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.