Fourier Analysis on Finite Abelian Groups

ISBN-13: 9780817649159 / Angielski / Twarda / 2009 / 159 str.

Bao Luong

Opis

Fourier analysis has been the inspiration for a technological wave of advances in fields such as imaging processing, financial modeling, algorithms and sequence design. This unified, self-contained book examines the mathematical tools used for decomposing and analyzing functions, specifically, the application of the discrete] Fourier transform to finite Abelian groups. With countless examples and unique exercise sets at the end of each section, Fourier Analysis on Finite Abelian Groups is a perfect companion to a first course in Fourier analysis. The first chapter provides the fundamental material that is a strong foundation for all subsequent chapters. Special topics including: * Computing Eigenvalues of the Fourier transform * Applications to Banach algebras * Tensor decompositions of the Fourier transform * Quadratic Gaussian sums. This book introduces mathematics students to subjects that are within their reach, but it also has powerful applications that may appeal to advanced researchers and mathematicians. The only prerequisites necessary are group theory, linear algebra, and complex analysis.

Recenzje

From the reviews:

"The book under review covers, qua orientation, a pretty broad spectrum ... . The author, Bao Luong, targets well-prepared upper-division students and certain 'outsiders' (scientists and engineers) and has taken pains to make his presentation accessible. ... this compact book is indeed very readable ... . there are fifty-six exercises scattered throughout the text, generally quite sporty." (Michael Berg, The Mathematical Association of America, October, 2009)

"The presentation is entirely theoretical ... . What the book does do is cover the Fourier transform (FT) on finite abelian groups, with some emphasis on Gaussian quadratic sums (eigenvalues of the FT) and eigenspaces of the FT operator. There are 56 exercises of varying difficulty spread throughout the book. ... may be helpful for that student's review at the end of the course and for the instructor, mathematicians, and many scientists and engineers." (Colin C. Graham, Mathematical Reviews, Issue 2011 e)

Spis treści

Preface.- Overview.- Chapter 1: Foundation Material.- Results from Group Theory.- Quadratic Congruences.- Chebyshev Systems of Functions.- Chapter 2: The Fourier Transform.- A Special Class of Linear Operators.- Characters.- The Orthogonal Relations for Characters.- The Fourier Transform.- The Fourier Transform of Periodic Functions.- The Inverse Fourier Transform.- The Inversion Formula.- Matrices of the Fourier Transform.- Iterated Fourier Transform.- Is the Fourier Transform a Self-Adjoint Operator?.- The Convolutions Operator.- Banach Algebra.- The Uncertainty Principle.- The Tensor Decomposition.- The Tensor Decomposition of Vector Spaces.- The Fourier Transform and Isometries.- Reduction to Finite Cyclic Groups.- Symmetric and Antisymmetric Functions.- Eigenvalues and Eigenvectors.- Spectrak Theorem.- Ergodic Theorem.- Multiplicities of Eigenvalues.- The Quantum Fourier Transform.- Chapter 3: Quadratic Sums.- 1. The Number G_n(1).- Reduction Formulas.

Blurb

Fourier analysis has been the inspiration for a technological wave of advances in fields such as imaging processing, financial modeling, cryptography, algorithms, and sequence design. This self-contained book provides a thorough look at the Fourier transform, one of the most useful tools in applied mathematics.

With countless examples and unique exercise sets at the end of most sections, Fourier Analysis on Finite Abelian Groups is a perfect companion for a first course in Fourier analysis. The first two chapters provide fundamental material for a strong foundation to deal with subsequent chapters.

Special topics covered include:

* Computing eigenvalues of the Fourier transform

* Applications to Banach algebras

* Tensor decompositions of the Fourier transform

* Quadratic Gaussian sums

This book provides a useful introduction for well-prepared undergraduate and graduate students and powerful applications that may appeal to researchers and mathematicians. The only prerequisites are courses in group theory and linear algebra.