Fourier Analysis on Finite Abelian Groups

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.

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)