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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 $L_p$, $H_p$, 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 $\Phi$ 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 $\Phi$. Typical examples of such $\Phi$'s are the rational function $ \Phi (\cdot) = (1 + \vert\cdot\vert^2)^{-N} $ and the Gaussian function $ \Phi (\cdot) = e^{-\vert\cdot\vert^2}. $ 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

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