L Bernstein Estimates and Approximation by Spherical Basis Functions

The purpose of this paper is to establish Lp error estimates, a Bernstein inequality, and inverse theorems for approximation by a space comprising spherical basis functions located at scattered sites on the unit n-sphere. In particular, the Bernstein inequality estimates Lp Bessel-potential Sobolev norms of functions in this space in terms of the minimal separation and the Lp norm of the function itself. An important step in its proof involves measuring the Lp stability of functions in the approximating space in terms of the p norm of the coefficients involved. As an application of the Bernstein inequality, we derive inverse theorems for SBF approximation in the LP norm. Finally, we give a new characterization of Besov spaces on the n-sphere in terms of spaces of SBFs.