文档介绍：AN INTRODUCTION TO
HARMONIC
ANALYSIS
Yitzhak Katznelson
Third Corrected Edition
Preface
Harmonic analysis is the study of objects (functions, measures, etc.),
defined on topological groups. The group structure enters into the study
by allowing the consideration of the translates of the object under study,
that is, by placing the object in a translation-invariant space. The study
consists of two steps. First: finding the "ponents" of
the object, that is, objects of the same or similar class, which exhibit
the simplest behavior under translation and which "belong" to the ob-
ject under study (harmonic or spectral analysis); and second: finding
a way in which the object can be construed as bination of its
ponents (harmonic or spectral synthesis).
The vagueness of this description is due not only to the limitation
of the author but also to the vastness of its scope. In trying to make it
†
clearer, one can proceed in various ways ; we have chosen here to sac-
rifice generality for the sake of concreteness. We start with the circle
group T and deal with classical Fourier series in the first 2ve chap-
ters, turning then to the real line in Chapter VI ing to locally
compact abelian groups, only for a brief sketch, in Chapter VII. The
philosophy behind the choice of this approach is that it makes it easier
for students to grasp the main ideas and gives them a large class of con-
crete examples which are essential for the proper understanding of the
theory in the general context of topological groups. The presentation of
Fourier series and integrals differs from that in [1], [7], [8], and [28] in
being, I believe, more explicitly aimed at the general (pact
abelian) case.
The last chapter is an introduction to the theory mutative
Banach algebras. It is biased, studying Banach algebras mainly as a
tool in harmonic analysis.
This book is an expanded version of a set of lecture notes written
†
Hence the indefinite article in the title of t