Resumen de The multiresolution structure of pairs of dual wavelet frames for a pair of Sobolev
We study the multiresolution structure of wavelet frames. It is known that the internal structure of almost any nontrivial overcomplete dyadic tight wavelet frame’s underlying multiresolution analysis (Vj )j∈Z is degenerated in L2(R). More precisely, the relation W0 ⊕ V0 = V1, that would hold for wavelet bases, collapses into W0 = V1, where W0 is the closed linear span of the wavelets’ integer shifts.
In the present paper, we extend the latter result in three ways: First and most significantly, we don’t require a tight wavelet frame and verify that the result still holds for a pair of dual wavelet frames. Secondly, we allow for general scaling matrices. Thirdly, the pair of dual wavelet frames is not required to form a frame for L2(Rd) but only for a pair of dual Sobolev spaces (Hs(Rd), H−s(Rd)). Thus, the dual refinable function does not have to be in L2(Rd). Finally, we construct pairs of dual wavelet frames for a pair of dual Sobolev spaces from any pair of multivariate refinable functions.
