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.

A generalized filter construction is used to build non-MRA Parseval wavelets for dilation by 2 in L2(R2). These examples have the same multiplicity functions as wavelet sets, yet can be made to be Cr with Cr Fourier transform for any fixed positive integer r.