Characterizations of function spaces on the sphere using frames

Dai, Feng

doi:10.1090/S0002-9947-06-04030-X

Characterizations of function spaces on the sphere using frames
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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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