi n, 5, pp. 259-2601): if XI and X, are complex Banach spaces and R E L(X,,X,) then A is a surjective isometry if A is injective and if the transpose, A', is an isometry. q.e.d.

3.1 Non-synthesizable phenomena

213

The fact that A is surjective is the major issue in Theorem 3.1.8. 3.1.17 A characterization of synthesis preserving mappings. In light of Theorem 3.1.8 and the problem of transferring S or non-S-sets between different groups, it is natural to ask when a map a : rl -+ T z determines a bijective isometric isomorphism A : AI(E2)-+ A,(El),where E, c r, is compact. The proof of the following result establishes criteria of this kind and uses Theorem 3.1.8; the proof is difficult.

Theorem3.1.9.([DeLeeuwandHerz, l]).Letrl a n d r , b e L C A G u n d l e t a : ~-+r, , be an injective continuous homomorphism. Assume that El c rl is compact and set E2 = aEl. Then El is an S-set (for A(T,)) ifand only ifEz is an S-set Cfor A(T2)). Related results were proved independently by [Spector, 1, e.g. Proposition 81 (cf. [Spector, 21). [Reiter, 3; 41 initiated research on the problem solved by Theorem 3.1.9; (in the notation of Theorem 3.1.9) he proved that if a is also a relatively open map of rl onto a closed subgroup of Tz then El E rl is an S-set if and only if E, c r2is an S-set. G l i c k s b e r g proved one direction of Theorem 3.1.9, viz. if E, is an S-set then El is an S-set, independently of D e Leeuw and H e n . Actually, the hypothesis in Theorem 3.1.9 that a be injective can be weakened to the assumption that a be injective on a neighborhood of El. On the other hand, the following observation [De Leeuw and Herz, 1, p. 2241 shows that it is not sufficient to assume that a is a continuous homomorphism, injective on El. Example 3.1.1. Define a : R3 +. R 2 by the map a(yl,y2, y3) = (yl, yz). Let El = {(rl, y2 y 3 ) E R3:7: + y: + y: = 1 and y3 2 O}. Thus, E, = {(yl, y,) E R2:y: + 7: d 1) = B2.tl is injective on E l , El is a non-S-set in R3 because of Schwartz's example, and E2 is an S-set in R2 by Proposition 2.5.5.

rl and T 2 are locally isomorphic if there are neighborhoods V , c r l of 0 E r[,i = 1,2, and a homeomorphism a of V , onto V, such that V

A,y E

Vl,

A + y E Vl

.(A)

+ a(y) = a(A + y).

With regard to the homomorphism problem discussed in Section 2.4.9, Spector [Spector, 1, Proposition 91 proved the following result using the same methods that he employed in his contribution to Theorem 3.1.9 : assume that rl and T z are locally are algebraically isomorphic (where 0, isomorphic; then A(rl)]j({Ol})and ,4(r2)/j({0,}) is the O-element of r,).It is natural to ask about the converse situation: if A(rl)/j({Ol}) and ,4(r,)/j({O,}) are algebraically isomorphic is it true that rl and r2are locally isomorphic ? Jeri s o n has provided an affirmative answer when rland rzare locally connected metrizable groups; a fancy topological contribution to the problem is found in [Jerison, Siegel, and Weingram, I]. 3.1.18 The structure of A(T) and the union of Helson sets. We now use Theorem 3.1.8 as the point of departure for a brief exposition of an important recent theorem giving

214

3 Results in spectral synthesis

information on the structure of A ( T ) ; basically we shall just introduce the reader to some of the literature. We rephrase Theorem 3.1.8 in the following way Theorem 3.1.8'. Let rl and r2be LCAG, let a : rl +-T 2 be an injective continuous homomorphism, and let El be compact so that E2 = aE, is compact. For each cp E A(T1) and for each r > 1 there is I) E A(T2)such that ll$llA(r2)

G rll~ll~cr,)

and

v Y E E,,

$(Y)

= cp 0 a - w

Proof. Let r = 1 + 6 and take 0 < E < 6)1VllA(r,). Let 9 E A(E,, r,)have the property that cp = 4 on E l . By Theorem 3.1.8, R - 9 E A(E2,I',) and

I/ 4-l @IIA(E2.r*) = l l ~ l l A ( E l , r l ) .

9 on E2 and

Take $ E A(T2)with the property that J/ = II$IIAtr2,

< IIR-'

@IIA(E2.r2)

+ 6. q.e.d.

If E 5 f then W E )= ~uP~ll~IIA(E)/IIcpllc,ca,:cp E is the Helson constant of E (where IIcpIlc,cs

\{OH

= sup Icp(7)l). H(E) is finite if and only if E YE

E

is an Helson set. The following generalization of Theorem 3.1.8' for the case that H(E2)< 03 is important because of the information obtained about the behavior of the functions outside of E2. Theorem 3.1.10. Let rl a n d r 2 be LCAG, let a : rl +-T 2be a continuous homomorphism, and let E2 E T 2 be an Helson set. Assume that

(3.1.44)

B:E2 +

rl

is a continuous function with the property that (3.1.45)

a o B:E2 -+ E2

is the identity function. Thenfor any cp E A(T,), r > H(E2),E > 0, and U a neighborhood of0 E r2,there is I) E A V 2 ) such that I I $ I I A c r 2 ) 6 r211cpllA(rl),

v Y E E2,

Irp

0

B(r) - $(r)l< 8,

3.1 Non-synthesizable phenomena

215

and V

Il(l(r)l ~ r ' s u p { l r p ( ~ ) l : a ( ~ ) ~U}. y+

Conditions (3.1.44) and (3.1.45) are obviously satisfied in Theorem 3.1.8'. In the above form, Theorem 3.1.10 is due to [Herz, 111. The major application of Theorem 3.1.10 is to prove Theorem 3.1.11. There is a continuous function w : (0,1] [I, a) with the following property. Given r, an Helson set E E r, r > H(E), cp E C,(E), and a closed subset F E r disjoint from E. Thenfor each E E (0,1] there is $ E A(f) such that --f

ll$llAm

rZllcpllm4E),

v Y E E,

l(l(Y)

v Y E F,

ll(l(Y)l

= cp(Y),

and r 2 EIIVllm

(cf. Remark 3 after Example 2.5.3). The main motivation for the development of Theorem 3.1. I1 was to prove that the finite union of Helson sets in r is an Helson set. For the case that r is discrete this union problem was solved by [Drury, 21 by using essentially a special case of Theorem 3.1.10. V a r o p o u l o s then extended the theorem to metrizable f and L u s t made the final step to every r. Stegeman and H e r z are responsible for sharp estimates on

44. In any case, D r u r y ' s idea has been the major influence on this subject. [McGehee, 31 has an historical note and discussion on these matters; and [McGehee, 21 contains a detailed exposition of [Herz, 111. In light of the proof ofTheorem 3.1.6 it is interesting to note that the union of two Kronecker sets (in T)is not necessarily a Kronecker set, e.g. [Bernard and Varopoulos, 11. Exercises 3.1 3.1.1 Non-synthesis of discrete spaces Malliavin's theorem and Exercise 1.4.3 combine to yield: r is discrete ifand only i f E is an S-set for each closed set E E f. A natural question is whether this result is valid for every commutative semi-simple Banach algebra X (instead of A(T)) with maximal ideal space X" (instead of r). a) Let X be the Banach algebra defined in Exercise 1.1.2. Prove that X" = Z contains non-S-sets (e.g. [Mirkil, I]). b) Let X be a commutative semi-simple Banach algebra whose maximal ideal space X" is a discrete abelian group. Assume i) {cp E X:cardsuppcp < a}is dense in X ;

216

3 Results in spectral synthesis

ii) For each cp E X and each x E @, (.,x)cp(.) E X.

Prove that each E E Xmis an S-set. Note that the example of part a) satisfies condition i). With regard to this example we also mention [Graham, 5 ; Wermer, 11. 3.1.2 A(r)andjinitely generated closed ideals Prove that A(R) doesn’t have any non-0 countably generated closed ideals (we distinguish, for example, between “principal closed ideals” and “closed principal ideals”). Compare this with D o u a d y ’ s result: let X b e any Banach algebra and let IE X be an ideal; iff isfinitely generated then I = f. 3.1.3 Multipliers and spectral synthesis

k E L(LD(T),LD(T)),p E [l, to], is a convolutor on Lp(T) if V cp e L D ( r and ) V x E G,

k(r,cp) = T, k(cp).

Consequently, the “L1multiplier theorem” quoted in the suggested proof of Exercise 2.4.8casserts that M ( r )is the set of all convolutors on L’(T). In 1923, F e ke te essentially established the multiplier problem, viz. to determine the convolutors of Lp(r). For the case ofp = 1, the multiplier problem is related in the obvious way to the homomorphism problem discussed in Section 2.4.9 (cf. Exercise 2.4.8). Regarding this nexus, a fundamental result which can be used to prove the above L’ multiplier theorem, and which can be derived using Bochner’s theorem and Exercise 2.5.lc, is the following: if X z B(G) is a commutative ring of functions (on G ) under pointwise multiplication and if A(G) is an ideal in X then X = B(G) (the result remains valid, if instead of the hypothesis that A(G) is an ideal in X, we assume that X . A ( G ) E B(G)); and some fundamental papers are [Edwards, 1; Helson, 3; Wendel, 1 ;21. R. E. E d w a r d s established the “modern” approach to the multiplier problem in the late 1950’s and [ H o r m a n d e r , 11 appeared in 1960(in this regard, see [Fi g8- T a l a m a nc a and G a u d r y, 11) ;expositions of the field are found in [Brainerd and Edwards, 1 : Edwards, 5,11, Chapter 16; Larsen, I]. a) For each p E [l, w ] let Ak(T) be the set of elements T E A’(T) for which (E3.1.1)

3 KT,D> 0 such that V cp E C,(T)c A’(T) IIT* d l p < KT.~II(PIID*

Prove that if T EAk(T) and cp E L p ( r ) ,where p E [l, to], then T is a convolutor on Lp(T)and T * cp E LD(T)(for p = to, T * cp E C,,(T)). b) Let A E L(LD(T),Lp(T)),p T E A ; ( f ) such that V cp E L”(r),

E

[l, to), be a convolutor on LD(r).Prove that there is

R(cp) = T * cp.

3.1 Non-synthesizable phenomena

217

(Hint: First do the case p = 2; then use the Riesz-Thorin theorem). f’is the multiplier corresponding to the convolutor T. Let A,(T), 1 < p < m, be the space of continuous functions m

(E3.1.2)

@=

2 cpr * $&,

(pU E L P ( f )and

$&€Lq(r),

k-1

A p ( T ) = LP(T) (3L Q ( f )

is a Banach space normed by

where the infimum is taken over all representations of @ having the form of (E3.1.2). A,(T) = C,(f) (resp. A2(T)= A(T)) and the corresponding I( ([-normis the supnorm (resp. [I IIA norm). [Figh-Talamanca, 11 proved that i f A l ( T ) , 1 < p < UJ,defined in part a), is given the canonical norm topology inducedfrom L(LP(T),LP(T)) then Ab(T) is the dual ofA,(T). Later, by an adjustment of a technique in [Herz, 71, e.g. [Eymard, 2, ThCortme 11, Herz proved that A,(T), 1 < p < 00, is a Banach algebra (cf. [Herz, 101 for a virtuoso performance, as well as other work by H e r z on Ap(T)). c) Prove that A y - ) = A p ) = M ( T ) c A ; ( T ) = A;(T) 5 A;(T) = A‘(T),

1 1 wherepE(1,co)and-+-= 1. P 4 The spectral synthesis problem for the (A,(T),AL(T)) duality was posed by E y m a r d in 1968 (e.g. [Eymard, 1 ; 21). He determined values ofp and n for which S”-’ 5 R” is and is not a “p S-set”, e.g. [Eymard, 2, Thtorkme 51. Later, [ L o h o u t , 11 showed: there is cp E A ( f ) such thatfor each p E (1, m), le,with the closure taken in the A,(T) topology, is not self-adjoint; in particular, there is a closed set E G T such that,.for each p E (1, co), E is not a “p S-set” (the remark and references about differentiation at the beginning of Section 3.1.2 are relevant here). In this regard, and related to Meyer’s work discussed in Remark 1 after Example 2.5.2, see [ L o h o u t , 21. 3.1.4 A topology for which synthesis always h o l h

[Schwartz, 31, [Ditkin, 11, Beurling, [Pollard, 21, and [Herz, 5, Theorem 4.5, pp. 210-212 and pp. 225-2261 have proved the following: let T be R or W; then there is a topology on C,(G), finer than the K topology, such that spectral analysis and spectral synthesis always hold (and, yet, spectral synthesis fails in R and R2 for the weak * topology!). This result is strengthened in [Herz, 61; in this regard and with regard to

218

3 Results in spectral synthesis

the technique introduced in Section 3.2.2, we refer to [Herz, 91.Prove that if E G W is the surface defined by sin : + sin2 2 71

Yz

L

L

+ sin2 L = r , Y3

r E (o,+),

then E is not an S-set for the topology of pointwise convergence, e.g. [Herz, 51.

