𝐿^{𝑝} Bernstein estimates and approximation by spherical basis functions

Mhaskar, H.; Narcowich, F.; Prestin, J.; Ward, J.

doi:10.1090/S0025-5718-09-02322-9

$L^p$ Bernstein estimates and approximation by spherical basis functions
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by H. N. Mhaskar, F. J. Narcowich, J. Prestin and J. D. Ward;

Math. Comp. 79 (2010), 1647-1679

DOI: https://doi.org/10.1090/S0025-5718-09-02322-9

Published electronically: December 2, 2009

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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 $\ell ^p$ 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.

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