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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
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