This self-contained overview examines the decomposition and analysis of functions within Fourier transforms and Abelian groups. Readers will learn the fundamentals, which are illustrated by examples and unique exercises at the end of each section.

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.

Examines the mathematical tools used for decomposing and analyzing functions, specifically, the application of the [discrete] Fourier transform to finite Abelian groups Provides countless examples and unique exercise sets at the end of each section A perfect companion to a first course in Fourier analysis Includes special topics such as computing Eigenvalues of the Fourier transform, applications to Banach algebras, tensor decompositions of the Fourier transform and quadratic Gaussian sums Introduces mathematics students to subjects that are within their reach but have powerful applications that also appeal to advanced researchers and mathematicians. The only prerequisites are group theory, linear algebra, and complex analysis Includes supplementary material: sn.pub/extras

Inhalt

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.