  In this paper we introduce a polynomial frame on the unit sphere $\sph$ of
$\mathbb{R}^d$, for which every distribution has a wavelet-type decomposition.
More importantly, we prove that many function spaces on the sphere $\sph$, 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 $\sph$. 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'', {\it Amer. J. Math.} {\bf 114}
(1992), no. 4, 737--785).
