TY - GEN AU - Ceccherini-Silberstein,Tulio AU - Fabio Scarabotti AU - Filippo Tolli TI - Discrete harmonic analysis: representations number theory expanders and the fourier transform T2 - Cambridge Studies in Advanced Mathematics SN - 9781107182332 U1 - 515.2433 23 PY - 2019/// CY - United Kingdom PB - CUP KW - Analysis KW - Harmonic Analysis KW - Fourier Transforms KW - Finite Groups KW - Finite Fields N1 - 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 N2 - 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 ER -