## Fourier Analysis on Finite Groups and ApplicationsThis 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 |

421 | |

437 | |

### Common terms and phrases

abelian group additive group adjacency matrix adjacency operator Aff(q algebra Cayley graph Chapter character table Chung compute congruence conjugacy classes consider convolution defined Definition denotes Diaconis diagonal discrete Fourier transform eigenfunctions eigenvalues elements equation euclidean graphs example Exercise Figure finite abelian group finite analogues finite circle finite field finite group finite upper half Fourier analysis Frobenius function f Gauss sum Gelfand pair group G Heisenberg group Hint induced representation inner product integer isomorphic k-regular Kloosterman sums Laplacian Lemma linear Matlab matrix entries multiplicative group norm Note number theory odd prime orthogonal polynomial proof Prove quadratic reciprocity law Ramanujan graphs random number random walk representation of G ring says Selberg Selberg trace formula Show spectral spectrum spherical functions subgroup Suppose symmetric space Terras Theorem trace formula upper half plane vertices x e G ye G Z/nZ Z/pZ zeta function

### Popular passages

Page 432 - Discrete Fourier Transform When the Number of Data Samples is Prime," Proceedings of the IEEE 56, 1107-8 (1968) 131.

Page v - ... It is the largest in existence. Can you imagine the thrill of turning it to some new corner of the heavens to see something never before seen from earth? I actually like that he is busy with the Royal Society and his club, for when I finish my other work I can spend all night sweeping the heavens. Sometimes when I am alone in the dark, and the universe reveals yet another secret, I say the names of my long lost sisters, forgotten in the books that record our science — Aglaonice of Thessaly,...