3.1.5 Malliavin's theorem-the completion of Varopoulos' proof a) Define the map a : D , -+ T3 which takes an element y

E(yl,.

..) E D ,

into the element

(27t(.rlr4r7 . . . ( 2 ) ) ,27t(.rzr5r8...(2)), 2n(.r3r6r9 . . . ( 2 ) ) )E T3 where r, = eyI= enkj,for k, = 0 or k, = 1, .abc.. .(2) indicates the binary expansion of an element I E [0,1], and 27t(.abc.. .(2)) = 27tI E [0,27t].Prove that a is a continuous bijection, where the injection follows once we eliminate a set of measure 0 from both D , and T 3 . b) Because of part a) we induce the homomorphisms ~ ( 1 3 )

--f

c(D,)

-+

L~TJ),

from a and a-' ;and consequently we have the well-defined homomorphisms (E3.1.3)

V(T3)

--f

6L"(T3).

V(D,)

+ La(T3)

Using (E3.1.3),prove that the first homomorphism of (E3.1.3), viz. V(T3) --f V(D,), is an isometric homomorphism; and that, as such, we obtain a continuous map (~3.1.4) D, x D,

-+

1 3

x

1 3

between the corresponding maximal ideal spaces. c) Prove that if E E T3 x T3is a non-S-set for V(T3), then its inverse image by (E3.1.4) is a non-S-set for V(D,). (Hint:Suppose the result is false and use (E3.1.3) to obtain synthesis in L"(T3) 6 L"(T3);then regularize the situation in L"(T3)6L"(T3) to provide synthesis in V(T3), the desired contradiction). As indicated in Section 3.1.15, this completes the proof of Malliavin's theorem. In light of our discussion of the Reiter, D e Leeuw, H e r z theorem, it is instructive to state the general result which [Varopoulos, 81 gave in order to prove Exercise 3.1.5~;the proof is easy. Let XI and Xzbe regular Banach algebras with units, rp, and ( p 2 , let 4:X I + Xz be an isometric homomorphism onto a closed subalgebra f o r which 4 c p J = rp2, and let a : Xl;l+ XP be the induced continuous transpose of A (cf. Proposition 2.4.1); then

i)

a is injective (by the regularity)

3.1 Non-synthesizable phenomena

219

and

ii)

i f E , G X;l is closed and E2 = a-’ El then

k(E1) = A-’(k(Ez)).

If A satisfies two further conditions, which it does in the case of Exercise 3.1.5c, then we can conclude that iii)

jo= A --‘(A&))

(“k” and “j” are defined in the expected way).

3.1.6 Conditionsfor which A(EI,I‘,) = A(&, r,) Fill in the details for the suggested proof of Theorem 3.1.8. 3.1.7 A relation between A(R) and A(T), and a r e d t in synthesis a) We’ve indicated a connection between Theorem 3.1.7a and a theorem by Wiener. Actually, W i e n e r proved the following theorem (for the case of 8 = 1): Let cp: [-n,n] +. C be bounded and assume that there is 6 > 0 for which cp(y) = 0 when - 6 < IyI n,

cp(Y)

o

and let h:R + C satisfy the following conditions: h 2 0 is increasing on {x:x > 0}, h(-x) = h(x) for each x E R, and there is a constant k such that h(2x) < kh(x).

V x E R,

Then cp(y)

=

2 a,,e““’

where

2 la,, Ih(n) < co

if and only if m

$(y) =

f ( x ) elxydx

where

Prove this result. b) Let?= cp E A(T) satisfy the condition that (E3.1.5)

2’\f(n)l log In1 <

Prove that if cp f -

( I)0 then

00.

=

A(T) whenever cp(I) = cp(y) = 0 for -n

A(T); and using part a) prove that

< I < y < n,e.g. [Kahane, 2; Wik, 21.

(PX(~,,,)E

220

3 Results in spectral synthesis

3.2 Synthesizable phenomena 3.2.1 Integral representation and spectral synthesis. In this section we shall prove results aimed at determining whether or not a given pseudo-measure is synthesizable or not. The techniques that we use are related to ideas developed by Beurling. It turns out that there is a fundamental relationship between the existence of an integral representation for the operation ( T , cp), where T E A'(T) and cp E A(T), and the synthesizability of Tor cp. We shall be more specific on this point, shortly, but, first, we establish the setting and some notation. For technical convenience we shall generally work on T = T and with the space A' = ( T E A ' ( T ) : T ( O ) = O } .

By working in A' there is no restriction in the generality of any of our results since one can always add in a S and/or make a translation without affecting a synthesizable phenomenon. If T E A' has a Fourier series representation T then F

- 2' -2

c,,einY

;el"*

E

L~(T)

since {c,,:nE 2) is bounded; and F ' = T,

distributionally. In this situation, we shall write F-T

or

F-TEA'

for the correspondence between F and T ; obviously, then, if F then ( F - k) T.

N

T and k is a constant

N

We know that each p

E

M(T) is synthesizable and so

V cp E k(supp PI,

If F

N

p

E A'

(P,cp> = 0.

f l M(T),then F is a function

of bounded variation and

T

BV(T) will denote the class of functions F having bounded variation for which F p E A' fl M(T). In this case, it is easy to verify that rp d F = 0 from first definitions. In

N

3.2 Synthesizablephenomena

221

fact, F i s constant on any open interval contiguous to suppp and so when we write out the approximating sums for the Stieltjes integral, cp dF, we see that

1

t

2 dt,)(F(Y,) - F(Y,-

1))

=0

when y, and y, - are in the same contiguous interval. In the limit, the fact that q~ = 0 on supp p assures that the other terms in such sums are 0. Consequently, we see that if the inner product (T, cp), T E A’(T) and cp E A(T), has a “Stieltjes integral” representation and cp = 0 on suppT then there is a good chance that synthesis holds, i.e., (T, cp) = 0. One approach to rigorize this idea on integral representation is given in Section 3.2.2Section 3.2.4.

3.2.2 The Beurling integral and a criterion for synthesis. The basic and essentially cp E A(T), elementary calculation that we use is the following: for F- T E A’ and!= (3.2.1)

(T, cp) = 2 F(n)f(n) = i 2 ng(n)f(n)

=i

=L II ] - ! r2 -(2g(n)&n)f(n)(einr-

2‘p(n)f(n)n

where H EA’ is the “conjugate distribution”

-

i z’sgnn elnY,

sgnn = n/lnl;

note that by Parseval’s formula the expression (3.2.1) becomes

B(q,F)is the Beurling integral of cp with respect to F. Since

the above calculation is valid, and we have

-

Proposition 3.2.1. Given F T E A’ and cp E A(T). Then (3.2.3.)

B(cp, F) = ( T ,cp).

- sin2:‘”dr)

l)(e-Inr- l))dr,

0

!f

(I yn,[

222

3 Results in spectral synthesis

The integral B(cp,F) exists for 1; T E A' and cp E C(T) if N

In order to prove Theorem 3.2.1, below, we need to introduce the following quantities :

and V cp E C(T)

and

V F E LZ(T),

0

Clearly, llBll(rp,F)may be infinite, and llBll(cp,F)< m if and only if

0

Also V cp E C(T) and v F E L2(T),

IB(cp, F)I G IBI(cp, F) G IlBlKcp, F).

For any function cp EL'(T) we define (as [Beurling, 6; 11; 121 has done) the circular contraction

where p > 0. A straightforward calculation shows that

(3.2.4)

V P > O and V L Y ,

l%(Y+a-%(Y)l<

Icp(Y+4-cp(4\.

Proposition 3.2.2. Given F - T EA' and cp E C(T) for which IBl(cp,F) < Q). Then IBl(cp,,F) < m for each p > 0, and

(3.2.5)

lim IBl(cp,, F)= 0. P+O

Proof. IBl(cp,,F) < IBl(cp,F) because of (3.2.4). The result follows by the definition of cp, and the Lebesgue dominated convergence theorem. q.e.d.

3.2 Synthesizablephenomena

223

We are now in a position to relate these calculations to the problem of synthesis. Take F - T E A’ and cp E A ( T ) for which s u p p T Zcp, ~ and assume that IBl(cp,F) < to. Because of Proposition 3.2.1 and Proposition 3.2.2 we have (3.2.3) and (3.2.5). On the other hand, cpp = cp on a neighborhood of supp T. In light of (3.2.3) it is not unreasonable to expect that with a strengthening of our hypothesis, lB((cp,F)< co, we could conclude that (3.2.6)

B(v, F ) = B(cpo, F ) ;

and consequently we would obtain the annihilation, (T, cp) = 0. Note that rpp E C(T) when rp E A ( T ) ; and that if we knew that cpp E A(T) then the condition, ISl(rp, F) < co, would be sufficient to ensure that (T, cp) = 0 (cf. Exercise 3.1.7). Theorem 3.2.1. Given F then (T, cp) = 0.

-

T E A‘ and cp E A(T)for which suppTS Zrp. IfllBJl(rp,F)< 00

Proof. In light of what we’ve done it is sufficient to prove that IlBll(rp,F) < co implies (3.2.6). Set $ = rp, - cp so that $ Ej(Zrp). Let 8, E A(T) be an approximate identity for which $, = $ * 8, ~j(Zrp). For example, let 4K (3.2.7) 8, = - A , for (I1 < K, 3,

where A, was defined in Exercise 1.2.6. Since llO,II,# < 1, we have

w,,rllZ= 4 2 l i ( n ) h,(n)Iz n

nr

Sin2- G W ,~ I V . 2

Becausewe are assuming llBll (cp,F) < 00 we can use the Lebesgue dominated convergence theorem, and compute that lim B(t,hA- JI, F)= 0. 1 -+O

A ( T ) and so B($,,F) = (T, I ),) from Proposition 3.2.1 ; thus B($A,F)= 0 since $, E j(Zrp) and supp T E Zrp. Hence B($,F)= 0, and this is (3.2.6). q.e.d.

$A E

3.2.3 Technical lemmas. In order to apply Theorem 3.2.1 we consider some elementary conditions in order that 2%

(3.2.8)

1

1

-p IF, 41 IIcp, 41dT c a. 0

224

3 Results in spectral synthesis

-

Proposition 3.2.3. Given F E L'(T)for which p(0)= 0. If F T E A' then llF,rll= O(r1I2),

r

j.

0

(c$ Exercise 2.1.2a).

Proof. Since F - T E A'and llF, r1I2= 4 2'(&)I2

sin'

nr

2

for F EL2(T)(by Parseval's formula) we have

The Fourier series on the right-hand side of (3.2.9) represents the even function cp(r) = r2

nr - -on [0, n). 2

q.e.d.

The technique of Proposition 3.2.3 is strengthened in Exercise 3.2.1 to characterize pseudo-measures in terms of quantities such asllF,r 11. Notationally, we set o,(cp,4 = SUP

{IlTrcP

- cpII,>

O
and so llcp,lII (3.2.10)

< 02(cp,4.Also, cp E Lip,(T) if and only if cp E C(1)and

@,(I)

= ~ ~ ( c 1) p , = O(Ia),

I + 0,

where I > 0. o,(cp,2) is an increasing function as either p or I increase, and lim op(cp, 2) = 0.

V cp E LD(T),

l+O

Further, if cp E C(T) then

Proposition 3.2.4. a) Giuenf= cp E A(T)for which

2 If(n)l'Inl

< 03.

Then 2x

Va<0

3.2 Synthesizablephenomena

225

b) Given cp E Lip,(T), a > 112. Then

Proof. a) Take a < 312.

2x

2 If(n)12sin2 2

2rr

dl

0

0

Q

(I

1 2 - 2 ad l r

[

I

If(n)12

T

112

sin2 + d l ]

T

KZ If(4l’14. b) The result is immediate from (3.2.10). q.e.d. 3.2.4 Applications of the Beurling integral. Define, as we did in Section 2.4.10, 2n

(3.2.11)

1 A(cp) = 2x

P I 2 lip, 111 d l 0

for cp €L2(T).Theorem 3.2.1 and Proposition 3.2.3 yieldTheorem 3.2.2. Given cp E A(T). ZfA(cp) < m then cp is synthesizabfe.. As we noted in Section 2.4.10, A(cp) < m for cp E L2(T) actually implies that cp E A(T).

There are two classes of functionswhich come to mind immediately as possible applications of Theorem 3.2.2: first, we’d like to test various A(T) r l Lip,(T), a > 0; and second we’d like to test A(T) n BV(T) since, as is well known, cp E BV(T) if and only if (3.2.12)

Il.r,cp - vII1 = O(lrl),

r

-+

0

(e.g. [Benedetto, 121). For the first case we use Proposition 3.2.4b and obtainCorollary 3.2.2.1. If (3.2.13)

cp E Lip.(T),

a > 1/2,

then cp is synthesizable (of course, if& > 1 /2 then Lipa(T) 2 A(T) by Bernstein’s theorem, as we noted in Exercise 2.5.2).

