Preface by Walter Rudin
In classical Fourier analysis the action takes place on the unit circle, on the integers and on the real line. During the last 25 or 30 years, however, an increasing number of mathematicians have adopted the point of view that the most appropriate setting for the development of the theory of Fourier analysis is furnished by the class of all locally compact abelian groups. The relative ease with which the basic concepts and theorems can be transferred to this general context may be one of the factors which contributes to the feeling of some that this extension is a dilution of of the classical theory, that it is merely generalization for the sake of generalization.
However, group-theoretic considerations seem to be inherent in the subject. They are implicit in much of the classical work, and their explicit introduction has led to many interesting new analytic problems (it is one of the aims of this book to prove this point) as well as to conceptual clarifications. To cite a very rudimentary example: In discussing Fourier transforms on the line it helps to have two lines in mind, one for the functions and one for their transforms, and to realize that each is the dual group of the other.
Also, there are classical subjects which lead almost inevitably to this extension of the theory. For instance, Bohr (1) noticed almost 50 years ago that the unique factorization theorem for positive integers allows us to regard every ordinary Dirichlet series as a power series in infinitely many variables. The boundary values yield a function of infinitely many variables, periodic in each, that is to say, a function on the infinite-dimensional torus Tω. It then becomes of interest to know the closed subgroups of Tω, and it turns out that these comprise all compact metric abelian groups. Once we agree to admit these groups we have to admit their duals, i.e., the countable discrete abelian groups, and since the class of all locally compact abelian groups can be built up from the compact ones, the discrete ones, and the euclidean spaces, it would seem artificial to restrict ourselves to a smaller subclass.
The principal objects of study in the present book are the group algebras L1(G) and M(G); L1(G) consists of all complex functions on the group G which are integrable with respect to the Haar measure of G, M(G) consists of all bounded regular Borel measures on G, and multiplication is defined in both cases by convolution. Although certain aspects of these algebras have been studied for non-commutative groups G, I restrict myself to the abelian case. Other Lp-spaces appear occasionally, but are not treated systematically.
The development of the general theory, given in Chapter 1, is based on some simple facts concerning Banach algebras; these, as well as other background material, are collected in the Appendices at the end of the book. It seems appropriate to develop the material in this way, since much of the early work on Banach algebras was stimulated by Fourier analysis. Chapter 2 contains the structure theory of locally compact abelian groups. These two chapters are introductory, and most of their content is well known.
The material of Chapters 3 to 9, on the other hand, has not previously appeared in book form. Most of it is of very recent vintage, many of the results were obtained only within the last two or three years, and although the solutions of some of the problems under consideration are fairly complete by now, many open questions remain.
My own work in this field has been greatly stimulated by conversations and correspondence with Paul J. Cohen, Edwin Hewitt, Raphael Salem, and Antoni Zygmund, and by my collaboration with Henry Helson, Jean-Pierre Kahane, and YitzhakKatznelson. It is also a pleasure to thank the Alfred P. Sloan Foundation for its generous financial support.