Copyright © 2010 Bao-Huai Sheng and Dao-Hong Xiang. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
It is known that in the field of learning theory based on reproducing kernel
Hilbert spaces the upper bounds estimate for a -functional is needed. In the present paper, the upper bounds for the -functional on the unit sphere are estimated with spherical harmonics approximation. The results show that convergence
rate of the -functional depends upon the smoothness of both the approximated
function and the reproducing kernels.