Crystallographic fast Fourier transforms

This paper presents methods for incorporating crystallographic symmetry properties into complex Fourier transforms in a form particularly well suited for use with the Cooley-Tukey fast Fourier transform algorithm. The crystallographic transforms are expressed in terms of a small number of one-dimensional special cases. The algebra presented here has been used to write computer programs for both Fourier syntheses and Fourier inversions. Even for some quite large problems (7000 structure factors and 149000 grid points in the asymmetric unit) the rate-limiting step is output of the answers.