Interpolation by polynomials and radial basis functions on spheres

The paper obtains error estimates for approximation by radial basis functio ns on the sphere. The approximations are generated by interpolation at scat tered points on the sphere. The estimate is given in terms of the appropria te power of the fill distance for the interpolation points, in a similar ma nner to the estimates for interpolation in Euclidean space. A fundamental i ngredient of our work is an estimate for the Lebesgue constant associated w ith certain interpolation processes by spherical harmonics. These interpola tion processes take place in "spherical caps" whose size is controlled by t he fill distance, and the important aim is to keep the relevant Lebesgue co nstant bounded. This result seems to us to be of independent interest.