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Tight wavelet frames in Lebesgue and Sobolev spaces

Lasse Borup 1 Rémi Gribonval 2 Morten Nielsen 1 
2 METISS - Speech and sound data modeling and processing
IRISA - Institut de Recherche en Informatique et Systèmes Aléatoires, Inria Rennes – Bretagne Atlantique
Abstract : We study tight wavelet frame systems in Lp(Rd), and prove that such systems (under mild hypotheses) give atomic decompositions of L p(Rd) for 1 < p < +infty. We also characterize Lp(Rd) and Sobolev space norms by the analysis coefficients for the frame. We consider Jackson inequalities for best m-term approximation with the systems in L p(Rd ) and prove that such inequalities exist. Moreover, it is proved that the approximation rate given by the Jackson inequality can be realized by thresholding the frame coefficients. Finally, we show that in certain restricted cases, the approximation spaces, for best m-term approximation, associated with tight wavelet frames can be characterized in terms of (essentially) Besov spaces.
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Lasse Borup, Rémi Gribonval, Morten Nielsen. Tight wavelet frames in Lebesgue and Sobolev spaces. Journal of function spaces, Hindawi, 2004, 2 (3), pp.227--252. ⟨inria-00567343⟩

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