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 Rd can be bisected by a single hyperplane) and point partitions of a Tverberg-type (generalizations of Radon's theorem: any d+2 points in Rd 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.