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
The single underlying method of averaging the wavelet functional over translates yields first a new completeness criterion for orthonormal wavelet systems, and then a unified treatment of known results on characterization of wavelets on the Fourier transform side, on preservation of frame bounds by oversampling, and on equivalence of affine and quasiaffine frames. The method applies to multiwavelet systems in all dimensions, to dilation matrices that are in some cases not expanding, and to dual frame pairs.