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. This text 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.
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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.Review:
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)
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