On the construction of frames for Triebel-Lizorkin and Besov spaces
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- by George Kyriazis and Pencho Petrushev PDF
- Proc. Amer. Math. Soc. 134 (2006), 1759-1770 Request permission
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 $\theta (x)=\exp (-|x|^2)$. 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.References
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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
- MR Author ID: 138805
- Email: pencho@math.sc.edu
- 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
- © Copyright 2005 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 134 (2006), 1759-1770
- MSC (2000): Primary 42C15, 46E99, 46B15, 41A63, 94A12
- DOI: https://doi.org/10.1090/S0002-9939-05-08199-2
- MathSciNet review: 2204289