Fourier Analysis on Finite Groups and Applications

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Cambridge University Press, Mar 28, 1999 - Mathematics - 442 pages
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This book gives a friendly introduction to Fourier analysis on finite groups, both commutative and noncommutative. Aimed at students in mathematics, engineering and the physical sciences, it examines the theory of finite groups in a manner 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 analog of higher dimensional Euclidean space. The second part emphasizes matrix groups such as the Heisenberg group of upper triangular 2x2 matrices with 1's down the diagonal and entries in a finite field. The book concludes with an introduction to zeta functions on finite graphs via the trace formula.
  

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Contents

Congruences and the Quotient Ring of the Integers mod n
8
The Discrete Fourier Transform on the Finite Circle ZnZ
30
Graphs of ZnZ Adjacency Operators Eigenvalues
46
Four Questions about Cayley Graphs
70
Finite Euclidean Graphs and Three Questions about Their Spectra
83
Random Walks on Cayley Graphs
97
Applications in Geometry and Analysis Connections between Continuous and Finite Problems Didos Problem for Polygons
114
The Quadratic Reciprocity Law
128
Finite Nonabelian Groups
237
Fourier Transform and Representations of Finite Groups
240
Induced Representations
267
The Finite ax + b Group
281
The Heisenberg Group
293
Finite Symmetric Spaces Finite Upper Half Plane Hq
302
Special Functions on Hq AfBessel and Spherical
329
The General Linear Group GL2 F
362

The Fast Fourier Transform or FFT
151
The DFT on Finite Abelian Groups Finite Tori
161
ErrorCorrecting Codes
187
The Poisson Sum Formula on a Finite Abelian Group
197
Some Applications in Chemistry and Physics
212
The Uncertainty Principle
223
Selbergs Trace Formula and Isospectral Nonisomorphic Graphs
385
The Trace Formula on Finite Upper Half Planes
394
Trace Formula for a Tree and Iharas Zeta Function
408
References
421
Index
437
Copyright

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About the author (1999)

Audrey Terras is Professor of Mathematics at the University of California, San Diego.