Fourier analysis on finite groups and applications

Abstract

Introduction Cast of characters Part I: 1. Congruences and the quotient ring of the integers mod n 1.2 The discrete Fourier transform on the finite circle 1.3 Graphs of Z/nZ, adjacency operators, eigenvalues 1.4 Four questions about Cayley graphs 1.5 Finite Euclidean graphs and three questions about their spectra 1.6 Random walks on Cayley graphs 1.7 Applications in geometry and analysis 1.8 The quadratic reciprocity law 1.9 The fast Fourier transform 1.10 The DFT on finite Abelian groups - finite tori 1.11 Error-correcting codes 1.12 The Poisson sum formula on a finite Abelian group 1.13 Some applications in chemistry and physics 1.14 The uncertainty principle Part II. Introduction 2.1 Fourier transform and representations of finite groups 2.2 Induced representations 2.3 The finite ax + b group 2.4 Heisenberg group 2.5 Finite symmetric spaces - finite upper half planes Hq 2.6 Special functions on Hq - K-Bessel and spherical 2.7 The general linear group GL(2, Fq) 2.8. Selberg's trace formula and isospectral non-isomorphic graphs 2.9 The trace formula on finite upper half planes 2.10 The trace formula for a tree and Ihara's zeta function.