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 in...

Baer characterized capable finite abelian groups (a group is capable if it is isomorphic to the quotient of some group by its center) by a condition on the size of the factors in the invariant factor decomposition (the group must be noncyclic and the top two invariant factors must be equal). We provide a different characterization, given in terms of a condition on the lattice of subgroups. Name...

Gabor frames for signals over finite Abelian groups, generated by an arbitrary lattice within the finite time-frequency plane, are the central topic of this paper. Our generic approach covers both multi-dimensional signals as well as non-separable lattices, and in fact the multi-window case as well. Our generic approach includes most of the fundamental facts about Gabor expansions of finite sig...

For positive integers n and k, the anti-van der Waerden number of Zn, denoted by aw(Zn, k), is the minimum number of colors needed to color the elements of the cyclic group of order n and guarantee there is a rainbow arithmetic progression of length k. Butler et al. showed a reduction formula for aw(Zn, 3) = 3 in terms of the prime divisors of n. In this paper, we analagously define the anti-va...