On the Existence of Wavelets for Non-expansive Dilation Matrices

Sets which simultaneously tile Rn by applying powers of an invertible matrix and translations by a lattice are studied. Diagonal matrices A for which there exist sets that tile by powers of A and by integer translations are characterized. A sufficient condition and a necessary condition on the dilations and translations for the existence of such sets are also given. These conditions depend in an essential way on the interplay between the eigenvectors of the dilation matrix and the translation lattice rather than the usual dependence on the eigenvalues. For example, it is shown that for any values |a| > 1 > |b|, there is a (2×2) matrix A with eigenvalues a and b for which such a set exists, and a matrix A′ with eigenvalues a and b for which no such set exists. Finally, these results are related to the existence of wavelets for non-expansive dilations.