We generalize the Second Oversampling Theorem for wavelet frames and dual wavelet frames from the setting of integer dilations to real dilations. We also study the relationship between dilation matrix oversampling of semi-orthogonal Parseval wavelet frames and the additional shift invariance gain of the core subspace. Oversampling of wavelet frames has been a subject of extensive study by several authors dating back to the early 1990s. The first oversampling results are due to Chui and Shi [16, 17], who proved that oversampling by odd factors preserves tightness of dyadic affine frames. This is now the central result of the subject known as the Second Oversampling Theorem. Its higher dimensional generalizations to integer matrix dilations were studied by Chui and Shi [18], Johnson [27], Laugesen [30], and Ron and Shen [31]. In particular, these authors introduced (in several equivalent forms) the class of oversampling matrices ‘relatively prime’ to a fixed dilation A and they established several oversampling results for (not necessarily tight) affine frames. Dutkay and Jorgensen [23] shed a new light on these results by showing that oversampling of orthonormal (or frame) wavelets by such matrices leads to orthonormal (or frame) superwavelets, respectively. Chui and Sun [22] have completed the understanding of the case of integer dilations by showing that the class of ‘relatively prime’ matrices is optimal for the Second Oversampling Theorem; that is, if an oversampling matrix falls out of this class, then the oversampling does not preserve a tight frame property in general. However, it is possible to give a characterization of oversampling matrices preserving tightness once affine frame generators are chosen. These
In light of the results of Chui and Sun @cite_23 , it is not surprising that, for integer dilations, all of the previously studied conditions on tightness preserving lattices @math are equivalent to our newly introduced condition . We state these conditions in the proposition below.
@cite_23
2008731398
Abstract Let M be a dilation matrix, Ψ a finite family of L 2 -functions, and P the collection of all nonsingular matrices P such that M, P, and P M P −1 have integer entries. The objective of this paper is two-fold. First, for each P in P , we characterize all tight affine frames X ( Ψ , M ) generated by Ψ such that the over-sampled affine systems X P ( Ψ , M ) relative to the “over-sampling rate” P remain to be tight frames. Second, we characterize all over-sampling rates P ∈ P , such that the over-sampled affine systems X P ( Ψ , M ) are tight frames whenever the affine system X ( Ψ , M ) is a tight frame. Our second result therefore provides a general and precise formulation of the second over-sampling theorem for tight affine frames.