For the second case, (3.2.10) and (3.2.11) combine to yield-

226

3 Results in spectral synthesis

Corollary 3.2.2.2. I f c p

r:

E A(T)

n Lip,(T) n BV(T), where a > 0, then cp is synthesizable.

If q E A(T) is an element of Lipll,(T) or BV(T) then the integral, (3.2.1 l), has the form

-.

In both of these cases the LLquantity, llcp,I11, is inappropriate for the functions we

0

are considering, and there is no reason to expect the integral representation, B(cp, F),to provide information about synthesis. Before further discussion on the method of integral representation we shall settle the specific problems concerning Lipll,(T) (Paragraph 3.2.5) and BV(T) (Paragraph 3.2.7). Remark. The contraction method in spectral synthesis is due to Beurling, but important contributions are found in [Kinukawa, 1; 21 (cf. [Kinukawa, 3, Theorem 31). Beurling's theorems provide synthesis for weaker than weak * topologies, and parallel results without the theory of contraction are found in [Benedetto, 111. In this regard, see the remark at the end of [Beurling, 6, Section 21 and [Domar, 31. 3.2.5 The hurling-Pollard theorem. Given a closed set E s T and recall the definition of E, in (3.1.12) and the related remark in (3.1.13). The following result is essentially due to Beurling and [Pollard, 21 (cf. [Kahane, 10, pp. 77-78]).

Theorem 3.2.3. Given cp E A(T) and set E = Zcp. Each of the following conditions implies that cp is synthesizable :

b) SUP Iq(y)l = O(I1lz),I + 0,A > 0. YE€,

c) cp E Lip,(T), a 2 1/2. Proof. It is obviously sufficient to prove that cp is synthesizable when condition a) is satisfied. Given T E A'(E) we use (3.2.7) and set T , = T * gA.

Note that TAE C(T) and suppT, c E,. Clearly, llTAllA,< IITIIA. and so by the Lebesgue dominated convergence theorem, T, + Tin a(A'(T),A(T)). Thus,

(3.2.14)

( T , cp) = lim (T,, cp) L -0

= lim A -0

J E,\E

TA(y)cp(y) dy.

3.2 Synthesizable phenomena

227

By (3.2.14), (3.2*16)

P>l 6

l
11

112

(sup Idy)()(nl(EA\ E ) ) ” 2 . YEEA

(3.2.14), (3.2.15), (3.2.16), and a combine to give the result. q.e.d. The theorem can be made slightly finer by using only the L2-estimate in (3.2.16). 3.2.6 Constant values of the primitive of a pseudo-measure. Proposition 3.2.5b is a technical fact which we need in what follows. The statement itself is intuitively reasonable but some proof is required.

Proposition 3.2.5. a) Let F E L’[O, 1] be real-valued and assume that

/

F(y) d Y ) dy = 0

6

for all continuousfunctions rp which vanish at 0 at 1 . Then F = 0 a.e.

-

b) Given F T E A’ and assume that T = 0 on the open interval (A, y ) c R, considered as an open subset of T. Then there is a constant k such that F = k a.e. on (A, y). Proof. a) Assume the result is not true so that IF1 > 0 on some set XE (0,l) having positive measure. Let

There is an open set U,c (0,l) such that E

J

(3.2.17)

(0.1)

IF(Y)I dy < i’

\ u,

and we take U,large enough so that m(X f l U,)> 0. Set X+ = {y E U,:IF(y)l > 0) and X - = { y E 17,: IF(?)[ < O}; then X, are measurable and either mX+ > 0 or mX- > 0 (or both). From measure theory there are compact sets K,

c X,

for which (0, I } f l K , = 0

and (3.2.18)

m(X,

\ K,) < -.

E

161lFlli

228

3 Results in spectral synthesis

(3.2.17) and (3.2.18) combine to yield 1

&

I W l d~ < 4’

IF(y)ldy -

(3.2.19)

K+UK-

0

and so

j

(3.2.20)

I m l dY ’T ‘ 3E

K+UK-

Choose continuous functions $* to be + I on K* and 0 on K,; further, assume that 0 < $* 6 1 and let $,(O) = $,(l) = 0. Define q = $++ $-. Then 1

J F(Y) cp(Y)dY = J

0

(0.1)

F(Y) cp(Y) dY +

\ U8

j

IF(r)ldY + r,,

K + UK-

where Ire[ < 4 4 because of (3.2.18). Thus, by (3.2.17),

I 1

(3-2.21)

F(Y)cp(Y)dY =

j

I W l dY + r ,

Irl < 42.

K+UK-

0

By our hypothesis and (3.2.21) we conclude that

the desired contradiction. b) Let X

= {cp E

C(T) :SUPP cp

(4Y ) )

and Y = {$ E X:3 0 E X such that 8 ’ = $},

where we take the ordinary derivative in the definition of Y. Y is a subspace of X and, by hypothesis and the definition of the distributional derivative, we have (3.2.22)

4 ‘ $ E Y,

( F , $ ) = 0.

Clearly, X \ Y # 0 and we choose cpo

E

X \ Y for which

cpo(y)dy = 1. T

3.2 Synthesizablephenomena

229

Given cp E X we define

j

$ = cp - 'PO cp(v)dy and S

$(r)dr,

O(s) =

s E [0,2rr).

0

It is easy to check that $, O E X and, of course, 8' ='$. Consequently, each cp E X has the representation (3.2.23)

cp = ccpo

+ $,

c E C and $ E Y.

+

If (cl - cz)cpo - $& = 0, where c, E C and $, E Y, we see that c, = c2 and $, = t,bZ since cpo E X \ Y. Thus, the representation (3.2.23) is unique. Therefore, because of (3.2.22) and (3.2.23), we have (3.2.24)

V cp E X ,

( F , cp>

=c

J* F(y)c p ~ ( y )dy =

k d y ) dy

I

I X

since c = cp(y)dy (here, k = F(y)q,(y)dy).

Now, is dense in the set W of continuous functions on [A, y ] which vanish at A and y. Therefore, from (3.2.24), F - k E L'[;l,y]has the property that

q.e.d. Let E G T be a closed set and write the complement as m

I

E"=UZ,sT, 1

is a disjoint family of open intervals. We write I, 1 Haar measure m on T is such that E, = mZ, = -(y, - A,) and so 2n

where { I , : j = 1,. ..}

E FI

m(E") =

2ml, = 2E, < 1.

Because of Proposition 3.2.5b1if F (3.2.25)

F= i k , ~ , , I

on E - .

-

T E A' fl A'(E) then

= (;l,,y,).

230

3 Results in spectral synthesis

3.2.7 Katznelson’s theorem: cp E A(T) n BV(T) is synthesizable. The following theorem is due to Katznelson.

Theorem 3.2.4. If cp E A(T)

n BV(T) then cp is synthesizuble.

Proof. i) It is sufficient to prove that (T, cp) = 0, where F - T E A’ and T E A’(2cp). We use the notation of (3.2.7) and Theorem 3.2.3. Therefore, setting FA= F * 8,,

we compute that F ; = T , = T*9,,

pointwise and distributionally (cf. [Schwartz, 5, ThCortme 111, p. 541). Thus, since cp E BV(T), we integrate by parts and obtain (3.2.26)

(T,, cp)

=

I

J

T,(Y) cp(Y) dy = - FAY) drp(y).

T

T

ii) Suppose F E C(T). Then lim \IF, - FII, = 0, A > 0. Because ,-PO

y~~ (T, - T,cp)

= 0,

A > 0 ( as we noted in the proof of Theorem 3.2.3), (3.2.26) yields

From Proposition 3.2.5b and the definition of the Stieltjes integral, we see that the integral in (3.2.27) vanishes since F takes constant values on the intervals contiguous to zq. iii) If F is not necessarily continuous we use the notation of Section 3.2.6 and write (3.2.28)

-VA9 cp) =

5

J-1

Fdy)dcp(y) II

(instead of (3.2.26)), where (2q)-= 1

Let m,(A) = F,((yj j,

Ij.

+ A,)/2) and define the functions HA, = FA- mj(1) On Ij .

Clearly, for each j, (3.2.29)

FA(y) IJ

ddy)

=

I IJ

J

(y) d d y ) *

3.2 Synthesizable phenomena

231

In part iv) we shall prove that

(3.2.30)

lHA,,(y)l = K < 03.

SUP SUP J . 1 YEIj

Because of (3.2.29),(3.2.30),and the fact that cp E BV(T), we can use the Lebesgue dominated convergence theorem and take the limit (as 1+ 0) under the summation sign in (3.2.28). Since F is constant on I, we proceed as in part ii) to see that Vj,

lim

A -0

j FA(?)dcp(y) = 0,

1 > 0.

IJ

Consequently, (T, cp) = 0. iv) It remains to prove (3.2.30). An easy calculation (e.g. Exercise 1,2.6)shows that 3 M > 0 such that V 1> 0,

llTA Ilm Q

M

T.

If mZJ < 21, then, by the mean-value theorem,

v y E I,,

IHA.,(y)l

=

l H A , j b ) - H,,,((Y,

+ 1~)/2)li2M.

If mI, > 21 we compute that HA,,= 0 on [A, + 1,y, - 11; and we again obtain that lHA,,(y)l i2Mfor y E I., by using the mean-value theorem for the difference H J y ) H,,,(1, + 4(rev. H ~ Y- H ) J Y , - 4) if Y E (A, 1, + 4(resp. Y E (7, - 1,1~,)l q.e.d.

3.2.8 Spectral synthesis and the Kempisty-Denjoy integral. We now return more directly to the integration problem introduced in Section 3.2.1 (cf. (3.2.27)). As a warning to the reader, we note that this section and Section 3.2.9are more speculative than not; of course, the integration problem itself is important. Let a E (O,+), take F E L'(T) to be real-valued, and let I c T be a closed interval in A. Define m(l, a) = sup { y:m{y :F(y) < y} < rmf} and M ( / , a ) = inf{y:m{y:F(y) > y } Q arnl}. Note that if y , < y2 then m{v :F(Y) < Y l > Q m{r :F(Y) < Y2> and

232 3 Results in spectral synthesis

and so am1 2 m{y:F(y) < y 2 }

-

am1 2 m{y:F(y) < y d

and am1 2 m{y:F(y) > y,}

=-

aml 2 m{y:F(y) > yz}.

Proposition 3.2.6. Let a E (O,+), take F E L1(T) to be real-valued, and let I S T be a closed interval in FI. Then a) m(l, a) < M(l, a). b) -cn < infF(y) < m(l,a) and -cn < m(l,a). YE1

c) M(Z,a) < supF(y) < m and M(l,a) < m. VSl

d) ZfF= k on Zthen m(l,a) = M(l,a) = k. Proof. a) Note that card{u:m{y:F(y)= u } > 0)6 xo, and thus {y:m{y:F(y)= y } = 0) is dense. Take such a y which satisfies

(3.2.31)

aml2 m{y:F(y)< y } .

Also let z satisfy

aml 2 m{y :F(y) > z}. If y > z then

rn{y:F(y)> y } 6 m{y:F(y)> z } 6 aml. Combining this with (3.2.31) gives

2aml 2 m{y :F(y) < y } + m{y:F(y) > y } = mZ, a contradiction, since a < 1/2. b) Let y = inf{F(y):y E Z} (y could be fa)so that y < m(l,a), since m(y E I :F(y) < y } = 0. To show that --a, < m(l,a). If there is no z for which m{y:F(y) < z} < aml then for all z, m{y:F(y) < z} > aml. Now, as z + -03, { y :F(y) < z} decreases, and so

m X = lim m{y :F(y) < z } 2 aml, 0'--4)

where X = 0{ y :F(y) < z}. z

Consequently, m{y:F(y) = -m} > 0 since F(y) = -a if y E.'A This is the required contradiction. c) is proved in a fashion analogous to b). d) If y G k, m{y E l:F(y) < y} = 0; and if y > k, m{y :F(y)< y } = mZ > aml.

3.2 Synthesizablephenomena

233

Thus k = m(l,a). A similar argument works for M(Z,a). q.e.d. Given p E C(T), and define the upper and lower sums,

v, F, p)

n

=

W )= 2 MJ,,4 ( d y h - dY,- 1) 1

and n

s(P, F, cp)

= s(P) =

z:

W

I Y a) (cp(Y,) -

d Y ,- l)),

1

for any partition P:yo < yI < , .. < yn and J , = [y,- l,y,]. F is integrable with respect to p with integral

1

L = (K) F d p

if for each E > 0 there is a partition P, such that

(3.2.32)

)S(P)-LI < E

for each partition P finer than P,. Conditions equivalent to (3.2.32) are also valid in terms of s(P) and in terms of a Cauchy criterion. Further, if (K) F d p exists for some a E (0,1/2) then it exists with the same value for all a E (0,1/2).

