Introduction to Harmonic Analysis and Generalized Gelfand Pairs

The book under review starts by recalling the elementary theory of Fourier series (from the basic definitions through criteria for convergence and uniform convergence up to Parseval’s theorem) and Fourier integrals (convolution product, Fourier transform, the inversion theorem, Plancherel’s theorem, and functions of bounded variation). It then proceeds to prove Poisson’s summation formula, with an application to the functional equation for the theta functions which, as the author remarks, is used in number theory to prove the functional equations for various zeta or L-functions. The introductory chapters end with Bochner’s theorem and a few lines sketching how to extend the results to Rⁿ, remarking that the proofs are usually more difficult.

The middle part of the book, from chapter three to chapter five, is devoted to abstract harmonic analysis on locally compact (abelian) groups, as pioneered by H. Cartan and R. Godement. Chapter three reviews the necessary facts on topological groups, quotients, local compactness, direct products and functions on locally compact groups. Chapter four introduces Haar measures following Weil and Bourbaki, quoting the Riesz representation theorem for the connection to the set-theoretic approach. The existence of a left or right invariant Haar measure is assumed, referring the reader to Bourbaki (Intégration, Hermann, 1965) or Weil (L’Intégration dans les Groupes Topologiques et ses Applications, Hermann, 1953) but uniqueness is proved. Then we have Weil’s formula relating the Haar measures on a group, a closed normal subgroup and the corresponding quotient. This chapter ends with a generalization of the convolution product for complex-valued functions on a locally compact group equipped with a left invariant Haar measure.

Chapter five, on harmonic analysis on locally compact (abelian) groups with a left invariant Haar measure, begins with the elementary properties of unitary representations and irreducible representations (with examples given by the left or right regular representation on any group and the one-dimensional representations of abelian groups). Again, some results are just quoted, for example that any representation of a compact group on a Hilbert space is equivalent to a unitary representation. Next, some results on functional analysis are reviewed, from the Alaoglu’s theorem to the Krein-Milman’s theorem. The last sections of chapter five are devoted to the main properties generalizing the classical results now in the realm of locally compact abelian groups. Thus we find here, after recalling the necessary facts for the unitary characters of one such group and Pontryagin duality, Bochner’s theorem, the inversion theorem and some of its corollaries, and Plancherel’s theorem.

The last part of the book, from chapter six to chapter nine is devoted to the author’s approach to abstract harmonic analysis, generalizing the techniques of Cartan, Godement and Weil to the setting of Gelfand pairs (G, K) where G is a locally compact group and K is a compact subgroup; the classical theory is recovered when K is the trivial subgroup. Thus, we find in chapter six analogs of Bochner’s theorem, the inversion theorem, and Plancherel’s theorem. Examples of Gelfand pairs are studied in detail in chapter seven.

In chapters eight and nine a further generalization of some parts of the theory, due to E. G. F. Thomas (“The Theorem of Bochner-Schwartz-Godement for Generalized Gelfand Pairs”, in Functional Analysis: Surveys and Results III, 291-304. Elsevier, 1984) and the author, is developed, now for pairs (G, H) where H is just a closed subgroup of G. This part of the book, actually about half of it, is the main contribution to the large literature on abstract harmonic analysis, collecting in an expository form results that are dispersed on several papers and making them accessible to a graduate student. The requisites for these last chapters jump a bit, requiring some familiarity with Lie groups and Lie algebras, distribution theory, and topological vector spaces.

This is a well-written book, and although rather terse, a motivated reader will have no trouble filling in the details, but a willingness to do that is presupposed. Examples, although scarce, are well chosen. The book has no exercises labeled as such, but I believe that filling in details more than makes up for that.

Felipe Zaldivar is Professor of Mathematics at the Universidad Autonoma Metropolitana-I, in Mexico City. His e-mail address is fzc@oso.izt.uam.mx