$L^p$ Bernstein estimates and approximation by spherical basis functions

Abstract

The purpose of this paper is to establish L p 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 L p Bessel-potential Sobolev norms of functions in this space in terms of the minimal separation and the L p norm of the function itself. An important step in its proof involves measuring the L p stability of functions in the approximating space in terms of the l p norm of the coefficients involved. As an application of the Bernstein inequality, we derive inverse theorems for SBF approximation in the L P norm. Finally, we give a new characterization of Besov spaces on the n-sphere in terms of spaces of SBFs.