Characterizations of function spaces on the sphere using frames

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).