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arXiv · 0810.5075

L^p Bernstein estimates and approximation by spherical basis functions

H. N. Mhaskar·F. J. Narcowich·J. Prestin·J. D. Ward
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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.

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H. N. Mhaskar, F. J. Narcowich, J. Prestin, J. D. Ward. 2008-10-28. L^p Bernstein estimates and approximation by spherical basis functions. https://arxiv.org/abs/0810.5075

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