Applications of Harmonic Analysis

389 Pages·1964·15.976 MB·English

by  I. M. Gel'fand and N. Ya. Vilenkin (Auth.)

GENERALIZED FUNCTIONS
VOLUME 1, Properties and Operations—I. M. Gerfand and
G. E. Shilov
Translator: Eugene Saletan
VOLUME 4. Applications of Harmonic Analysis
—I. M. Gel'fand and N. Ya. Vilenkin
Translator: Amiel Feinstein
In Preparation
VOLUME 5. Integral Geometry and Representation Theory
—I. M. GelTand, M. 1. Graev, and N. Ya.
Vilenkin
Translator: Eugene Saletan
Future Volumes
VOLUME 2. Function and Generalized Function Spaces—
I M Gel'fand and G E Shilov
. . . .
VOLUME 3. Theory of Differential Equations
—I. M. Gel'fand and G. E. Shilov
GENERALIZED
FUNCTIONS
Volume 4
Applications of Harmonic Analysis
I. M. GEL'FAND
and
N. Ya. VILENKIN
Academy of Sciences, U.S.S.R.
Translated by
AMIEL FEINSTEIN
1964
ACADEMIC PRESS · New York and London
COPYRIGHT © 1964, BY ACADEMIC PRESS INC.
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NO PART OF THIS BOOK MAY BE REPRODUCED IN ANY FORM,
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LIBRARY OF CONGRESS CATALOG CARD NUMBER: 63-16960
PRINTED IN THE UNITED STATES OF AMERICA
GENERALIZED FUNCTIONS:
VOLUME 4: APPLICATIONS OF HARMONIC ANALYSIS
THIS BOOK IS A TRANSLATION OF
NEKOTORYE PRIMENENIYA GARMONICHESKOGO ANALIZA.
OBOBSHCHENNYE FUNKTSII, VYPUSK 4.
GOSUDARSTVENNOE IZDATEL'STVO FIZIKO-MATEMATICHESKOI LITERATURY,
MOSCOW, 1961.
Translator's Note
This translation differs from the Russian edition of 1961 mainly in
that the authors have revised the proof contained in Section 3.2 of
Chapter 1, as well as a major portion of Section 3.6 of the same chapter
(in particular, the proof of Theorem 7). The symbols τ and A in the
margin indicate the beginning and end of a few passages in which I have
ventured to deviate (in a not completely trivial way) from the original,
for the purpose of eliminating a gap or an obscurity, and in one case
(in Section 5.1, Chapter 4) of reducing a ten-line proof to the obvious
one-line proof. In addition, I have allowed myself the luxury of a number
of remarks, which appear in footnotes marked by a dagger (t). Any
comments regarding these or other portions of this volume will be welcome.
June, 1964 AMIEL FEINSTEIN
Foreword
This book is the fourth volume of a series of monographs on functional
analysis appearing under the title "Generalized Functions." It should
not, however, be considered a direct sequel to the preceding volumes.
In writing this volume the authors have striven for the maximum inde­
pendence from the preceding volumes. Only that material which is
discussed in the first two chapters of Volume 1 must be considered as
the indispensable minimum which the reader is required to know. In
view of this, certain topics which were discussed in the preceding volumes
are briefly repeated here.
This book is devoted to two general topics: recent developments in
the theory of linear topological spaces and the construction of harmonic
analysis in Az-dimensional Euclidean and infinite-dimensional spaces.
After the appearance of a theory of topological spaces, the question
arose of distinguishing a class of topological spaces, defined by rather
simple axioms and including all (or nearly all) spaces which arise in
applications. In the same way, after a theory of linear topological spaces
was created, it became necessary to ascertain which class of spaces is
most suitable for use in mathematical analysis. Such a class of linear
topological spaces—nuclear spaces—was singled out by the French
mathematician A. Grothendieck.
