Principles of Harmonic Analysis

First, I will compare this book with its predecessor, Deitmar’s A First Course in Harmonic Analysis, 2nd edition, (Springer-Verlag 2005), which is a short and very readable book that I would recommend for self-study. It would be perfect preparation for a course using the book under review, though it is not needed for background.

Principles of Harmonic Analysis is an excellent and thorough introduction to both commutative and non-commutative harmonic analysis. It is suitable for any graduate student with the appropriate background: “knowledge of set theoretic topology, Lebesgue integration, and functional analysis on an introductory level. For the convenience of the reader, all necessary ingredients from these areas have been included in the appendices.”

The first two chapters (Haar Integration and Banach Algebras) provide background material. Chapters 3 and 4 cover key results about locally compact abelian (LCA) groups needed in harmonic analysis, including the Plancherel theorem, Pontryagin duality, the Poisson summation formula, and the structure theorems. Pontryagin duality and the Plancherel theorem were mentioned, but not proved, in Deitmar’s First Course book.

The remainder of the book focuses on harmonic analysis on locally compact groups that are not necessarily abelian. Chapter 5, on operators on Hilbert space, includes the functional calculus, Hilbert-Schmidt operators and Trace Class operators. Chapter 6 concerns representations of L¹(G) and the interplay between these algebra representations and unitary representations on the underlying group G. Chapter 7 on compact groups includes the Peter-Weyl theorem and an analysis of the representations of SU(2).

After some work, Chapter 8 describes direct Hilbert integrals on Hilbert bundles. This leads to a noncommutative Plancherel theorem for certain locally compact groups. For a proof, the reader is referred to Dixmier’s classical book on C^*-algebras and their representations. Concrete examples are given for the Heisenberg group (Chapter 10) and SL₂(R) (Chapter 11). The Selberg trace formula is introduced and proved in Chapter 9. It is also applied to SL₂(R) in Chapter 11.

The last chapter, on Wavelets, would be an intimidating and difficult introduction for someone with no prior acquaintance with wavelets. However, it does contain some more interesting theorems about representations.

The appendices are well done, with enough details to actually be useful. I applaud the authors’ free use of nets, rather than filters, in topology.

In summary, this is a superb book. It covers a great deal of important material, but it is extremely readable and well organized. Graduate students, and other newcomers to the field, will greatly appreciate the authors’ clear and careful writing.

Kenneth A. Ross (rossmath@pacinfo.com) taught at the University of Oregon from 1965 to 2000. His research area of interest was commutative harmonic analysis, especially where it has a probabilistic flavor. He is the author of the book Elementary Analysis: The Theory of Calculus (1980, now in 14th printing), co-author of Discrete Mathematics (with Charles R.B. Wright, 2003, fifth edition), and, as Ken Ross, the author of A Mathematician at the Ballpark: Odds and Probabilities for Baseball Fans (2004).