London Mathematical Society Student Texts 43: Fourier Analysis on Finite Groups and Applications (Softcover)

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Also from the back cover: "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.

"The author divides the book into two parts. In the first part, she 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 3x3 matrices with 1s down the diagonal and entries in a finite field, and it also includes a comparison of the finite and infinite versions of Selberg's trace formula. The book concludes with an introduction to zeta functions on finite graphs via the trace formula."

Part I: Finite Abelian Groups

1. Congruences and the Quotient Ring of the Integers mod n
2. The Discrete Fourier Transform on the Finite Circle Z/nZ
3. Graphs of Z/nZ, Adjacency Operators, Eigenvalues
4. Four Questions about Cayley Graphs
5. Finite Euclidean Graphs and Three Questions about Their Spectra
6. Random Walks on Cayley Graphs
7. Applications in Geometry and Analysis. Connections between Continuous and Finite Problems. Dido's Problem for Polygons
8. The Quadratic Reciprocity Law
9. The Fast Fourier Transform or FFT
10. The DFT on Finite Abelian Groups — Finite Tori
11. Error-Correcting Codes
12. The Poisson Sum Formula on a Finite Abelian Group
13. Some Applications in Chemistry and Physics
14. The Uncertainty Principle

Part II: Finite Nonabelian Groups

15. Fourier Transform and Representations of Finite Groups
16. Induced Representations
17. The Finite ax + b Group
18. The Heisenberg Group
19. Finite Symmetric Spaces — Finite Upper Half Plane H_q
20. Special Functions on H_q — K-Bessel and Spherical
21. The General Linear Group GL(2,F_q)
22. Selberg's Trace Formula and Isospectral Non-isomorphic Graphs
23. The Trace Formula on Finite Upper Planes
24. Trace Formula For a Tree and Ihara's Zeta Function