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Transactions of the American Mathematical Society

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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
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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  • [BR] C. de Boor, A. Ron, Fourier analysis of the approximation power of principal shift invariant spaces Constr. Approx., 8 (1992), 427-462. MR 94c:41023
  • [Da] I. Daubechies Ten Lectures on Wavelets, SIAM, Philadelphia, 1992. MR 93e:42045
  • [De] R. DeVore, Nonlinear approximation Acta Numer. (1998), 51-150. MR 2001a:41034
  • [DJKP] D. Donoho, I. Johnstone, G. Kerkyacharian and D. Picard, Wavelet shrinkage asymptotia J. Royal Stat. Soc. Ser. B. 57 (1996), 301-369. MR 96g:62068
  • [DL] R. DeVore and G. G. Lorentz, Constructive Approximation, Springer, New York, 1993. MR 95f:41001
  • [Do] D. Donoho, Unconditional bases are optimal bases for data compression and for statistical estimation Appl. Comp. Harm. Anal., 1 (1993), 100-115. MR 94j:94011
  • [FJ1] M. Frazier, B. Jawerth, Decomposition of Besov Spaces Indiana Univ. Math. J., 34 (1985), 777-799. MR 87h:46083
  • [FJ2] M. Frazier, B. Jawerth, A discrete transform and decompositions of distribution, J. of Functional Analysis 93 (1990), 34-170. MR 92a:46042
  • [FJW] M. Frazier, B. Jawerth, and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS 79 (1991), AMS. MR 92m:42021
  • [FS] C. Fefferman and E. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. MR 44:2026
  • [JM] R.-Q. Jia, C. Micchelli, Using the refinement equations for the construction of pre-wavelets II: powers of two, Curves and Surfaces, Academic Press, 1991. MR 93e:65024
  • [K1] G. Kyriazis, Wavelet coefficients measuring smoothness in $H_p({\mathbb R}^d)$, Appl. Comp. Harm. Anal. 3 (1996), 100-119. MR 97h:42016
  • [K2] G. Kyriazis, Approximation of Distribution Spaces by Means of Kernel Operators J. Four. Anal. Appl. 2 (1996), 261-286. MR 97d:46041
  • [K3] G. Kyriazis, Unconditional bases of function spaces, preprint.
  • [M] Y. Meyer, Ondelettes et Opérateurs I: Ondelettes, Hermann Éditeurs, 1990. MR 93i:42002
  • [Pee] J. Peetre, New thought on Besov spaces, Duke Univ. Math. Series. Durham, N.C., 1993. MR 57:1108
  • [Pek] A. Pekarskii, Chebyshev rational approximation in a disk, on a circle, and on a segment, Mat. Sb., 133 (175) (1987), 86-102, (English translation in Math. USSR Sbornik, 61 (1988), 87-102). MR 89e:30060
  • [Pet] P. Petrushev, Bases consisting of rational functions of uniformly bounded degrees or more general functions J. Funct. Anal., 174 (2000), 18-75. CMP 2000:13
  • [PP] P. Petrushev, V. Popov, Rational approximation of real functions, Cambridge University Press, Cambridge, 1987. MR 89i:41022
  • [T] H. Triebel, Theory of Function Spaces, Birkhäuser, 1983. MR 86j:46026

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Additional Information

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

P. Petrushev
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

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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