Table Of ContentErrata for Benedetto, Spectral Synthesis
page line
94 1 erase “i.e.,” “Thus \k E C,(n).”
1
127 6 “u” instead of “s” i.e., “. . . I: ,
n>x n
127 -5 “- lim ” instead of “ lim ”, two times
X+- X+-
145 -1 insert “, supp q C_ K, F(0) = 0,” after “compact” and
before “and”.
178 -6 6‘ y=nx” instead of ‘‘y E qx’’
202 1 “3.1.4” instead of “3.1.5”.
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, re-
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copying 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. II. Series.
QA3.P8 VOI6. 6 [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
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 syn-
thesis;
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 success-
fully 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 3.2.1 5-3.2.19) 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
r.
G will always be a locally compact abelian group with dual group 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,
r
and = T = R/2d, 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. Warner who introduced me to the spectral synthesis
problem. There has also been the generous and often ingenious assistance of
G. Helzer, R. Johnson, and G. Salmons who were always willing to discussmath-
ematics 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. Antman, A. Atzmon, L. GBrding,
C. Graham, J.-P. Kahane, Y. Katznelson, R. Kaufman, C. McGehee,
D. Niebur, J. Osborn, F. Ricci, D. Sweet, and P. Wolfe. 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. Hoffman and E. Vesentini of these institutions
for the hospitality extended to me. My final thanks go to G. Maltese 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
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 A,(f) 1.1.6 A(E) 1.1.8
Aj(E) 1.1.8 A'(f) 1.3.1 AA(f) 1.3.4
A(U) 1.3.6 A'@) 1.3.13 A&!?) 1.3.13
A:(R) 2.1.3 AS 2.1.4, E2.1.2 AME) E2.1.1
AP(G) 2.2.8 AE(f) 2.5.10 A(E,X) 3.1.16
A' 3.2.1 A; 3.2.10 A"(E) 3.2.14
B(f) 1.1.4 B'"=R'" 2.5.7 bT 3.1.16
gf 3.1.16 BV(T) 3.2.1 BMO 3.2.10
B(X) 3.2.17 B(X,E) 3.2.17
C Introduction Co(f) 1.1.4 C,(C) 1.1.5
C E1.1.4 Cm(T) E1.1.4 Co(E) 1.3.13
Cm(fin) E1.3.6 C?(R") E1.3.6 Cz(fi") E1.3.6
C(R") -E1.3.6 C"(T) E1.4.5 Cb(G) 2.2.1
C(E1) @ C(E2) 3.1.9 C(E,X) 3.1.11
Do(f) 1.3.5 D(fi") E1.3.6 D,(R") E1.3.6
D,(R") E1.3.6 D,O(R") E1.3.6 D(cp,Z) 2.5.1
Dm 2.5.2 D(k,j) 2.5.2 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 Gd 3.1.16
Z,,, 1.4.7, Iw(M) 2.1.5
Zloc( a) 2.4.2
j(E) 1.1.8
k(E) 1.1.8
L'(C) 1.1.2 L"O(G) 1.1.2 L(Y, Y) 1.4.3
Lip.(T) E2.5.2
16 Index of notation
M(G) 1.1.4 MO(f) 1.3.4 M,(r) 1.3.4
M(I) 1.3.12 M(E) 1.3.13 MT 1.4.1
Mf(E) 1.4.5
O(@) 2.2.8
Q Introduction
R Introduction II Introduction R+ Introduction
supp Introduction, 1.3.6, 1.4.1 spy 1.4.1
sp@ 1.4.1 SnsIIntl E1.4.4 SP,@ 2.1.3
SUPP~ 2.1.3,2.1.5 supp. 2.2.4 SUPP,~ 2.2.8
SUPP, E2.2.1
T Introduction 9 1.4.1 Fsp 1.4.1
5 a 1.4.1 5 ° C 1.4.11
V El .4.5 Va E1.4.5 V 2.5.9
V(E) 3.1.9 V(E1,EZ) 3.1.9
WAP(G) 2.2.8
X” 1.1.3 2 j.1.3 Xi @ X2 2.4.12
6
XI X2 3.1.10 Xi @ I Xz 3.1.11
Z Introduction Z+ Introduction Z(X) 1.1.8
Zp 1.1.8
11.
II I11 1.1.2 I1 Ilm 1.1.2 II IIA, 1.3.1
11 IIA”(E) 3-2.14 11 I(m E2.1.2 II IIu 2.5.9
II Ilv 2.5.9 I1 IIQ 3.1.9 Ilg,rll 3.2.2
LCAG 1.1.2 (RH) 2.3.15 (PNT) 2.3.9
(A) 2.3.3 (C,1) 2.3.3 (R,2) 2.1.10
R2 2.1.10 f 1.1.4 j 1.1.4
1.3.1 f 1.2.1 f- 1.3.7
T*S 1.3.1 T*p 1.3.1 D’T E1.3.6
L(T)(s) E1.3.6 T2O 1.3.5 T= 0 on U 1.3.6
.s,T 1.3.3 ryf 1.1.9 Tp 1.3.7
C,p E1.3.5 B(p,F) 3.2.2 IBl(v,F) 3.2.2
Ilell(cP,F) 3.2.2 p,, 3.2.2 F- T 3.2.1
F- TEA ’ 3.2.1 a topology 2.2.2 /!tIo pology 2.2.1
K topology 2.2.1 H(E) 3.1.18 ~ ~ ( E62.)5.2
a,( p, 1) 3.2.3 X” Introduction int X Introduction
e. 2.5.4 fi E1.2.6 di E1.2.6
V, E1.2.6 S 1.1.4 6, 1.3.3
m 1.1.2 ((3) 2.3.7 n(x) 2.3.8
A(n) 2.3.8 ~ ( x )2 .3.8 y 2.3.11
d(n) 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 ofharmonica nalysis 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),
r.
written additively, with Haar measure and dual group Haar measure is translation
invariant on the Bore1 sets of G and such measures are unique up to multiplicative
e r f' r
constants. We write = so that = G. Haar measure on G or 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)i s the space of all m-measurable functions CP for which a co-n stant
a, > 0 exists such that {x E G:I CP(x)I > a,} is locally null. We write @ Y if
{x: @(x) # Y(x)}i s locally null, and so L"(G) = Ym(G)/.-. Any non-discrete non-6-
compact LCAG contains a locally null set A with mA = w.) We designate the usual
norms on L'(G) and Lm(G)b y 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
