An Equivariant Fast Fourier Transform Algorithm

This paper presents a generalization of the Cooley-Tukey fast Fourier transform algorithm that respects group symmetries. The algorithm, when applied to a function invariant under a group of symmetries, fully exploits these symmetries to reduce both the number of arithmetic operations and the amount of memory used. The symmetries accommodated by the algorithm include all of the crystallographic groups. These groups arise in crystallographic structure analysis, which was the motivating application for the algorithm presented in this paper.