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Full Description
This book gives a friendly introduction to Fourier analysis on finite groups, both commutative and non-commutative. Aimed at students in mathematics, engineering and the physical sciences, it examines the theory of finite groups in a manner that is both accessible to the beginner and suitable for graduate research. With applications in chemistry, error-correcting codes, data analysis, graph theory, number theory and probability, the book presents a concrete approach to abstract group theory through applied examples, pictures and computer experiments. In the first part, the author parallels the development of Fourier analysis on the real line and the circle, and then moves on to analogues of higher dimensional Euclidean space. The second part emphasizes matrix groups such as the Heisenberg group of upper triangular 2x2 matrices. The book concludes with an introduction to zeta functions on finite graphs via the trace formula.
Contents
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.
