Multiscale interpolation on the sphere: Convergence rate and inverse theorem

In this paper we study the convergence rate and inverse theorem for spherical multiscale interpolation in Lp   and Sobolev norms. The multiscale interpolation is constructed using a sequence of scaled, compactly supported radial basis functions restricted to the unit sphere SnSn. For the interpolation scheme the problem called “native space barrier” is considered. In addition, a Bernstein type inequality is established to derive an inverse theorem for the multiscale interpolation, and some numerical experiments to illustrate the theoretical results are given.