Remark. We use the notation “ ( K ) I F d p ” because of Kempisty’s theory of integral (for the case p(y) = 7) in 1925. Actually, [Denjoy, 11 proved the equivalence of the Kempisty integral and one of his own (vintage 1919) in 1931. Setting en(y)= elnyas we did in Proposition 2.5.3, it is easily checked that if F then

(3.2.33)

j

(K) Fde,

= -f(n).

Also, if F E BV(T) then

(3.2.34)

‘if pE

C(T),

I

1

(K) F d p = F d p ,

by Proposition 3.2.6 and the definition of the Stieltjes integral; in this case {s(P,F, en),S(P,F, en):n E Z and P) is uniformly bounded because of (3.2.34).

-

T E A‘

234 3 Results in spectral synthesis

-

Proposition 3.2.7. Giuen cp E A(T) and F T E A’. a) If{s(P,F, en):n E Z and P } is uniformly bounded then

b) Zf(3.2.35) is valid and T E A’(Zcp) then (T, cp} = 0. P r o o f . a) follows by hypothesis and (3.2.33). b) Choose a partition P,, containing all of the endpoints of the first ncontiguous intervals of zq. We further demand that P,, should satisfy the following property: if

Y E P.n zcp, then there are endpoints 0, and 8, from the first n contiguous intervals of Zq, such that Pmn(e,,ek) ‘Zq* Because of Proposition 3.2.6, s(P,,,F,cp) = 0. q.e.d.

3.2.9 The general problem of integration and synthesis. The situation in Section 3.2.8 appears quite technical, but we shall try to clear things up a bit by the following explanation in terms of determining A;(T). Given a closed set E E r and T E A‘@). Assume that there is p E M ( E ) and K(y, * ) : E +. A;(E)

such that

j

T = K(Y9 4d A 4 E

(in the sense of the Bochner integral). Then

v cp E k(E),

(T,cp> = 0.

In Proposition 3.2.7a we started with the specific and simple measure p = S (assuming for the discussion that 0 E E); and as a result were forced into the situation of imposing rigid boundedness conditions on s(P, F, cp), whose analogue in the above scheme is K( , ). By allowing greater flexibility in the choice of p the type of conditions required on K can be weakened so that the resulting “Proposition 3.2.7a” can have more useful hypotheses. To say anything more definite would require more specific information on the given T EA’(E). In any case, from this point of view, the integration problem becomes one of constructing useful kernels K and choosing appropriate measures p.

3.2 Synthesizable phenomena

235

3.2.10 A characterization of pseudo-measures in terms of Lp. We now consider a variation on the Beurling-Pollard theme which allows us to determine information about F when F- T E A'. We shall use the Hausdorff-Young theorem, e.g. [Katznelson, 5, Chapter IV]: 1 1 Theorem 3.2.5. Given p E [ 1,2] and define q by theformula - + - = 1. P 4 a) IfF E LP(T)(G L'(T))then P = @ E La(Z)and 11@11a

b) If@

G IIFllP.

E Lp(Z)then

IlFlla G

2 @(n)einY converges in La(T)to afunction F E La(T)for which

II@IIP.

Theorem 3.2.6. Given F T E A' where T a) Then F E f l Lp(T)and N

1 1 where-+-=

- 2'

c,, elnY.

1.

P 4 b) I f E = suppT and m E = 0 then (using the notation of(3.2.25)) m

(3.2.36)

T = 2 k,(6,,

- by,)

on C'(T),

1

where

(3.2.37)

V r < 1/(2ellTllA.),

2 erIkJ'

&J

< 03.

J-1

Proof. a) If p > 1 then 2'Ic,,/nlp< 00, and so we apply Theorem 3.2.5b to obtain F E f l {La(T):q< a}and '

b.i) Since mE = 0 we have m

F = 2 kJ x I , a.e. 1

(3.2.36) follows by the definition of the distributional derivative.

236 3 Results in spectral synthesis

ii) We compute, for q < co, that

since we can use the Lebesgue dominated convergence theorem and since

Combining (3.2.38) and (3.2.39) we obtain

),,,.

iii) We now estimate 2'l/lnlp ( n Clearly, if p > 1, dt 1"

2

1 p-1'

1

Thus,

By the ratio test

Consequently,

2 (rKp)4/q!converges, and a

by the usual properties of interchanging

3.2 Synthesizable phenomena

237

sums in absolutely convergent series, we have I - 1

J-la-O

q.e.d. The estimate in (3.2.37) combines with Theorem 3.2.1 to yield an assortment of expected corollaries. There areeasy examples and well-known methods (e.g. [Bary, 2,11, pp. 243 ff.]) to show that ( r l Lp(T))\Lm(T)# 0 (cf. the interesting potpourri composed of [Banach, 1, pp. 203-204; G r o t h e n d i e c k , 1; H a n d e l s m a n and Lew, 1; Rickert, 1; S h a p i r o , 11). On the other hand, if F- T E A', it is not transparent that F is ever unbounded (cf. (E1.3.4)). For convenience, let A ; = { T EA': F ELm(T)when F - T EA'].

In Exercise 3.2.3d we show that there are elements T E A'\A; for which card supp T = No. In the opposite direction we have

Proposition 3.2.8. Given

- 2 cne"nY m

(3.2.40)

T

E A'

n-1

where { k :n = 1, ...} is a lacunary sequence (defined in Exercise 1.1.4). Then F E A(T), where F - T, andso T EAZ(T) (as we show in Exercise 3.2.5).

P r o o f . If k,,,/kn b 1 > 1 for each n, then

Thus,

q.e.d. Naturally, given (3.2.40) we can verify directly that TEA:(T).

Remark. f= cp E H1(T) if f ( n ) = 0 for each n < 0 and f~ L1(Z). Clearly, H1(T)is a closed ideal in A(T), and there is a fundamental theorem by Hardy, e.g. [Katznelson, 5, p. 911, which says: (3.2.41)

f= cp E H'(T)

-5 1

If(n)l/n < Q).

238

3 Results in spectral synthesis

In 1961, John and Nirenberg made a study of functions FEL'(T)for which 1

(3.2.42)

SUP I

ml

1

IF(y) - FIJdy = IIFII* < 00,

I

where Z E T is a closed interval in R and 1

1

FI = - F(y)dy ; mZ I

such functions are said to have bounded mean oscillation and if (3.2.42) is satisfied we write F E BMO. Fefferman has proved that BMO is the dual of H'(T).The major recent work on this topic is [Fefferman and Stein, 11. We mention BMO since

LYT) E BMO 5 n LP(T), and because, by Theorem 3.2.6a and (3.2.41), F-TEA'

=- F E B M O .

There are functions F E BMO for which i@ Ink(n)I = co (cf. Exercise 3.2.3). 3.2.11 Measure theoretic criteria for a pseudo-measure to be a measure. With regard to

Example 2.5.3~we now prove Proposition 3.2.9. Let E E T be a closed set for which mE = 0. Then

(A,(T) was defined in (2.5.9)). Proof. Set C; = Cm(T)il A,(T) and let T EA'(T) have the property that Vcp~Cz,

(T,cp>=O.

Since cp = 1 E C; we see that T E A' and so

-

where F T. The function cp used in the proof of Proposition 3.2.5 can easily be adapted to have the form $', supp $ 2 (0,l). Thus, if ZC E" is an open interval in R we use Proposition 3.2.5a in conjunction with (3.2.43) to conclude that F = 0 a.e. on Z. Since m E = 0 we see that F = 0 a.e. and so T = 0. The result follows by the HahnBanach theorem. q.e.d.

3.2 Synthesizable phenomena

239

Proposition 3.2.10. Let E E T be a closed set for which mE = 0 and let T E A‘(E). Then T E Y(de$ned before Proposition 2.5.6) ifand only i f T E M(E). Proof. The result follows by Theorem 2.5.5 and Proposition 3.2.9. q.e.d. With regard to Proposition 3.2.10, it is shown in [Benedetto, 6, Theorem 2.61 that Proposition 3.2.11. Let E c T be a closed set for which mE = 0 and let F’ = T distributionally, where F E L’(T),f(0)= 0, and supp T c E. Then T E Y ifandonly ifT E M(E). Proposition 3.2.1 1 can’t be extended to any first order distribution Ton T (the order of a distribution was discussed in Exercise 2.5.58) as is seen by the example m

T=

2 and;/nr I

2

Ion

I *.

A condition for membership in Y is given in [Benedetto, 111. Also, by standard measure theory, Proposition 3.2.12. Given a closed totally disconnected set E E T and T E A‘(E).Assume that

for every disjoint family {F,:j= 1,. ..} C.F( using the notation of Section 2.5.9). Then T EY .

There are closed totally disconnected sets E E: T, mE > 0, and T E A’(E) \ M(E) such that T EY [Katznelson and Rudin, 11. 3.2.12 Kronecker’s theorem and the condition that A’(E)= M(E). As we promised in Section 1.3.13 we shall construct uncounlable sets E c T of strong spectral resolution. Such a set necessarily has the property that mE = 0 (cf. Sections 3.2.10 and 3.2.1 I). The construction is due to [Kahane and Salem, 11 and uses Kronecker’s theorem. Of course, in light of Example 2.5.3, it is really only necessary to exhibit a perfect Kronecker set (referenceswere given after Theorem 3.1.6‘); and that would be a slightlyeasier task than the Kahane-Salem construction, although also depending ultimately on Kronecker’s theorem.

We made use of a form of Kronecker’s theorem in Section 2.4.8, and we shall now prove a stronger version in Theorem 3.2.7 assuming the facts about the Bohr compactification that we listed in Section 3.1.16. Kronecker’s theorem is a result in Diophantine approximation and we refer to [Benedetto, 61 for references and a discussion in classical terms. The proofs of Theorem 3.2.7 and Theorem 3.2.8 are obviously true in a more general setting.

240

3 Results in spectral synthesis

Theorem 3.2.7. Let E c T be aJinite set and let E E (0,l). a) There is N > 0 such that for every homomo;phism rp:T

--f

T

there is an integer nQ E [-N, N ] such that y

E,

Id?)- (yYnQ)l

<

b) If E is independent over Q then there is N > 0 such that for every function q:T

-+

T,

with the property

v YE

€9

IdY)l

= 1,

there is an integer nQE [-N, N ]for which y

E,

Id?)-

(79

nQ) l

c) I f E is independent over Q then there is N > 0 such that

Proof. a) Recall that PZ is the Bohr compactification of Z. If $ E PZ then by the definition of the topology on PZ,

is an open neighborhood of $. Thus {Ne($):$ E PZ} is an open cover of the compact space PZ and we let {N,($,) : j = 1,. ..,k} be a finite subcover. By the definition of PZ, it is possible to choose n,EN,($,)n Z, and we let N = max{ In,\ : j = 1,. ,,,k}. b) Let (E) c T be the group generated by E. Since E is independent each y E (E) has a unique decomposition, y = 2k,A, k, E Z. IEE

Thus the function $Q:(E) -+ T, where vyE(E),

+Q(y)=

n

I€€

is a well-defined homomorphism.

q(A>kA,

3.2 Synthesizable phenomena

241

There is a standard procedure, paralleling the proof of the Hahn-Banach theorem, to show that i,bw extends to a homomorphism qe:T -+ 1. The result now follows from part a). c) Take N from part b). Assume, without loss of generality, that if p E M ( E ) then SUPP p = E. If p E M(E) then p = 2 aySyand we define cp :E + T as p(y) = a,/la,l. Theconditions of YE€

part b) are satisfied and for the appropriate nw E [-N, N] we have ‘dYEE,

E’Icp(Y)-(Y.nw,)l

v Y E E,

Re [cp(Y)(Y,ne)l 1 - 4

Thus,

-

=I’P(Y)(YJw)-ll 2 111 -Re[dY)(Y,naJl.

’

and so

This gives the result. q.e.d. The format for the following construction is to construct a perfect set E c T such that for each T E A ’ ( E ) there is a canonical procedure to write down a sequence {pk:k = 1,. ..} c M,(E) for which

(3.2.44)

II

SUP Pklll

<

and

(3.2.45)

fir

--t

in the K topology.

We conclude that T E M(E) by the Alaoglu theorem. Recall that if the norm in (3.2.44) were replaced by 11 I(A, then (3.2.44) and (3.2.45) are equivalent to the statement that the sequence {pk:k = 1 , . ..} converges to T in the j? topology. We mention this now because of the discussion in Section 3.2.13 and Section 3.2.14.

Theorem 3.2.8. There are non-empty perfect sets, E E T for which M(E) = A’(E). Proof. i) We shall write E = fl E,,, where {En:n = 1,. ..} is determined inductively and each Enis a disjoint finite union, E n = U E;, J

of compact neighborhoods EJ”. Take El c T to be any compact neighborhood and take F1E int El to be a non-empty finite independent subset.

