Methods of equivariant topology have been successfully applied in recent years to a variety of problems in geometric combinatorics, especially to those concerning measure equipartitions (generalizations of the ham sandwich-theorem: any $d$ masses in $\mathbb{R}^d$ can be bisected by a single hyperplane) and point partitions of a Tverberg-type (generalizations of Radon's theorem: any $d+2$ points in $\mathbb{R}^d$ can be partitioned into two sets with overlapping convex hulls). Reformulating these problems and their topological reductions in terms of harmonic analysis on finite groups, we will show how a variety of both classical and new partition theorems can be obtained as the annihilation of prescribed Fourier transforms.