ar X iv : 0 81 0 . 50 75 v 1 [ m at h . FA ] 2 8 O ct 2 00 8 L p Bernstein Estimates and Approximation by Spherical Basis Functions ∗ †

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