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Journal of Function Spaces and Applications
Volume 2, Issue 3, Pages 227-252
http://dx.doi.org/10.1155/2004/792493

Tight wavelet frames in Lebesgue and Sobolev spaces

1Department of Mathematical Sciences, Aalborg University, Fredrik Bajers Vej 7G, 9220 Aalborg East, Denmark
2IRISA-INRIA, Campus de Beaulieu, 35042 Rennes CEDEX, France

Received 1 March 2003

Academic Editor: Hans Feichtinger

Copyright © 2004 Hindawi Publishing Corporation. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

We study tight wavelet frame systems in Lp(d) and prove that such systems (under mild hypotheses) give atomic decompositions of Lp(d) for 1p. We also characterize Lp(d) and Sobolev space norms by the analysis coefficients for the frame. We consider Jackson inequalities for best m-term approximation with the systems in Lp(d) 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.