242

3 Results in spectral synthesis

Assume we have E,,. . ., En- and Fl,. ..,F,,-1. Let F,,= {y,,,:j= 1,. ..} E int En- have the properties that F,,- G F,,, F,, is finite and independent, and card(F,, fl EJ-’) = 2 for each j. We define En. Choose cp,, E A(T) such that 2,

IIqJ.nllA<

cp,,

,, = 1 on a compact neighborhood V,,,, and y,,,,

Vjjzk,

and suppcp,,,,

E int

V,,,,,

S U P P ~ ~ , , ~ ~ S U P P ~ ~ ~ . ~ = ~ ,

= int En-l.

Take E,, > 0 with the property that ~,(1

+ 2cardF,,) < l/n.

Let N,, > n be the integer corresponding to F,,and E = 1/2 in Theorem 3.2.7. For each integer m E [-N,,,N,,] we choose i,bm E A(T), as in Exercise 2.5.lb, such that

i,b,

= (y,.,,,

m) on a compact neighborhood W,,,,

and II$m

- (. ,m)II, < E n .

Define

and En=U E;. I

From our definition, E is a non-empty perfect set. ii) Take T E A’(E). Preserving the notation of part i) we define Pn=

2 ( T ,( P J . ~ ) ~ Y I . ~ J

Thus, p,,E M(F,) a d (3.2.46)

IlPnlllG 2(cardFn)IITII,,.

Set T,, = T - p,,. iii) We prove that

(3.2.47)

V rn E [-Nn, Nn],

( T n , $ m )= 0.

243

3.2 Synthesizablephenomena

iv) We next prove that (3.2.48)

V k E Z,

limp&) n

= ?(k).

From (3.2.47) we have

v

[ - N ~ , N ~ I , fin(m)=pn(m)-
+m>,

and so, from (3.2.46), (3.2.49)

V mE

[-In,

Nn I,

I T n (m)I = I (Tn ( <~ ~

' 9

m) - $ m )I E n I TI~A, ~ + 2cardFn). ~ T ~ ~ ~

~

(

1

Fix k E Z.For any n 2 Ik I we have

I f n (k)I G I T I A S I n by (3.2.49) and since Nn > n. v) Because Fn is independent we use Theorem 3.2.7 to estimate 4IIPnII1 G suP{lfin(m)l:mE [-Nn,NnI}

+ sup{lf",(m)l : m E [ - ~ n , ~ n ] ) G IIT(IA*+ EnIITIIA,(l + 2cardFA G IITIIA,

where the last inequality follows by (3.2.49).

vi) The estimate of part v) yields the II I(,-norm boundedness of { p n : n= 1,. ..}. Consequently, T E M ( E ) by (3.2.48) and the Alaoglu theorem. q.e.d. Theorem 3.2.8 should be compared with [McGehee, 1, Section 71. The above method leads to a very natural question: for any non-empty perfect set E c T, does there exist a non-empty perfect subset F E E for which A'(F) = M(F)?

244

3 Results in spectral synthesis

Kaufman’s method, referred to after Theorem 3.1.6’, yields a positive solution to the question, and, in fact, F can be chosen as a Kronecker set.

3.2.13 Bounded spectral synthesis. A closed set E c T is a set of bounded synthesis, a bounded S-set, if such that V T E A ‘ ( E ) 3 { p“ :n = 1 , . ..} C M,(E) lim p. = T n

in

a(A’(T), A(T)).

Bounded S-sets are important from the point of view of actually constructing the process to approximate a synthesizable phenomenon, as well as bounding the 11 [IA,norms of the approximating measures. If E is an S-set we can always approximate T E A’(E) by a directed system from M,(E) but we have no control over the 11 11”’norms of the elements from this system. Consequently, it is an interesting problem to determine whether or not a given S-set is a bounded S-set. In light of Theorem 2.2.1 and Theorem 3.2.9 it is worthwhile to note that this problem is “dual“ to the “bounded spectral analysis” problem discussed in Section 2.2.7.

Theorem 3.2.9. Let E c_ T be a compact set. E is a bounded S-set V T E A ’ ( T ) 3 {p,,:n= 1, ...} c M,(E)

o

such that

lim in= f in the /Itopology

(the topology was defined in Section 2.2.1). P r o o f . (>) The fact, supIIpnIIA,< a, is a consequence of the Banach-Steinhaus theorem. By our hypothesnes and properties of Z,it is immediate that fl --f f in the K topology. (-=) This direction is clear by the Lebesgue dominated convergence theorem. q.e.d. The above result is true much more generally; the “0’ direction is always trivial whereas the ‘ ‘ 5 ’ ’direction can require a little more effort.

3.2.14 A characterization of bounded-S-sets. It turns out that there are closed sets E G T which are S-sets which are not bounded S-sets. The basic examples are given in [Varopoulos, 1 1 , Theorem 3; L i n d a h l and Poulsen, 1, Chapters XI and XI1 by V a r o p o u l o s ; K a t z n e l s o n and McGehee, 2, Theorem VI]. In order to describe a closely related problem we define

where the supremum is taken over M(E) \ (0) and where E c T is closed. A(E)is a ( ~ ~A ( E ) C A”E). Banach space normed by I1 I ~ A ”and

3.2 Synthesizable phenomena

245

Further, Proposition 3.2.13. Let E c T be closed and take cp E C(E). cp E A(E) o there is a sequence {cp, :n = 1,. ..} c A(E)for which (3.2.50)

limcp.

= cp,

uniformly on E

and (3.2.51)

SUPII(P~I/A(E)= n

K < 03.

Proof. (=.) Let Y be A ( E ) along with all uniform limits of sequences {cpn:n= 1,. ..} c A(E) for which (3.2.51) holds. Y is closed in C(E). Take cp E x ( E ) .If cp E C(E)\ Y then there is p E M ( E ) \ (0) for which 1(p, cp) I > 0 and (P,y> = 0. Since en(y)= elnyis an element of A ( E ) we contradict the hypothesis that cp E x ( E ) . (c) Let cp satisfy (3.2.50) and (3.2.51). If E > 0 and p E M ( E ) then for large n

Thus, I(p, cp>l

< KIIpIIA.and Kis independent of p ; hence cp E A@). q.e.d.

It is clear that A ( T ) = J(T). On the other hand, examples for which A ( E ) #A(E)have been given by [Katznelson and McGehee, 1; V a r o p o u l o s , 101. The relation of this topic to the problem of finding bounded S-sets is given byTheorem 3.2.10. Let E G T be closed. A ( E ) is closed in A(E) + E is a bounded S-set. This result is a theorem in functional analysis and follows from a classical theorem in [Banach, 1, p. 2131 (cf. [Dixmier, 11). 3.2.15 The structure problem. Given T EA’(T). It can be quite difficult to determine whether or not T E A;(T). Until now any weakening of this problem has been on T, e.g. by considering topologies weaker than the weak * topology in which to approximate T with elements from M(suppT). Another way to simplify the problem and a t the same time to pose the “structure” problem mentioned in the Introduction is to look more closely at the behavior of f’. The structure problem, then, is to investigate the relation between the behavior of f’ and the support properties of T (among other things); and, from our point of view, to gain information about the synthesizability of Tin terms of the behavior of p. 3.2.16 The balayage problem and spectral synthesis. In order to pose some specific problems in terms of our remark in Section 3.2.15 it is convenient for perspective to comment on Beurli ng’s balayage problem.

246

3 Results in spectral synthesis

Let X G G and E E r be closed. Balayage is possible for X and E if Vp

E M(G)

3v

such that D = ji on E .

E M(X)

The notion of balayage stems from P o i n c a r e ' s balayage process in potential theory; and historically E was a collection of potential theoretic kernels. Assume that E c r is a compact S-set and that V ~ E E and

V V G ~ ,

a compact neighborhood of y, 3 pEM,,(VnE);

then Be u r 1i n g (1959-60) proved that balayage is possible for such an E and any closed X if and only if 3 K ( X , E ) > Osuch that V

T = Q E Cb(G)for which supp T c E ,

we can conclude that

SUP I

X€C

6 K ( X , E ) SUP I Q(4 I X€X

(cf. [Beurling, 14; H. L a n d a u , 11). This led to the natural dual problem; find closed sets X G G and E G r for which it is possible that (3.2.52)

V Q E Cb(G) 3 3 = P ' E C,(G)for which supp S E E, such that

v x E x,

Q(x) = Y(x).

We shall refer to (3.2.52) as the Beurling interpolation problem (cf. [Helson and Kahane, 2; Kahane, 13, Chapter 10.6-Chapter 10.1 I]).

-

3.2.17 The structure problem and spectral synthesis. Let B(X, E ) be the restriction to X of M ( E ) and let B ( X ) = B ( X , T ) . Given T E A'(1). In light of our remarks in Section 3.2.15 it is interesting to know where (in Z)Tmight look like an element of B(Z). The main problem is tofind those sets X E Zfor which f is in the a(L"(X),L ' ( X ) ) closure of B(X, suppT). This is the ordinary synthesis problem for X = Z. A simpler problem is the following. Given T E A'(1). Find the sets X c Z and the strongest possible topology T for which f is in the T closure of B ( X ) .Of course, if T is the a(L"(X),L'(X))topology then the approximation can be made for any X E Z since T is an S-set. It turns out that ?' is in the uniform closure of B ( X ) i f and only i f whenever {fn :n = 1,. ..} G L'(X) has the properties that

3.2 Synthesizable phenomena

247

and

(3.2.54)

limlJfnII, = 0

then

lim n

2

f,,(rn)f(rn)=O

mex

[Hartman, 11 (cf. Exercise 2.2.5). 3.2.18 Refinements of Bochner's theorem and the structure problem. E L"(2) may not only be close to an element of B(2) on certain sets X c 2 but it might happen that ? E B(X, E ) . Given T E A'(T). The problem is to find conditions on T, X,and E for which we can conclude that f~ B(X, E ) . Of course if X = 2 and E = T then we have Bochner's theorem (cf. [Lumer, 11).

A generalization of Bochner's theorem in one direction is the following, e.g. [Rosenthal, 2; Doss, 11: Theorem 3.2.11. Given X such that

SUPllPnlll n

c 2. Zf

there is K > 0 and a sequence {pc,:n= 1,. . .} c M,(T)

K

and

v m E X,

lim i n ( m )= f ( m ) n

then ? E B( X)and the corresponding measure p (for which p = ? on X ) has the property that ilpII1 ,< K .

In the other direction we have Theorem 3.2.12. Given a closed set E 5 T and T E A'(T). PEB(2,E ) o whenever :n = 1,.. .} E L'(2) has the properties that

{fn

(3.2.55)

II II

SUP i n

m

<

and

(3.2.56)

V y E E,

limfn(y) = 0,

n.+m

then

(3.2.57)

lip 2fn(m)f(rn)= 0. m

Proof. ( 2 ) Let ?E B ( 2 , E ) and take { f n : n= I,.. .} E L ' ( 2 ) which satisfies (3.2.55) and (3.2.56).

248

3 Results in spectral synthesis

Since (T,fn) = 2 fn(m)f'(m) m

and T E M ( E ) we obtain (3.2.57) from (3.2.55) and (3.2.56). (-=)i) We first prove that F E B(Z). By Bochner's theorem it is sufficient to prove that if the sequence of trigonometric polynomials, k(n)

2 aJ,n(~,~J,n),n=l,...,

Vn(Y)=

J-1

has the property that

then

We set k(n)

f n

=

2

aJ,n'(nIj,m)~

J-1

so that f,,= cpn. Thus, {f,:n = 1, ...} satisfies (3.2.55) and (3.2.56) and we conclude with (3.2.57); but the left-hand side of (3.2.57) is precisely the left-hand side of (3.2.58), and so f" E B(Z). ii) Suppose supp T 0 E and let cp E A(T) have the property that F f l E = 8 and (T, cp) # 0, where F = supp cp. Since T E M(T)we have T(F)# 0, where T is considered as a set function. n = 1,. ..} be a sequence of compact neighborhoods of F for which U,, fl E=O Let { U,,: and let {f,, = cpn: n = 1,. ..} E A ( T ) be a sequence of functions for which supp cpn G U,,, 0 < cpn < 1, and cp,, = 1 on F. Then, since T E M(T), (3.2.59)

lim (T, cp,)

n-rm

= T(F).

(3.2.55) and (3.2.56) are satisfied for {fn:n = 1,. ..} and so, by (3.2.57), lim ( T , cp.) n+m

= 0.

This contradicts (3.2.59). q.e.d.

3.2 Synthesizable phenomena

249

There are more complicated proofs of Theorem 3.2.12; and the above proof, generalized to any G and r is in [Frieberg, 11. The condition (3.2.56) can obviously be replaced by (3.2.54); and Theorem 3.2.12 should be compared with Hartman’s result in Section 3.2.17. 3.2.19 Helson sets and the structure problem, The considerations of Section 3.2.18 lead to an important result concerning Helson sets in Z.For perspective we begin by stating the well-known fact (e.g. [Rudin, 5, Section 5.731) Proposition 3.2.14. X G Z is an Helson set o E B(X).

