## New bases for Triebel-Lizorkin and Besov spaces

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- by G. Kyriazis and P. Petrushev
- Trans. Amer. Math. Soc.
**354**(2002), 749-776 - DOI: https://doi.org/10.1090/S0002-9947-01-02916-6
- Published electronically: October 3, 2001
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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 $\Phi (\cdot ) = (1 + |\cdot |^2)^{-N}$ and the Gaussian function $\Phi (\cdot ) = e^{-|\cdot |^2}.$ This paper also shows how the new bases can be utilized in nonlinear approximation.## References

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## Bibliographic 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
- MR Author ID: 138805
- Email: pencho@math.sc.edu
- 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.
- © Copyright 2001 American Mathematical Society
- 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
- MathSciNet review: 1862566