Fourier Analysis on Finite Groups and Applications

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.