Weighted Approximation of Functions on the Unit Sphere

The direct and inverse theorems are established for the best approximation in the weighted L^p space on the unit sphere of R^d+1, in which the weight functions are invariant under finite reflection groups. The theorems are stated using a modulus of smoothness of higher order, which is proved to be equivalent to a K-functional defined using the power of the spherical h-Laplacian. Furthermore, similar results are also established for weighted approximation on the unit ball and on the simplex of R^d.