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The purpose of this paper is to establish L^p error estimates, a Bernsteininequality, and inverse theorems for approximation by a space comprisingspherical basis functions located at scattered sites on the unit n-sphere. Inparticular, the Bernstein inequality estimates L^p Bessel-potential Sobolevnorms of functions in this space in terms of the minimal separation and the L^pnorm of the function itself. An important step in its proof involves measuringthe L^p stability of functions in the approximating space in terms of the l^pnorm of the coefficients involved. As an application of the Bernsteininequality, 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 interms of spaces of SBFs.

L^p Bernstein estimates and approximation by spherical basis functions