Bessel sequences and affine frames

Abstract

We formulate several criteria on square-integrable functions in terms of certain smoothness and rate of decay that guarantee that these functions generate Bessel sequences. As a consequence, we show that one can obtain affine frames by arbitrarily oversampling any of the well-known wavelets. On the other hand, we also show that for any integer scaling parameter a, oversampling of any affine frame by an integer factor n preserves the frame bounds, provided that n and a are relatively prime; consequently, for tight frames, and more generally frames with duals, the frame series representations remain valid for such oversampling. A corresponding oversampling theorem for Weyl-Heisenberg frames is also established. © 1993 Academic Press Inc.