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Characterizations of function spaces on the sphere using frames


Author: Feng Dai
Journal: Trans. Amer. Math. Soc. 359 (2007), 567-589
MSC (2000): Primary 41A63, 42C15; Secondary 41A17, 46E35
DOI: https://doi.org/10.1090/S0002-9947-06-04030-X
Published electronically: June 13, 2006
MathSciNet review: 2255186
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Abstract: In this paper we introduce a polynomial frame on the unit sphere $ \mathbb{S}^{d-1}$ of $ \mathbb{R}^d$, for which every distribution has a wavelet-type decomposition. More importantly, we prove that many function spaces on the sphere $ \mathbb{S}^{d-1}$, such as $ L^p$, $ H^p$ and Besov spaces, can be characterized in terms of the coefficients in the wavelet decompositions, as in the usual Euclidean case $ \mathbb{R}^d$. We also study a related nonlinear $ m$-term approximation problem on $ \mathbb{S}^{d-1}$. In particular, we prove both a Jackson-type inequality and a Bernstein-type inequality associated to wavelet decompositions, which extend the corresponding results obtained by R. A. DeVore, B. Jawerth and V. Popov (``Compression of wavelet decompositions'', Amer. J. Math. 114 (1992), no. 4, 737-785).


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Additional Information

Feng Dai
Affiliation: Department of Mathematical and Statistical Sciences, CAB 632, University of Alberta, Edmonton, Alberta, Canada T6G 2G1
Email: dfeng@math.ualberta.ca

DOI: https://doi.org/10.1090/S0002-9947-06-04030-X
Keywords: Spherical frames, wavelet decomposition, spherical harmonics, Besov spaces, nonlinear approximation
Received by editor(s): October 20, 2004
Published electronically: June 13, 2006
Additional Notes: The author was supported in part by the NSERC Canada under grant G121211001.
Article copyright: © Copyright 2006 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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