V T E A‘(T),

One direction of Proposition 3.2.14 does not require G to be discrete. Proposition 3.2.15. Let X G G be closed and let B( X ) be the spuce of restrictions of B(G) to X . If cb(X)= B( X ) then X is an Helson set. Proof. We let B ( X ) have the canonical quotient norm 11 [IBcx) turn has the total variation norm from M(T)). The identity map

B(X)

from B(G) (which in

cb(x)

--f

is a continuous bijection when cb(x) is given the sup norm topology. Thus, by the open mapping theorem, there is K > 0 such that

(3.2.60) Take p

where v,

V@ E

E

Cb(X),

Il@llBtx,<

KII@IIm.

M ( X ) = Co(X)’.Then C,(X) G B ( X ) , by hypothesis, and

E

M ( r ) has the property that

v x E X,

i,(X)

If we choose V~ so that IIv,II1 IlplI1 < ( K

= @(x).

< Il@llBcx,

+ 1, then (3.2.60) and (3.2.61) combine to give

+ 1)IIPIIm. q.e.d.

[DCchamps-Gondim, 11 has proved that if X C Z is an Helson set and E G T is a closed set for which int E # 8 then V T E A’(T),

?E B ( X , E ) .

250

3 Results in spectral synthesis

Exercises 3.2 3.2.1 A characterization of A' Given the forms T

-

x c , ern? and F

- 2'? In

-

elnY.Prove that F T E A' (resp. T E

M(T)) if and only if

where the supremum is taken over all finite sequences { [&,yk] :k = 1,. . .,m} of nonoverlapping intervals in T. The details are carefully written out in [Edwards, 5, Section 16.5.31. 3.2.2 Hadamard measures

F E LZ(T),p(0) = 0, is a function of bounded deviation if

This notion was introduced by H a d a m a r d (1892) to generalize the notion of bounded variation. a) Fix a > 0 and prove that F(y)=y"sin(l/y"),

where y

E

[0,27r),

satisfies (E3.2.2) but that it does not have bounded variation. References and discussion of functions of bounded deviation are found in [Benedetto, 101. b) Prove that if F is a function of bounded deviation then F E L"(T). Note that there are F T EA' for which cardsuppT= X, but for which F .$L"(T) (e.g. Exercise 3.2.3d).

-

T E A' is a Hadamard measure if F is a function of bounded deviation for F What are the spectral synthesis properties of Hadamard measures ?

N

T E A'.

3.2.3 Arithmetic progressions in the construction of pseudo-measures a) Let {c,:n = 1,. ..} be a decreasing sequence of non-negative numbers and assume that there is a constant K such that V n 2 1, nc,

< K.

Prove that VyEEandVN,

2 InT,

c,sinny < K ( n + l ) .

251

3.2 Synthesizablephenomena

This fact was used in Exercise 1.3.2. The bound K can be refined in some cases, e.g. [Katznelson, 5, p. 221.

-2 m

b) Given T

c,,einy E A’(T),where cn 2 0.

1

Prove that if m

21 cntn diverges then T EA’(T)\ M(T).This fact is used in [Benedetto, 91. c) Let A, y E 1,where y # 0. Define

O
Using part a) prove that IlPNllA. 6 2(n + 1).

Thus, if a closed set E c T contains a sequence of finite arithmetic progressions A,, card A , = k , then E is not an Helson set. Substantial refinements of this fact and the construction of elements in A’(T)\ M(T)are found in [ K a h a n e and Salem, 2; 31 (e.g. see [Benedetto, 6, Section 4.3 and Section 7.21 for exposition and references). In fact, K a h a n e and Salem have shown that the Cantor set C E T is not an Helson set; and [Katznelson, 31 has proved, even further, that C E T is a set of analyticity (cf. our remarks on the Katznelson conjecture in Exercise 2.4.2).

-

1 T E A’ where suppT c E = (0, - :n = 1 , . . .} c [0,2n) and n 1 F # L“(T). In particular, prove that E is not an Helson set. Note that ( 0 , X : n= 1 , . . .} 2 is an Helson set, e.g. [Benedetto, 6, p. 1121. (Hint: A finite arithmetic progression in E can have the form { j / K !: j = 1 , . . . , k } , where K is fixed and k < K.Let En G E be a finite arithmetic progression with 2M,, 1 terms such that if y E En+,then y < 1 for each 1 E En. Choose M,, so that

d) Using c), construct F

+

1

and let pn E M(E,,) have mass 0 at the center of En and mass l / j (resp. -11) at thejth point (of En) to the right (resp. to the left) of the center. By part c), ~ ~ p n l<~2(n A r 1).

-

If Gn pn then IGn(y)l=

+

M.

2 l / j for each y in the two intervals contiguous to the center 1

252

3 Results in spectral synthesis

of E,. Set

x

Thus, IlvnllA. < -(n 3 (E3-2.3)

+ 1) and for each 1 < j < n,

IFn(Y)l = l G , ( ~ ) l / j ~ k j

on the two intervals contiguous to the center of E,. A subsequence of {v, : n = 1,. ..} converges to T E A'@) by the Alaoglu theorem and p(0) = 0. It is easy to choose qnE AE(T) so that lim I(T, q,)( = 03 and 0 Q qn< 1 ; thus T 4 M(E). If F T then from (E3.2.3) we see that F 4 L"'(T); and this, of course, also tells us that T 4 M(E)). With regard to the multiplication defined in part e) we see that if T E A' \ A, and supp T is countable then T24 A'(T); and it is not difficult to construct T E A,for which T24 A'(T).

-

e) Given a closed set E E T for which mE = 0. If F the distribution

-

T E A' and G

-

S E A' we define

TS = (FG)',

where the derivative is taken in the distributional sense (cf. Exercise 1.4.5). With this multiplication we know that if A' n A'(E) is a Bunach algebra then E is an Helson set. 1 Verify this statement for the case E = { O , - : n = 1, ...}. (Hint: Adapt the example of

n

part d). In this regard Ricci observed that if A' f l A'(E) is a Banach algebra then A' fl A'(E) = A: fl A'(E); further, if the maximal ideal space of A'(E) satisfies certain natural conditions then A'@) is very close to being M ( E ) [Benedetto, 41. Of course, Korner's example, mentioned in Section 1.3.13, tells us that there are Helson sets which are not sets of strong spectral resolution. [Benedetto, 41 has been extended in an interesting way by [Ricci, 11. 3.2.4 Non-pseudo-measures with regular primitives Given a closed infinite set E E T for which mE = 0. Find m

=

21 k J x I ,

m

such that 2 k, converges and such that the distributional derivative T = F' of F is not 1 in A'(E), e.g. [Benedetto, 5, Theorem 3.11. Salmons has proved this result using the fact that the union of two countable Helson sets is an Helson set. 3.2.5 AXT) and derivatives of continuous functions In Exercise 2.1.2 we gave some properties of the space A:(T). We now continue this investigation (cf. Exercise 3.2.8). a) If cp E A(T) then it is obviously not necessarily true that the distributional derivative,

3.2 Synthesizable phenomena

253

cp’, is in A’(T). Prove that if cp E A(T) and cp’ = T E A’(T) then T E A:(T).

(Hint:Use only elementary facts about Ceshro summable series). The Hardy-Littlewood function (defined in Exercise 2.5.2) provides an example of a continuous function cp whose distributional derivative is an element of A’(T) \ A:(T). b) Analogous to our treatment in A’(F1) we define supp,T, for T E A’(T), to be 21 where! = cp E Z if and only if i

N

Prove that T E A’(T) is an element of A:(T) if and only if supp,T= 0. This fact is not true for r = F1 as we showed in Example 2.1.1b.

-

c) In the opposite direction to part a) note that (El .3.4) defines a discontinuous function F, such that F T E A‘ and T E A:(T). Find a non-trivial subclass X c BV(T) such that if cp’ = T is the distributional derivative of cp then V c p ~X,

cp€C(T) o TEA:(T).

-

m

d) Let X be the class of Fourier series, cp z f ( n )sin ny,where {f(n) :n = 1,. ..} decreases to 0. Prove that if cp‘ = T is the distributional derivative of cp then V cp E

X,

cp E C(T) o TEAA(T).

3.2.6 A(T) = L’(T) * A(T) Let L:(T) = {cp E L’(T) : cp 2 0 a.e.}. Prove that (E3.2.4)

A ( T ) = L i ( T ) * A(T).

(Hint: Use Cohen’s factorization theorem for compact spaces. (E3 4) can also be proved classically using properties of convex functions). 3.2.7 An element ofA’(T) \ M(T)and its Wiener support Define

1

a) Prove that T E A’(T)\ M(T).(Hint: First calculate P(n) in terms of X

--x

plus a term which tends to 0 as In( --t w ; thus T E A’(T). If T E M(T) we have that b

(E3.2.5)

3 K > 0 such that V (a,b) G (0,n),

[% a

sin y

6 K.

254

3 Results in spectral synthesis

On the other hand /*-2log-, sin y

b

a s a , b + 0.

a

a

Letting b -+ 0 exponentially and a + 0 polynomially we obtain a contradiction to (E3.2.5)). b) Prove that supp T = T whereas supp, T = (0)(cf. [Benede t t 0,111, where general criteria for this phenomena are given). 3.2.8 Sequence spaces and a strong form of uniqueness Let C2 be the set of non-zero even functions, w :Z + R+ such that {o(lnl) : n = 0,1,. . .} is decreasing and 2 o ( n ) < to; if w E a and 2 w(n)= 1 we write w E Q0. Define

L ~ = { @ : Z + c:Il@(n)lo(n)< 4

V'o~fl,,

(cf. [Beurling, 121 and the weighted spaces defined in Section 1.4.3), V @:Z

+

C,

11@11*

sup

=

2 I@(n)lw(n),

UEng

and the Banach space,

B={@:z

-+

c:pll*<m>.

Recall the definition of IITII" in Exercise 2.1.2; because of this, define

a) Prove that 1 1 0 1 1 "< 03 if and only if 11@11" b) Prove that

< 03.

B = n{L,::wEno)2Lyz). (Hint: E

a is a

complete metric space with metric p(wl,

n {L::o E ~2,).

To prove @ E B. Define @(m) Set

for I@(m)l< n for I@(m)l > n.

02) =

2 lwl(n) - w2(n)l.Let

3.2 Synthesizablephenomena

255

and L(o) = SupLn(0). n

Note that L(w) = 2 1910 so that L is lower semicontinuous. By Osgood's theorem there is a ball B(oo,r)E 62 such that K=

SUP

21@10<00.

cucB(a*,r)

The real vector space generated by s2 is complete and B(wo,r)- wo is a neighborhood of 0. Thus,

Il@lln < 4K/r). c) Prove that @ E B \ L"(Z) where @(n)=

i

0 0 2"

fornG0 forn#2" forn=2*.

d) Prove that L"(Z), normed by 11 I(=, is not complete. (Hint: If the result were true then by the open mapping theorem, (1 I(m and 11 (IA, are equivalent norms on "1); and thus by Exercise 2.1.2 A:(T) = AA(T), a contradiction). e) Prove that if A'(E), normed by (1 (Im, is complete then Eis a U-set, e.g. [Benedetto, 101. If E is finite then A'(E) normed by 11 Ilm is complete; are there infinite sets E for which A'@) normed by I( (Imis complete? f ) Show that L"(Z) # B, taken with the 1) Ilm-norm.(Hint: Use the example of part c), assume that lim

n-rm

I[@- @,,[Im

=0

for some { @,, :n = 1,. ..} E L"(Z), and obtain a contradiction). Norms such as 11 Itm have been used to study spectral synthesis problems by [Benedetto, 111 and [Beurling, 121. 3.2.9 A class of synthesizable pseudo-measures Given T E A'@) and assume that there is x > 0 such that f' has bounded variation on (-00, -x) U ( x , ~ )Prove . that T E A @ ) , e.g. [Atzmon, 41.

3.2.10 A definition of integral for spectral synthesis Let F E C(T)be real-valued. Define FJy) = sup { inf a E (0.t)

m(l,ct)}

256

3 Results in spectral synthesis

where the supremum is taken over all closed intervals I E T in R for which y E I and m E = e (m(l,a)was defined in Section 3.2.8). Since m(l,a,) 6 m(l,a,) when a1 < a, we have = sup { lim m(Z, a)}.

