Fourier Analysis on Finite Abelian Groups [electronic resource] / by Bao Luong.

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.

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.