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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the construction of frames for Triebel-Lizorkin and Besov spaces


Authors: George Kyriazis and Pencho Petrushev
Journal: Proc. Amer. Math. Soc. 134 (2006), 1759-1770
MSC (2000): Primary 42C15, 46E99, 46B15, 41A63, 94A12
Published electronically: December 15, 2005
MathSciNet review: 2204289
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Abstract: We present a general method for construction of frames $ \{\psi_I\}_{I\in \mathcal{D}}$ for Triebel-Lizorkin and Besov spaces, whose nature can be prescribed. In particular, our method allows for constructing frames consisting of rational functions or more general functions which are linear combinations of a fixed (small) number of shifts and dilates of a single smooth and rapidly decaying function $ \theta$ such as the Gaussian $ \theta(x)=\exp(-\vert x\vert^2)$. We also study the boundedness and invertibility of the frame operator $ Sf=\sum_{I\in\mathcal{D}} \langle{f,\psi_I}\rangle\psi_I$ on Triebel-Lizorkin and Besov spaces and give necessary and sufficient conditions for the dual system $ \{S^{-1}\psi\}_{I\in\mathcal{D}}$ to be a frame as well.


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Additional Information

George Kyriazis
Affiliation: Department of Mathematics and Statistics, University of Cyprus, 1678 Nicosia, Cyprus
Email: kyriazis@ucy.ac.cy

Pencho Petrushev
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
Email: pencho@math.sc.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-05-08199-2
PII: S 0002-9939(05)08199-2
Received by editor(s): July 6, 2004
Received by editor(s) in revised form: January 24, 2005
Published electronically: December 15, 2005
Additional Notes: The second author was supported by the National Science Foundation Grant DMS-0200665.
Communicated by: Andreas Seeger
Article copyright: © Copyright 2005 American Mathematical Society