F,(y)

a +o

a) If F = k E R on an open interval I prove that F,= k, 6 k on I. b) Given F T E A;. If F, T, prove that supp T, s supp T and

-

-

lim 11 F - F, Ilm = 0. t-rO

3.2.12 A property of A(T) a) Let cp E A(T) have the property that F o cp E A(T)

V F E A(T),

(cf. Exercise 2.4.4). Prove that s?pllel"@l/,,< 00. (Hint: Define the map A(T)

+

A(T)

F w- Focp and use the closed graph theorem). b) On the other hand, prove that the following property can not be valid for a given closed set E c T, e.g. [Benedetto, 81 : there is K > 0 such that for each n E Z we can find a real-valued element cpn E A(T) and a finite disjoint union N, of closed intervals covering E for which Vn,

lIcpnllA
and

v

EN,,

elmy--e

I@n(Y)

.

3.2.12 Non-synthesizable convergence criteria in A(T) a) Let f = cp E C(T) be absolutely continuous and assume that cp' E Lz(T). Prove that cp E A(T) and

(Hint: Note that If(0)l 6 ll(P'Il2

I lcpl

and

=(zlnf(n)lz)l'z;

apply Holder's inequality).

3.2 Synthesizable phenomena

257

b)FindTEA'(T)and(cp,:j= 1, ...} cC"(T)such that lim I CPJ IIm = 0, V j ,p ';

(E.3.2.6)

= 0 on

a neighborhood of supp T,

lirn~~cpJ1= 0,

and

lim I(T, cp,)l

> 0.

J +m

In light of part a), if we used the L2-norm in (E3.2.6) then we would have to conclude that lim(T,cp,) = O . (Hint: Use Exercise 3.2.3d to form F - T E A ' \ & where F = 2k J ~ , ,and , choose cp, to have the value 0 or l/kn,except for two small intervals, one of 1 which is an open set in 1.

(-,

I)

n J + 1 nJ

c) Prove that for each cp E A(T) there is a sequence {cpJ:j= 1,. . .} E Cm(T)for which limI!cp-cp,/I,=O and lim cp;

=0

a.e.

This Page Intentionally Left Blank

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Index of proper names N. H. A b e l 2.3.1 S. A g m o n 2.3.14, 2.5.1 E.A k u t o w i c z 2.4.5 I. A m e m i y a 1.2.4 S. A n t m a n Introduction R. A r e n s 2.4.6 L. A r g a b r i g h t E2.2.5 J. A r s a c 2.1.2 A. A t z m o n Introduction, 1.4.8,E1.4.4,

E2.1.6,3.1.2,E3.2.9 J. S. B a c h 1.5.2.4 S. B a n a c h 3.2.10, 3.2.14 N. B a r y E2.1.1,2.4.10,E2.4.4,2.5.5,

3.2.10 J. B e n e d e t t o 1.3.13, E1.3.1, E1.4.5,

A. C a l d e r o n

1.4.9,2.4.5,2.4.6,2.5.6,

E2.5.3 R. C a m e r o n 2.4.8 T. C a r l e m a n 1.4.2,1.5.4,2.4.5,2.5.1 L. C a r l e s o n 1.5.3,E2.1.6, 3.1.15 H.C a r t a n 2.4.13 P. C k z a n n e 1.5.2.2 K. C h a n d r a s e k h a r a n E1.4.4, 2.3.6,

2.3.14,E2.3.4, E2.5.7 M. C h a u v e 2.5.13 P. C h e b y s h e v 2.3.12 P. J. C o h e n E1.1.5,1.2.4,E1.4.1,2.4.6, 2.4.9,E2.4.3 H. C o l l i n s E2.2.1 J. C o n w a y E2.2.1 M. C o t l a r 2.5.1 H. C r a m k r E1.3.5, 2.1.3 C. C u c c i a 1.5.2.3 P. C u r t i s 1.4.11

2.1.1,2.1.5,2.1.6,2.1.7,E2.1.4,E2.2.2, 3.2.11, E2.4.5,2.5.10, 3.1.2,3.1.5,3.2.4, E3.2.2,E3.2.3,E3.2.4,E3.2.7,E3.2.8, E3.2.11 H. D a v e n p o r t 2.4.9 A. B e r n a r d 3.1.18 L. d e B r o g l i e 1.5.2.6 D. B e r n o u l l i 1.5.2.4 M. D e c h a m p s - G o n d i m 3.2.19 S. B e r n s t e i n E1.3.5, E2.1.2,E2.5.2 R. D e l a n g e 2.3.4 A. Berry E1.3.5 C. d e la V a l l k e - P o u s s i n 2.3.10,2.3.15 A. B e u r l i n g 1.1.8,1.3.10,1.4.2,1.4.3, 1.4.12, E1.4.5,2.1.6,E2.1.1,E2.1.6,2.2.1,K. D e Leeuw 3.1.12, 3.1.16, 3.1.17, E3.1.5 2.2.3,2.2.4,2.2.5,2.2.6,2.2.7,2.2.11, E2.2.4,E2.2.5,2.3.16,2.3.18,2.3.19, J. D e l s a r t e E1.4.5 3.2.8 2.4.8,2.4.9, 2.4.10,E2.4.8,2.5.1, E2.5.4, A. D e n j o y 2.4.1, 3.1.4,E3.1.4,3.2.1, 3.2.2,3.2.4,3.2.5, C. d e V i t o 1.4.3 W. D i e t r i c h E1.4.1 3.2.10, 3.2.16,E3.2.8 J. D i e u d o n n e E2.4.8 A. B l a n c h a r d 2.3.15 P. G. L. D i r i c h l e t 2.3.11 R. B o a s 2.4.10 2.1.4, 2.1.11, V. D i t k i n 2.2.11, 2.5.1,2.5.2, 3.1.7, S. B o c h n e r 1.3.10,E1.4.4, E3.1.4 2.2.11,2.3.6, E2.5.7 J. D i x m i e r E2.5.5,3.2.14 H. B o h r 2.2.8, 2.2.9 Y.D o m a r 1.4.3,2.2.2,2.2.5,3.2.4 E.B o m b i e r i 2.3.15 W. D o n o g h u e E2.1.1, E2.2.3,2.3.9, N.B o u r b a k i 1.3.5,1.3.9,2.1.3 2.3.19 B. B r a i n e r d E3.1.3 R.Doss 3.2.18 G. B r a q u e 1.5.2.2 A. D o u a d y E3.1.2 R. C. Buck 1.4.12, E2.2.1 S . D r u r y 1.2.7,2.5.13, 3.1.16, 3.1.18 R. Burckel E2.2.5

274 Index of proper names

C. Dunk1 E2.2.5, 2.4.7 F. D y s o n E1.3.5 W. Eberlein E1.3.5, E2.2.5, E2.4.8 R. E. Ed war d s E2.2.5, 2.5.3, E2.5.7, E3.1.3 H. Eggleston 2.2.8 L. E h r e n p r eis E1.4.5 M. Eisen 2.4.5 R. E l l io tt E1.4.5 L. E u l er 1.5.2.4, 2.3.7 P. Eymard E3.1.3

C. Fefferman 1.5.3, 3.2.10 M. Fekete E3.1.3 G. F e ld ma n 1.4.3 P. F e rma t E1.2.5 A. F i g a - Ta la man c a 1.4.11, E3.1.3 M. F i lip p i 2.5.8 S. F r i eb e r g 3.2.18 G. F ro b e n iu s 2.3.3 M. F u k a miy a 2.5.1 T. G a neliu s 2.3.16, E2.3.5 L. GArding Introduction, E2.2.3 A. G a r s i a 2.2.5, 2.2.6 G. G a u d r y E3.1.3 I. G e l f a n d E1.3.6, E1.4.5, 2.4.6 J. G i l b er t 1.2.3, E1.4.5 H. G i n d le r 2.4.5 G. G l aes er E1.4.5 M. Glasser 2.3.4 I. Glicksberg E2.4.8, 3.1.17 R. G o d e men t 1.2.2, 1.2.6, 1.3.10, 2.2.5, 2.2.7,E2.2.1, 2.5.1, 3.1.7 G. G o d i n 1.5.2.1 R. G o ld b e r g E2.1.2 J. G o ld in g 1.5.2.2 S. G o l d m a n 1.5.2.3 C. G r a h a m Introduction, E1.1.3, 2.4.11, 2.5.12, 3.1.16,E3.1.1 J. G r i s 1.5.2.2 A. G r o th e n d ie ck 3.1.9, 3.1.10, 3.2.10 J. H a d a m a r d 2.3.9, 2.3.10, 2.3.15, 2.4.4, E3.2.2 R. H a n d els ma n 3.2.10 M. H a n k i n s o n 1.5.2.4 G. H. H a r d y 2.1.9, 2.1.10, 2.1.11, 2.3.1, 2.3.4, 2.3.5, 2.3.6, 2.3.14, E2.3.2, E2.5.7, 3.2.10

S. H a r t m a n 3.2.17, 3.2.18 T. H edb erg E2.4.7 W. Heisenberg 1.5.2.6 H. H el mhol t z 1.5.2.4 H. H elson E2.1.6, 2.4.9, 2.4.11, 2.5.1, 3.1.7,E3.1.3, 3.2.16 G. Helzer Introduction C. H er z 1.4.2, 2.1.1, 2.1.6, E2.1.6, 2.2.2, 2.5.4, 2.5.6, E2.5.3, E2.5.5, 3.1.9, 3.1.12, 3.1.13, 3.1.16, 3.1.17, 3.1.18, E3.1.3, E3.1.4, E3.1.5 E. Hewitt E2.2.5, 2.4.7 E. Hille E2.2.4 K. H offm an Introduction, E1.4.5, 2.4.6 J. H of fm ann-Jsrgensen E2.2.1 L. H o r m a n d e r E.2.2.3,2.5.3, E3.1.3

S. I k e h a r a 2.3.14 A. E. I n gham

2.1.9, 2.3.4 T. I t 8 1.2.4 0. IvaSev-Mucatov E2.1.1

M. J er ison 3.1.17 F. J o h n 3.2.10 R. J o h n s o n Introduction J. J o s e p h s 1.5.2.4 M. K a c 1.5.4 J.-P. K a h a n e Introduction, E.1.3.5, E1.4.5,2.2.7,2.4.9,2.4.11,E2.4.2,E2.4.4, E2.4.7, 2.5.8, 2.5.13, 3.1.2, 3.1.4, 3.1.9, 3.2.5, 3.2.12, 3.2.16, E3.2.3 S. K aij ser 3.1.15 I. K a p l a n s k y E1.2.5, 2.5.1 J. K a r a m a t a 2.1.11, 2.3.4 H. K a r a n i k o l o v 2.4.9 Y. K at znel son Introduction, 2.4.11, E2.4.2, E2.4.7, 2.5.4, 2.5.8, 2.5.10, 3.1.2, 3.1.4, 3.1.7, 3.1.12, 3.1.16, 3.2.7, 3.2.11, 3.2.14, E3.2.3 R. K a u f m a n Introduction, 1.3.13, 3.1.4, 3.1.15, 3.2.12 Lord Kelvin 1.5.2.1 S. Kempisty 3.2.8 F. Keogh E1.1.4 M. K i n ukaw a 2.1.6, E2.1.6, 3.2.4 A. K onyushkov 2.4.10 P. Koosis 1.1.8, 2.2.5, 2.2.7 J. K o r e v a a r 2.2.5, 2.3.4, 2.3.16 T. K o r n e r 1.3.13, E2.4.2, 2.5.4, E3.2.3 E. K um m er E1.2.5

Index of proper names E. L a n d a u 2.3.5 H. L a n d a u 3.2.16 P. L a p l a c e 1.5.2.1 R. L a r s e n E3.1.3 Z. L e i b e n s o n 2.4.9 L. L e i n d l e r 2.4.10 N . L e v i n s o n 2.1.11,2.3.14,2.3.16,2.3.18 B. L e v i t a n 2.2.11 P. Levy El.3.5, 2.4.4, 2.4.9 J. Lew 3.2.10 L. L i n d a h l 3.2.14 J. L i n d e n s t r a u s s 3.1.9 J. E. L i t t l e w o o d 2.1.9, E2.1.1, 2.3.4, 2.3.5, 2.3.14, 2.4.9 E. Loewenstein 1.5.2.5 N. LohouC E3.1.3 L. Loornis 2.2.9, 2.2.11 S. L o z i n s k i i E1.3.1 G. Lurner 3.2.18 F. L u s t 3.1.18 N. L u z i n 2.4.1 S. L y t t k e n s 2.3.16 Y. L y u b i c h 1.4.3

