  We introduce a fast Fourier transform on regular d-dimensional lattices. We
investigate properties of congruence class representants, i.e. their ordering,
to classify directions and derive a Cooley-Tukey-Algorithm. Despite the fast
Fourier techniques itself, there is also the advantage of this transform to be
parallelized efficiently, yielding faster versions than the one-dimensional
Fourier transform. These properties of the lattice can further be used to
perform a fast multivariate wavelet decomposition, where the wavelets are given
as trigonometric polynomials. Furthermore the preferred directions of the
decomposition itself can be characterised.
