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LOCAL RADIAL BASIS FUNCTION APPROXIMATION ON THE SPHERE

Part of: Approximations and expansions Numerical approximation and computational geometry

Published online by Cambridge University Press:  01 April 2008

KERSTIN HESSE  and
Q. T. LE GIA
KERSTIN HESSE
Affiliation:
Department of Mathematics, Mantell Building, University of Sussex, Falmer, Brighton BN1 9RF, UK (email: [email protected])
Q. T. LE GIA
Affiliation:
School of Mathematics and Statistics, The University of New South Wales, Sydney NSW 2052, Australia (email: [email protected])
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Abstract

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In this paper we derive local error estimates for radial basis function interpolation on the unit sphere . More precisely, we consider radial basis function interpolation based on data on a (global or local) point set for functions in the Sobolev space with norm , where s>1. The zonal positive definite continuous kernel ϕ, which defines the radial basis function, is chosen such that its native space can be identified with . Under these assumptions we derive a local estimate for the uniform error on a spherical cap S(z;r): the radial basis function interpolant ΛXf of satisfies , where h=hX,S(z;r) is the local mesh norm of the point set X with respect to the spherical cap S(z;r). Our proof is intrinsic to the sphere, and makes use of the Videnskii inequality. A numerical test illustrates the theoretical result.

Keywords

error estimateinterpolationlocal approximationmesh normnative spacepositive definite kernelradial basis functionreproducing kernelSobolev spacespherespherical capVidenskii inequality

MSC classification

Secondary: 41A05: Interpolation 41A15: Spline approximation 41A25: Rate of convergence, degree of approximation 65D05: Interpolation 65D07: Splines
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 77 , Issue 2 , April 2008 , pp. 197 - 224
Copyright
Copyright © 2008 Australian Mathematical Society

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