Double coset decompositions and computational harmonic analysis on groups

In this paper we introduce new techniques for the efficient computation of a Fourier transform on a finite group. We use the decomposition of a group into double cosets and a graph theoretic indexing scheme to derive algorith ms that generalize the Cooley-Tukey FFT to arbitrary finite groups. We appl y our general results to special linear groups and low rank symmetric group s, and obtain new efficient algorithms for harmonic analysis on these class es of groups, as well as the two-sphere.