A generalization of multi-dimensional wavelet theory is introduced in which the usual lattice of translational shifts is replaced by a discrete subgroup of the group of affine, area preserving, transformations of Euclidean space. The dilation matrix must now be compatible with the group of shifts. An existence theorem for a multiwavelet in the presence of a multiresolution analysis is established and examples are given to illustrate the theory with two dimensional crystal symmetry groups as shifts.

Abstract. The interplay between the Hamilton-Jacobi theory of orthogonal separation of variables and the theory of group actions is investigated based on concrete examples.