The class of nuclear spaces includes all or nearly all linear topological
spaces which are presently used in analysis, and has a number of extremely
important properties: the kernel theorem of L. Schwartz is valid in
nuclear spaces, as is also the theorem on the spectral resolution of a
self-adjoint operator. Furthermore, any measure on the cylinder sets in
the conjugate space of a nuclear space is countably additive. The first
and fourth chapters of this book are devoted to the discussion of these
questions. In connection with spectral analysis, the concept of a rigged
Hilbert space is introduced, which turns out, apparently, to be very
useful also in many other questions in mathematics.
The second question which we study in this volume is the harmonic
analysis of functions in various spaces. Harmonic analysis in Euclidean
space (the Fourier integral) has already been discussed to some extent
in previous volumes. We have given up the idea of repeating here the
material in the preceding volumes which was devoted to the Fourier
integral (possibly, had all of the volumes been written at the same time,
many questions in the theory of the Fourier integral, for example the
Paley-Wiener theorem for generalized functions, would have found their
natural setting in this volume). We discuss here only questions of harmonic
analysis in Euclidean space which were left unclarified in the previous
vii
Viii FOREWORD
volumes. Namely, we consider the Fourier transformation of measures
having one or another order of growth (the theory of generalized positive
definite functions) and its application in the theory of generalized random
processes. The Fourier transformation of measures in linear topological
spaces is considered at the same time.
In the following, fifth volume, we single out questions of harmonic
analysis on homogeneous spaces (in particular, harmonic analysis on
groups) and intimately related questions of integral geometry on certain
spaces of constant curvature. This theory, which is very rich in the diversity
of its results (connected, for example, with the theory of special functions,
analytic functions of several complex variables, etc.) could not, of course,
be discussed in its entirety within the confines of the fifth volume. We
have restricted ourselves to discussing only questions of harmonic analysis
on the Lorentz group. It should be remarked that harmonic analysis on
the Lorentz group and the related homogeneous spaces is a considerably
richer subject than harmonic analysis in the "degenerate" case of a
Euclidean space. For example, in the case of a Euclidean space only the
smoothness of the Fourier transform of a function is influenced by
specifying one kind of behavior or another at infinity of the function
itself. But in the case of the Lorentz group, specifying the behavior of
the function at infinity leads to certain algebraic relations among the
values of its Fourier transform at different points. However, at the present
time these questions are only in the initial stages of investigation.
The material of this fourth volume represents a complete unit in itself,
and, as we have said, the exposition is practically independent of the
preceding volumes. In spite of the relation of one chapter to another,
one can begin a reading of this book with the first chapter, which contains
the general theory of nuclear and rigged Hilbert spaces, or with the second
chapter, which discusses the more elementary theory of positive definite
generalized functions.
We mention that certain chapters contain, together with general results,
others of a more specialized nature; these can be passed over at the first
reading.
The authors wish to express their deep gratitude to those who helped
them in working on this book: F. V. Shirokov, whose contributions far
exceeded the limits of ordinary editorial work, A. S. Dynin, B. S. Mityagin,
and V. B. Lidskii, whose valuable advice the authors used in writing up
various topics in the first chapter. They express their particular thanks
to S. A. Vilenkin, who took upon herself all the work connected with
preparing the manuscript for press.
I. M. GEL'FAND
N. YA. VILENKIN
CHAPTER I
THE KERNEL THEOREM. NUCLEAR SPACES.
RIGGED HILBERT SPACE
This chapter is devoted to the study of a class of countably normed
spaces^—so-called nuclear spaces. These spaces first appeared in con­
nection with the "kernel theorem," which will be used repeatedly in
this book. Later it became evident that nuclear spaces also play an
essential role in many other topics of functional analysis, namely, nuclear
spaces turn out to be the most natural class of spaces for the study of
the spectral decomposition of self-adjoint operators. These spaces were
already introduced in Volume III (Chapter IV, Section 3.1), in con­
nection with the consideration of spectral decompositions. However,
the definition of nuclearity which was given there is not entirely suitable
for the study of other questions. Therefore in this volume we will use
another definition of nuclear space, which in the essential cases is
equivalent to nuclearity in the sense discussed in Volume III. The
discussion in this chapter does not depend upon that in Volume III.
