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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

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 | References | Similar Articles | Additional Information

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
MR Author ID: 660750
Email: dfeng@math.ualberta.ca

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.