New bases for Triebel-Lizorkin and Besov spaces

Authors:
G. Kyriazis and P. Petrushev

Journal:
Trans. Amer. Math. Soc. **354** (2002), 749-776

MSC (1991):
Primary 41A17, 41A20, 42B25, 42C15

DOI:
https://doi.org/10.1090/S0002-9947-01-02916-6

Published electronically:
October 3, 2001

MathSciNet review:
1862566

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Abstract: We give a new method for construction of unconditional bases for general classes of Triebel-Lizorkin and Besov spaces. These include the , , potential, and Sobolev spaces. The main feature of our method is that the character of the basis functions can be prescribed in a very general way. In particular, if is any sufficiently smooth and rapidly decaying function, then our method constructs a basis whose elements are linear combinations of a fixed (small) number of shifts and dilates of the single function . Typical examples of such 's are the rational function and the Gaussian function This paper also shows how the new bases can be utilized in nonlinear approximation.

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Additional Information

**G. Kyriazis**

Affiliation:
Department of Mathematics and Statistics, University of Cyprus, P. O. Box 20537, 1678 Nicosia, Cyprus

Email:
kyriazis@ucy.ac.cy

**P. Petrushev**

Affiliation:
Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

Email:
pencho@math.sc.edu

DOI:
https://doi.org/10.1090/S0002-9947-01-02916-6

Keywords:
Triebel-Lizorkin spaces,
Besov spaces,
unconditional bases,
nonlinear approximation,
wavelets

Received by editor(s):
June 24, 1999

Published electronically:
October 3, 2001

Additional Notes:
This research was supported by ARO Research Contract DAAG55-98-1-0002.

Article copyright:
© Copyright 2001
American Mathematical Society