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Fourier Analysis on Finite Groups and Applications

Part of London Mathematical Society Student Texts

Author: Audrey Terras, University of California, San Diego

Date Published: June 1999
availability: Available
format: Paperback
isbn: 9780521457187

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About the Authors

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.
- An accessible introduction to the topic
- Area has become popular in recent years, due to the wide range of applications
- Author is recognised in the field
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Reviews & endorsements
'This book is likely to broader the mind of many a professional mathematician, and the list of over 400 references will be a valuable resource.' Peter Rowlinson, Bulletin of the London Mathematical Society

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Product details
- Date Published: June 1999
- format: Paperback
- isbn: 9780521457187
- length: 456 pages
- dimensions: 229 x 152 x 26 mm
- weight: 0.67kg
- contains: 64 b/w illus.
- availability: Available
Table of 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.
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Author
Audrey Terras, University of California, San Diego