Discrete Harmonic Analysis
Representations, Number Theory, Expanders, and the Fourier Transform
$80.00 USD
Part of Cambridge Studies in Advanced Mathematics
 Authors:
 Tullio CeccheriniSilberstein, Università degli Studi del Sannio, Italy
 Fabio Scarabotti, Università degli Studi di Roma 'La Sapienza', Italy
 Filippo Tolli, Università Roma Tre, Italy
 Date Published: August 2018
 availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
 format: Adobe eBook Reader
 isbn: 9781316865408
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This selfcontained 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 multiplicityfree 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.
Read more Provides a selfcontained, unified treatment of finite Abelian Groups, finite fields, Discrete and Fast Fourier Transforms, and applications to number theory, spectral graph theory and expanders, and representation theory of finite groups
 Includes a wellrounded set of examples and over 160 exercises
 Develops the reader's understanding from the fundamentals to the frontiers of current research
Reviews & endorsements
'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 selfcontained way that prepares the reader for cuttingedge research.' Alex Lubotzky, Hebrew University of Jerusalem
See more reviews'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 selfcontained 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 selfcontained 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, MAA Reviews
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×Product details
 Date Published: August 2018
 format: Adobe eBook Reader
 isbn: 9781316865408
 availability: This ISBN is for an eBook version which is distributed on our behalf by a third party.
Table of Contents
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 multiplicityfree triples
14. Representation theory of GL(2,Fq).
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