Characterizations of function spaces on the sphere using frames
Abstract
In this paper we introduce a polynomial frame on the unit sphere $\sph$ of
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
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).