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Discrete harmonic analysis: representations number theory expanders and the fourier transform/ Tullio Ceccherini-Silberstein, Fabio Scarabotti and Filippo Tolli

By: Contributor(s): Series: Cambridge Studies in Advanced Mathematics ; 172Publication details: United Kingdom: CUP, 2019Description: xiii, 573 pages, 23cmISBN:
  • 9781107182332
Subject(s): DDC classification:
  • 23 515.2433 C388
Contents:
Part I Finite Abelian Groups and the DFT -- Finite Abelian Groups -- The Fourier Transform on Finite Abelian Groups -- Dirichlet's Theorem on Primes in Arithmetic Progression-- Spectral Analysis of the DFT and number theory -- The Fast Fourier transform -- Part II Finite Fields and their characters -- Finite Fields -- Character Theory of Finite Fields -- Part III graphs and Expanders -- Graphs and Their Products -- Expanders and Ramanujan Graphs -- Part IV Harmonic Analysis on Finite Linear Groups -- Representation Theory of Finite Groups -- Induced Representations and Mackey Theory -- Fourier Analysis on Finite Affine Groups and Finite Heisenberg Groups -- Hecke Algebras and Multiplicity-Free Triples -- Representation Theory of GL (2 Fq) -- Appendix Chebyshev Polynomials -- Bibliography -- Index
Summary: 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.
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Includes Bibliographical references and index

Part I Finite Abelian Groups and the DFT --
Finite Abelian Groups --
The Fourier Transform on Finite Abelian Groups --
Dirichlet's Theorem on Primes in Arithmetic Progression--
Spectral Analysis of the DFT and number theory --
The Fast Fourier transform --
Part II Finite Fields and their characters --
Finite Fields --
Character Theory of Finite Fields --
Part III graphs and Expanders --
Graphs and Their Products --
Expanders and Ramanujan Graphs --
Part IV Harmonic Analysis on Finite Linear Groups --
Representation Theory of Finite Groups --
Induced Representations and Mackey Theory --
Fourier Analysis on Finite Affine Groups and Finite Heisenberg Groups --
Hecke Algebras and Multiplicity-Free Triples --
Representation Theory of GL (2 Fq) --
Appendix Chebyshev Polynomials --
Bibliography --
Index

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.

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