G. Mackey E2.2.1, 2.4.6, 2.5.1 D. M a c m i l l a n 1.5.2.1. B. M a l g r a n g e E1.4.5 P. Malliavin 1.4.8, E1.4.5, E2.4.2, 2.5.8, 2.5.13,3.1.1,3.1.2, 3.1.4,3.1.5 S. M a n d e l b r o j t 2.4.1, 2.4.5, 2.5.1 V. M a r r e n k o E2.2.4 J. M a r c i n k i e w i c z 2.1.10, E2.4.4 P. M a s a n i 1.5.2.3 V. M a t s a e v 1.4.3 C. M c G e h e e Introduction, E.2.4.2, 3.1.16,3.1.18,3.2.L2, 3.2.14 D. M e n ' s h o v E2.1.1 F. M e r t e n s 2.3.12 Y. Meyer 1.2.3, E1.4.5, 2.1.5, 2.1.9, E2.1.4,E2.2.3, 2.5.5, 2.5.1,E3.1.3 A. M i c h e l s o n 1.5.2.5 H. M i r k i l 2.4.6, E3.1.1 T. M o h E2.5.4 D. M u r a z E2.2.1 D. N e w m a n E2.1.6 1 . N e w t o n 1.3.10, 1.5.2.1, 1.5.2.6,2.1.6 D. N i e b u r Introduction L. N i r e n b e r g 3.2.10 E. N o e t h e r E1.2.5 B. Nyrnan 2.3.18

275

J. O s b o r n Introduction V. O s i p o v 3.1.8

R. P a l e y 2.3.18 A. Pelczynski

3.1.9 R. P h i l l i p s E2.2.4 P. P i c a s s o 1.5.2.2 S. P i c h o r i d e s 2.4.9 A. P i e t s c h 3.1.10 H. Pitt 2.3.5, 2.3.6, 2.4.7, 2.4.8 M. P l a n c k 1.5.2.6 H. P o i n c a r e 3.2.16 H. P o l l a r d 1.3.10, 2.1.6,,E2.1.6, 3.1.4,E3.1.4, 3.2.5, 3.2.10 F. P o u l s e n 3.2.14 A. P o v z n e r E2.2.4 A. Pringsheirn 2.3.1 V. P t a k 1.1.8

2.2.5,

D. R a i k o v E1.4.5, 2.4.6 J. R a i n w a t e r E2.4.5 A. R a j c h r n a n 2.5.11 D. Rarnirez E2.2.5, 2.4.7 H. R e a d 1.5.2.2 H. R e i t e r 1.4.8, E1.4.4, 2.2.11, 2.5.1, 2.5.6, 2.5.13,E2.5.3,E2.5.5, 3.1.7, 3.1.12, 3.1.17, E3.1.5 F. R icci Introduction, E3.2.3 I. R i c h a r d s 3.1.4 C. R i c k a r t E1.4.5 N. R i c k e r t 3.2.10 B. Riernann 1.3.4, 1.5.4, 2.1.10, E2.1.1, 2.3.10 F. R i e s z E1.1.4 M. R i e s z 1.1.4, 2.1.4 J. Riss E1.3.7, 2.5.1 W. R o g o s i n s k i 2.1.10 H. R o s e n t h a l 1.2.3, 1.2.4, 2.5.5, 2.5.7, E2.5.8, 3.2.18 B. R o s s i 1.3.10 L. R u b e l E.2.2.1 W. R u d i n 1.1.8, 1.4.8, E1.4.5, E2.2.5, 2.3.1, 2.4.9, 2.4.11, 2.5.10, 3.1.2, 3.1.5, 3.1.15, 3.2.11 R. R y a n E1.3.5 J. R y f f E2.2.1 S. S a e k i

1.2.2, 1.2.3, 2.5.6, 2.5.8, 2.5.13, 3.1.7, 3.1.15 R. S a l e m 1.1.8, E1.3.5, E2.1.1, 2.3.17, 2.5.5, 3.2.12, E3.2.3

276 Index of proper names D. Salinger E2.4.2 G. Salmons Introduction, E3.2.4 L. Schiff 1.5.2.6 R. Schmidt 2.1.9, 2.3.5 I. Schoenberg E1.3.5 B. Schreiber 1.2.3 E. Schrodinger 1.5.2.6, 2.1.4 L. Schwartz 1.3.5, E1.3.6, E1.3.7, 1.4.8, E1.4.5, 1.5.3, 1.5.4, 2.1.3, E2.2.2, 2.4.13, E2.5.5, 3.1.1, 3.1.9, 3.1.12, 3.1.17,E3.1.4, 3.2.7 I. Segal E1.1.3,E2.1.6, 2.4.6, 2.4.8,2.5.1, 3.1.5 F. Sentilles E2.2.1 G. Shilov E1.4.5, 2.4.6, E2.4.4, 2.5.1, 2.5.2 J. Siege1 3.1.17 A. Simon E1.3.5, E2.1.2 R. Spector 3.1.17 Y.Sreider 2.4.7 S.SteEkin E1.3.5, 2.4.10 J. Stegeman 3.1.7, 3.1.18 E. Stein 3.2.10 M. Stone E1.4.5 G. Sunouchi 2.4.10 D. Sweet Introduction 0. Szasz E1.3.5, E2.1.2

J. T a m a r k i n E2.2.4 A. T a u b e r 2.3.1, E2.3.1 E. Titchmarsh 2.3.9 F. Tricomi 1.5.4

N. Varopoulos E1.1.5, 1.4.8, E1.4.4, 2.4.11,E2.4.2, 2.5.7,2.5.8,2.5.10,2.5.11, E2.5.1, E2.5.5, 3.1.1, 3.1.4, 3.1.9, 3.1.10, 3.1.12, 3.1.13, 3.1.14, 3.1.15, 3.1.18, E3.1.5, 3.2.14 W. Veech 2.2.11, E2.5.8 E. Vesentini Introduction J. v o n N e u m a n n E2.2.1 A. Vretblad 1.4.3, E2.2.4, 2.3.18 H. Wallin 2.2.8 C. R. W a r n e r Introduction, 2.5.2, 2.5.6, E2.5.3 S. Weingram 3.1.17 J. Wendel E3.1.3 J. Wermer E3.1.1 H. Weyl 1.5.4 R. Wheeler E2.2.1 H. Whitney E1.4.5 D. V. Widder 2.3.6,2.3.11, 2.3.14,E2.3.4 H. Wielandt 2.3.4 N. Wiener 1.1.8, E1.3.1, 1.4.2, 1.5.2.3, 1.5.2.4, 1.5.2.5, 2.1.2, 2.1.5, 2.1.9, 2.1.11, 2.1.12, E2.1.1, E2.1.2, E2.1.5, E2.1.6, 2.3.4, 2.3.5, 2.3.6, 2.3.14, 2.3.16, 2.3.18, 2.4.1, 2.4.3, 2.4.7, 2.4.8, 2.5.1, 2.5.2, E2.5.7, 3.1.16, E3.1.7 I. Wik 1.2.3, 1.2.4, E1.2.6,E1.3.5, E2.4.8, E3.1.7 J. Williamson 2.4.7 A. Wintner E2.1.1 P. Wolfe Introduction A. Zygrnund E1.1.4, E1.3.5, 2.1.10, E2.1.1, E2.1.2,2.4.3,2.4.5, E2.4.4, E2.5.2

The following books are not referenced in the above list of proper names: [Benede t to, 2; 6; 12; D u n f o r d and Schwartz, 1; Edwards, 5; Hewitt and Ross, 1 ; Hewitt and S t r o m berg, 1; Horvhth, 1; Kahane, 13; K.ahane and Salem, 4; Katznelson, 5 ; Loomis, 1; Riesz and Sz.-Nagy, 1; Royden, 1; Rudin, 5; 61.

Index of terms Abel summable 2.3.3 algebraically primary ideal E1.2.5 almost periodic at y E r 2.2.9 --function 2.2.8 -- pseudo-measure 2.2.9 analyticity (set of) E2.4.2 annihilator ideal E1.4.5 approximate identity 1.2.1 arithmetic (fundamental theorem of) 2.3.7 autocorrelation 2.1.2

D(p, I)lemma 2.5.1 de la VallCe-Poussin kernel E.1.2.6 Dedekind domain E1.2.5 dichotomy (conjecture of) E2.4.2 Dirichlet set 2.5.13 discontinuous measure E2.1.1 distribution E1.3.6 distributional derivative El .1.4, E1.3.6, 2.1.8

Balayage 3.2.16 Bemstein’s inequality E2.2.3 Bessel function E1.4.4 Beurling integral 3.2.2 - interpolation problem 3.2.16 Bochner’s theorem I .I .4, 2.1.3 Bohr compactification 3.1.16 bounded approximate identity 1.2.1 - deviation E3.2.2 -mean oscillation 3.2.10 - Radon measure I . 1.4 - S-set 3.2.13 - synthesis (set of) 3.2.13 boundedly convergent series El .3.2 brightness of a signal 2.1.2

Equicontinuous convergence 3.1.1 1 Euler’s constant 2.3.1 1 exponential monomial E l .4.5 - polynomial E1.4.5 -type E2.2.3

C-set E1.2.1, E1.2.3, 1.4.9 C-set-S-set problem 2.5.4 Cantor-Lebesgue function E1.1.4 -measure E1.1.4 Cantor set E1.1.4 capacity E2.4.7 Cesaro mean E1.3.5 - summable 2.3.3 circular contraction 3.2.2 closed principal ideal 1.4.7 Cohen factorization theorem 1.1.8 compact neighborhood 1.1.5 continuous measure E1.3.1 - pseudo-measure 2.1.4 convolutor on L p ( f ) E3.1.3 covariance function 2.1.2

Fejtr kernel E1.2.6 finite decomposition 2.5.9 Fourier transform 1.1.4, 1.3.1, E1.3.6 Gelfand transform

1.1.3

Hadamard measure E3.2.2 Hardy-Littlewood function E2.5.2 harmonic spectrum E2.2.4 Helson constant 3.1.18 -set 1.3.1 3 Herz set 2.5.4 Hilbert Nullstellensatz E l .2.5 homomorphism problem 2.4.9 Idempotent measure 1.2.4 independent set 2.5.8 inversion theorem 1.1.4 Katmelson conjecture E2.4.2 Kronecker set 2.5.8 Kronecker’s theorem 2.4.8, 3.2.12 Lacunary sequence El. 1.4 Lambert series 2.3.1 1 Laplace transform El .3.6

278 Index of terms local belonging 2.4.2 locally a.e. 1.1.2 - isomorphic groups 3.1.17 -null 1.1.2 logarithmic capacity E2.4.7 Malliavin’s theorem 3.1.1 maximal ideal space 1.1.3 mean periodic function El .4.5 multiplicative element of X’ 1.1.3 multiplicity (set of) E2.1.1 multiplier E3.1.3 -problem E3.1.3 N-algebra E1.4.5 narrow-topology (or a topology) 2.2.2 Noetherian ring E1.2.5 non-spectral function 3.1.4 nuclear space 3.1.1 1 numerical Tauberian problem 2.3.16 Operates (a function) E2.4.2 operator neighborhood (strong and weak) E2.2.1 - topology (strong and weak) E2.2.1 orbit 2.2.8 order of a distribution E2.5.5 orthogonal transformation El .4.4 p-independent set 2.5.8 p S-set E3.1.3 Parseval-Plancherel theorem 1.1.4 perfect set 2.2.10 -symmetric set E1.1.4 Pisot number 2.5.5 polygon 2.5.7 positive definite distribution 2.1.3 -- function 1.1.4 potential E2.4.7 -kernel E2.4.7 power of a signal 2.1.2 primary ideal E1.2.4 prime ideal E1.2.5 -number theorem 2.3.9 projective tensor product 3.1.9 pseudo-function 1.3.4 - measure 1.3.1 Quasi-analyticity R-spectrum 1.4.3 radial function El -4.4

Radon measure 1.3.5 regular Banach algebra 1.1.3 -ideal 1.1.3 resolvent set 1.1.8 Riemann hypothesis 2.3.15 Riesz product El. 1.4 S-set E1.2.3, 1.3.13 -for V ( f ) 3.1.13 Saeki set 2.5.6 scattered set 2.2.10 semi-simple 1.1.3 Sidon’s theorem El. 1.4 singular support 2.5.3 slowly oscillating function 2.3.5 spectral distribution of energy 2.1.3 - resolution (set of) 2.5.8 - synthesis (set of) 1.3.13 Stirling’s formula 2.3. I 1 strict multiplicity (set of) E2.1.1 - topology (or /?topology) 2.2.1 strong Ditkin set 1.2.2 - spectral resolution (set of) 2.5.8 - U-set 2.5.13 support of T EA ’ ( f ) 1.3.6 synthesizable elements 1.3.13 1.4.4, 1.4.10 Tauberian condition 2.3.1 -remainder problem 2.3.16 tensor product 2.4.12, 2.4.13 torsion module 2.5.8 translate of Tby y 1.3.3 U-set E2.1.1 uniqueness (set of) E2.1.1 uniqueness in the wide sense (set of) E2.1.1 Variety El .4.5 Weakly almost periodic function 2.2.8 Wiener-Ditkin set E2.5.3 - -Uvy theorem 2.4.4 - spectrum 2.1.3 - support 2.1.3 Wiener’s inversion of Fourier series theorem 1.1.8 - Tauberian theorem 1.1.9, 1.2.6, 1.3.8, 2.1.11, 2.1.12, 2.2.5, E2.2.2, E2.2.3, E2.4.8, E2.5.4, E2.5.7

Zero set 1.1.8