In order to attain a complete independence of these treatments, we will
present in this chapter certain results on the spectral decomposition of
self-adjoint operators. Here, however, our main attention will be given
to the general aspects of theory, unlike the treatment in Volume III,
where a not inconsiderable part was played by the applications of these
results to specific differential operators.
An important role is played by the concept of a rigged Hilbert space.
^ We assume that the reader is familiar with the concept of a countably normed space
to the extent of Chapters I and II of Volume II of this series. Besides, a brief discussion
of the basic facts relating to a special case of such spaces—countably Hilbert spaces—is
given at the beginning of Section 3. Let us note that throughout this volume every
countably normed space considered will be taken, without special mention, to be complete.
Moreover, we will as a rule assume that the compatible norms || φ ||n, 1 < Λ < «,
which define the topolog>' in a countably normed space Φ are monotonically increa­
sing, i.e., that for every element ψΕΦ the inequalities
ΙΙ9ΊΙ1 < - < ΙΙ9ΊΙ„ < ···
hold.
2 THE KERNEL THEOREM Ch. I
This concept arises in considering nuclear spaces in which an inner
product is introduced in some way or another. The theory of rigged
Hilbert spaces is discussed in Section 4, where applications of this
theory to the spectral analysis of self-adjoint operators are presented.
Also related to the theory of nuclear spaces is the subject of measure
theory in linear topological spaces, discussed in Chapter IV. We will
show in that chapter that the nuclearity of a space Φ is a necessary and
sufficient condition for every measure on the cylinder sets in the space
Φ', conjugate to Φ, to be completely additive.
1. Bilinear Functionals on Countably Normed Spaces.
The Kernel Theorem
In this section we will study the general form of bilinear functionals^
on countably normed spaces. We will show that any bilinear functional
J5(<p, 0), continuous with respect to its arguments φ and φ which range
over countably normed spaces Φ and is continuous with respect to
certain norms || φ ||;^ and | i/r |^ in these spaces. In other words, we will
show that
\Β{φ,φ)\ <Μ\\φ\\^\φΙ
for all elements φ and φ of the spaces Φ and Ψ, where the numbers
M, m, η do not depend upon φ and φ.
Applying this result to various specific spaces, we obtain the general
form of bilinear functionals on these spaces. One of the most important
results thereby obtained is a description of the biUnear functionals on
the space Κ of infinitely differentiable functions with bounded supports.
It will be proved that these functionals have the form
Β{φ,φ) = {Ρ,ψ{χ)φ{γ)1
where F is a hnear functional on the space K2 of infinitely differentiable
functions of the variables χ = (jCj, ä:^.) and y = (y^ ...,>ά), with
^ A bilinear functional is a functional which is linear in both arguments ψ and φ.
Further on we will encounter functionals Β{φ, φ) which are linear in ψ and antilinear
in φ^ i.e.,
Β{αφ, + βφ^, φ) = αΒ{ψ„ φ) + βΒ{φ^, φ)
but
Β{ψ, αφ, + βφ^) = άΒ(φ, φ,) + βΒ{φ, φ^) .
Such functionals are called Hermitean-bilinear, or simply Hermitean functionals.
1.1 Bilinear Functionals on Countably Normed Spaces 3
bounded supports. This theorem is called **the kernel theorem"^ and
will be used frequently in this book.
1.1. Convex Functionals
The general properties of bilinear functionals are established by means
of certain theorems concerning convex functionals on countably normed
spaces.
A real functional ρ{ψ) ^ 0, defined on a linear space Φ, is called
convex if for any two elements φ and φ of Φ the relation
Ρ{ψ + ΦΧ Ρ{ψ) + Ρ{Φ)
is satisfied, and for any element ψ and complex number α the relation
ρ{(χψ) = I α \ρ{φ)
is satisfied.
