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Error estimates for scattered data interpolation on spheres. (English) Zbl 1042.41003

Summary: We study Sobolev type estimates for the approximation order resulting from using strictly positive definite kernels to do interpolation on the \(n\)-sphere. The interpolation knots are scattered. Our approach partly follows the general theory of Golomb and Weinberger and related estimates. These error estimates are then based on series expansions of smooth functions in terms of spherical harmonics. The Markov inequality for spherical harmonics is essential to our analysis and is used in order to find lower bounds for certain sampling operators on spaces of spherical harmonics.

MSC:

41A05 Interpolation in approximation theory
41A25 Rate of convergence, degree of approximation
41A30 Approximation by other special function classes
41A63 Multidimensional problems
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