Book Review: Discrete harmonic analysis

Abstract

What is discrete harmonic analysis? The book under review, by Tullio Ceccherini-Silberstein, Fabio Scarabotti, and Filippo Tolli, gives a comprehensive answer. It is a continuation of a series of wonderful books by the same authors, including Harmonic analysis on finite groups: Representation theory, Gelfand pairs and Markov chains [13] and Representation theory and harmonic analysis of wreath products of finite groups [14] (both published by Cambridge University Press), which witness the power of harmonic analysis on finite groups both in theory and in applications.The study of mathematical models involving infinite objects, via their approximation by finite counterparts and leading to a discretization of these models, is one of the features of modern mathematics. This also reflects a "competition" and interaction between pure and applied mathematics. The title of the famous article of Louis Auslander and Richard Tolimieri that appeared in 1979 in the Bulletin of the AMS [6] asks, "Is computing with the finite Fourier transform pure or applied mathematics?" The book of Ceccherini-Silberstein, Scarabotti, and Tolli gives an exposition and a clarification of many questions raised by Auslander and Tolimieri, and much more.In order to confirm the importance and popularity of discretization, let us recall a couple of examples. The statistical models on Z d , d ≥ 2 (including the Ising model), are studied by means of their approximations in finite "subdomains" [−n, n] d and by passing to the limit for n → ∞ (the van Hove limit). The spectral properties of the Laplacian on an infinite graph Γ can be retrieved from the spectral properties of a sequence (Γ n ) n∈N of finite graphs approximating Γ. Similarly, the L 2 -Betti numbers of an infinite residually finite group G can be computed via a sequence (G n ) n∈N of finite groups approximating G (the Lück approximation), and the Gibbs measures on such a group G are limits of Gibbs measures of the G n 's [34]. Finally, the classical Fourier transform on Z or on the unit circle S 1 can be approximated by the discrete Fourier transform (DFT for short) on the finite cyclic groups Z n . An additional advantage of this approach is the fact that the groups Z n are not only finite but self-dual (in the sense of Pontryagin), in contrast with Z and S 1 that are mutually dual.Jean-Baptiste Joseph Fourier [27], at the turn of the nineteenth century, showed that representing a function as a sum of trigonometric functions (the exponentials e 2iπnx ), namely its Fourier series, greatly simplifies the study of the heat transfer. The study of the decomposition process of a function into these trigonometric functions is called Fourier analysis, while the operation of building back the function from these pieces is known as Fourier synthesis. In parallel, the original concept of Fourier analysis has been extended to apply to more and more abstract and general situations: the general field is often known as abstract harmonic analysis.