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Bessel sequences and affine frames. (English) Zbl 0788.42011
Summary: 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.

42C15 General harmonic expansions, frames
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