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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.

##### MSC:
 42C15 General harmonic expansions, frames
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