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

DOI:
https://doi.org/10.1090/S0002-9939-05-08199-2

Published electronically:
December 15, 2005

MathSciNet review:
2204289

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Abstract: We present a general method for construction of frames 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 such as the Gaussian . We also study the boundedness and invertibility of the frame operator on Triebel-Lizorkin and Besov spaces and give necessary and sufficient conditions for the dual system 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:
https://doi.org/10.1090/S0002-9939-05-08199-2

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