Interpolation by Polynomials and Radial Basis Functions on Spheres

Abstract. The paper obtains error estimates for approximation by radial basis functions on the sphere. The approximations are generated by interpolation at scattered points on the sphere. The estimate is given in terms of the appropriate power of the fill distance for the interpolation points, in a similar manner to the estimates for interpolation in Euclidean space. A fundamental ingredient of our work is an estimate for the Lebesgue constant associated with certain interpolation processes by spherical harmonics. These interpolation processes take place in ``spherical caps'' whose size is controlled by the fill distance, and the important aim is to keep the relevant Lebesgue constant bounded. This result seems to us to be of independent interest.