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Inf-sup condition for spherical polynomials and radial basis functions on spheres

Authors: Ian H. Sloan and Holger Wendland
Journal: Math. Comp. 78 (2009), 1319-1331
MSC (2000): Primary 65D05; Secondary 41A05, 41A29
Published electronically: January 22, 2009
MathSciNet review: 2501052
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Abstract: Interpolation by radial basis functions and interpolation by polynomials are both popular methods for function reconstruction from discrete data given on spheres. Recently, there has been an increasing interest in employing these function families together in hybrid schemes for scattered data modeling and the solution of partial differential equations on spheres. For the theoretical analysis of numerical methods for the associated discretized systems, a so-called inf-sup condition is crucial. In this paper, we derive such an inf-sup condition, and show that the constant in the inf-sup condition is independent of the polynomial degree and of the chosen point set, provided the mesh norm of the point set is sufficiently small. We then use the inf-sup condition to derive a new error analysis for the hybrid interpolation scheme of Sloan and Sommariva.

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Additional Information

Ian H. Sloan
Affiliation: School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia
MR Author ID: 163675
ORCID: 0000-0003-3769-0538

Holger Wendland
Affiliation: Department of Mathematics, University of Sussex, Brighton, BN1 9RF, England
MR Author ID: 602098

Keywords: Inf-sup, sphere, radial basis function, spherical polynomial, hybrid approximation
Received by editor(s): December 17, 2007
Received by editor(s) in revised form: August 6, 2008
Published electronically: January 22, 2009
Article copyright: © Copyright 2009 American Mathematical Society