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${L}^{p}$ Bernstein estimates and approximation by spherical basis functions. (English) Zbl 1200.41019
Summary: 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.
##### MSC:
 41A27 Inverse theorems in approximation theory
##### Keywords:
Spherical basis functions; Bernstein estimates