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

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 $\ell^p$ 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.