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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Characterizations of function spaces on the sphere using frames
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by Feng Dai PDF
Trans. Amer. Math. Soc. 359 (2007), 567-589 Request permission

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
  • 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.
  • © Copyright 2006 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2255186