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.
Cite
@article{arxiv.0810.5075,
title = {L^p Bernstein estimates and approximation by spherical basis functions},
author = {H. N. Mhaskar and F. J. Narcowich and J. Prestin and J. D. Ward},
journal= {arXiv preprint arXiv:0810.5075},
year = {2008}
}