Harmonic Analysis on Finite Abelian Groups

We give a discussion of harmonic analysis on finite abelian groups, emphasizing various ways in which the structure of the group is encoded on various spaces of functions, ways in which the Fourier transform detects and preserves these structures. We discuss the major tools, like convolutions and Fourier transforms, along with some fundamental theorems, like the Plancheral, Parseval, Fourier inversion, and Poisson summation formulas, and fundamental heuristic principles, like the uncertainty principle. We originally intended to cover some applications Fourier analysis on finite abelian groups to number theory, and to cover the Fast Fourier transform, but did not get a chance to include these yet.