The multiresolution structure of pairs of dual wavelet frames for a pair of Sobolev
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M. Ehler [1]
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[1] University of Maryland
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- Localización: Jaen journal on approximation, ISSN 1889-3066, ISSN-e 1989-7251, Vol. 2, Nº. 2, 2010, págs. 193-214
- Idioma: inglés
- Texto completo no disponible (Saber más ...)
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Resumen
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.