As an example of a convex functional one can take the norm || ψ \\
in a normed space, since in view of the definition of a norm || φ + ^ ||
+ and \\ocφ\\= |α|||φ||.
We remark that finiteness of the value of the functional ρ{φ) for
every element 9? of Φ is not assumed in our definition, i.e., ρ{φ) may
assume the value +00, in which case we take λ · 00 = 00, α 7^ 0, and
CO a = 00 + 00 = 00. Obviously the set of elements φ for which
the functional ρ{φ) takes on a finite value forms a linear subspace* in Φ.
One could avoid using infinite values for ρ{φ) by considering the func­
tional ρ(φ) only on this subspace.
The concept of a convex functional is related to the geometric concept
of an absolutely convex set. A set Aim linear space Φ is called absolutely
convex if it contains, together with any two elements φ and φ, every
element of the form αφ + βφ, where OL and β are complex numbers such
that I α I + I j8 I < l.s
In order to establish the connection between the concepts of a convex
functional and an absolutely convex set, let us associate with the
functional ρ{φ) the set A consisting of every element φβΦ for which
ρ{φ) < 1. We show that this set is absolutely convex. In fact, let φ
'·* More precisely, "the kernel theorem for the space iC'*; the general kernel theorem
will be proved in Section 3.
* Nonclosed, generally speaking.
* In a real linear space the absolute convexity of a set is equivalent to its being convex
and centrally symmetric.
4 THE KERNEL THEOREM Ch. I
and φ be elements of the set A, i.e., ρ{φ) < 1 and ρ{φ) < 1. Then if
I α I + I j8 I < 1, we have
ρ{αψ + W < 1 α\ρ{φ) + | β \ρ{φ) < | α | + | ]8 | < 1
and so (χφ + βφ e Ay which proves the absolute convexity of A.
Conversely, if A is any absolutely convex closed set (we are assuming
now that Φ is a topological linear space), then there exists a convex
functional ρ{φ) such that A coincides with the set of elements φ for
which ρ{φ) < 1. This functional is defined by
Ρ{ψ) = Vq^
where q is the supremum of those values λ > 0 for which λφ e A, It
is easily seen that ρ{φ) < 1 for every element φ of the set A and that
ρ{φ) > 1 if φ does not belong to A. Moreover, from the absolute convexity
of the set A it follows that ρ{φ) is a convex functional.
We have thus estabHshed a correspondence between convex func­
tionals ρ{φ) and absolutely convex sets in the space Φ.
We remark that the functional ρ{φ) is finite on the smallest Hnear
subspace ¥^ in Φ which contains the set A. In particular the functional
ρ{φ) is finite on the entire space Φ, if for any element φ of Φ one can
find an element φ Φ 0 in A such that φ Hes on the Hne passing through
φ and the zero element (i.e., φ has the form φ = λφ). We will in this
case call the set A absorbing. Observe that if A is an absorbing set, then
every element φ of the space Φ belongs to one of the sets nA (we denote
by nA the set of elements of the form ηφ, φ Ε A),
We call a functional ρ{φ) lower semicontinuous if for any element
of the space Φ and any € > 0 there is a neighborhood U of <po such
that the inequality
Ρ{φ) > Ρ(ψο) - ^
holds at every point φ of this neighborhood.
Let us prove that if the functional ρ{φ) is lower semicontinuous, then
the set Ay consisting of those elements φ for which ρ(φ) < 1, is closed.
As a matter of fact, let φο be a limit point of the set A. Then in any
neighborhood of the point φο there is a point φ for which ρ{φ) < 1.
Consequently, in view of the lower semicontinuity of the functional
/)((p), ρΙΨΟ) < 1, which proves that A is closed.
Let us now prove that if the absolutely convex set A is closed, then
the corresponding convex functional ρ{φ) is lower semicontinuous. In
fact, the set A of points φ for which ρ{φ) < 1 being closed, the set of
points φ for which ρ{φ) > I is open. But then any set which is defined