This self-contained book introduces readers to discrete harmonic analysis with an emphasis on the Discrete Fourier Transform and the Fast Fourier Transform on finite groups and finite fields, as well as their noncommutative versions. It also features applications to number theory, graph theory, and representation theory of finite groups. Beginning with elementary material on algebra and number theory, the book then delves into advanced topics from the frontiers of current research, including spectral analysis of the DFT, spectral graph theory and expanders, representation theory of finite groups and multiplicity-free triples, Tao's uncertainty principle for cyclic groups, harmonic analysis on GL(2,Fq), and applications of the Heisenberg group to DFT and FFT. With numerous examples, figures, and over 160 exercises to aid understanding, this book will be a valuable reference for graduate students and researchers in mathematics, engineering, and computer science.

'Although the roots of harmonic analysis lie in the continuous world, in the last few decades the field has also started to play a fundamental role in the discrete one. This book gives a panoramic view of Discrete Harmonic Analysis - an area that touches many branches of mathematics, such as number theory, spectral theory, groups and their representations, and graphs. The authors open a door for the reader taking him or her on a beautiful tour of classical and modern mathematics All this is done in a self-contained way that prepares the reader for cutting-edge research.'

Alex Lubotzky - Hebrew University of Jerusalem

'This book collects a number of gems in number theory and discrete mathematics that have never been put under the same roof, as far as I know. A distinct feature is that it puts harmonic analysis in the foreground where most textbooks present it as ancillary results. The authors must be complimented for their taste in the selection of topics.'

Alain Valette - Université de Neuchâtel, Switzerland

'This impressive book unites the qualities of a textbook and a research monograph into one comprehensive text. The central theme is the character theory of finite groups and fields, along with various applications. It offers careful and self-contained introductions to all required basics, which can serve for a series of courses. At the same time, it conducts the reader through several modern research themes and results, ranging from Tao's uncertainty principle via expander graphs to Hecke algebras and a detailed study of the representation theory of linear groups over finite fields.'

Wolfgang Woess - Technische Universität Graz

'The book is split up into four parts … 'Finite abelian groups and the DFT', 'Finite fields and their characters', 'Graphs and expanders', and 'Harmonic analysis on finite linear groups'. So it’s clear that the book covers a lot of ground, and should indeed be of great interest to number theorists, fledgling and otherwise. … While the book is written 'to be as self-contained as possible' , requiring just linear algebra up to and including the spectral theorem, basic group and ring theory, and 'elementary number theory', the reader is exposed to a lot of serious mathematics, some even at or near the frontier.'

Michael Berg Source: MAA Reviews

‘The exposition of the book is kept elementary and is clear and very readable. The selection of topics assembled in this book is very appealing. The basics of harmonic analysis are laid out thoroughly and in detail and at several occasions they are complemented by non-standard applications and results which illustrate the efficiency of harmonic analysis. In all this is a beautiful and satisfying introduction to harmonic analysis, its methods and applications in the discrete case.’

J. Mahnkopf Source: Monatshefte für Mathematik

‘The book under review is a very good introduction … In a self-contained way (it requires just elementary undergraduate rudiments of algebra and analysis and some mathematical maturity) it leads the reader to cutting-edge research.'

Rostislav Grigorchuk Source: Bulletin of the American Mathematical Society

‘... a very good introduction, for researchers-in-training, to the study of discrete harmonic analysis, its various techniques, and its relationship to other branches of mathematics.’

Mark Hunacek Source: The Mathematical Gazette