Abstract
In this paper we introduce a polynomial frame on the unit sphere S d-1 of R d , for which every distribution has a wavelet-type decomposition. More importantly, we prove that many function spaces on the sphere S d-1 , such as L p , H p and Besov spaces, can be characterized in terms of the coefficients in the wavelet as in the usual Euclidean case R d . We also study a related nonlinear m-term approximation problem on S d-1 . In particular, we prove both a Jackson-type inequality and a Bernstein-type inequality associated to wavelet 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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