Discrete Harmonic Analysis
Representations, Number Theory, Expanders, and the Fourier Transform

A self-contained introduction to discrete harmonic analysis with an emphasis on the Discrete and Fast Fourier Transforms.

Tullio Ceccherini-Silberstein (Author), Fabio Scarabotti (Author), Filippo Tolli (Author)

9781107182332, Cambridge University Press

Hardback, published 21 June 2018

586 pages
23.5 x 15.5 x 3.6 cm, 0.96 kg

'... 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, The Mathematical Gazette

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.

Part I. Finite Abelian Groups and the DFT: 1. Finite Abelian groups
2. The Fourier transform on finite Abelian groups
3. Dirichlet's theorem on primes in arithmetic progressions
4. Spectral analysis of the DFT and number theory
5. The fast Fourier transform
Part II. Finite Fields and Their Characters: 6. Finite fields
7. Character theory of finite fields
Part III. Graphs and Expanders: 8. Graphs and their products
9. Expanders and Ramanujan graphs
Part IV. Harmonic Analysis of Finite Linear Groups: 10. Representation theory of finite groups
11. Induced representations and Mackey theory
12. Fourier analysis on finite affine groups and finite Heisenberg groups
13. Hecke algebras and multiplicity-free triples
14. Representation theory of GL(2,Fq).

Subject Areas: Calculus & mathematical analysis [PBK], Number theory [PBH], Discrete mathematics [PBD]