Local radial basis function approximation on the sphere

Abstract

In this paper we derive local error estimates for radial basis function interpolation on the unit sphere double-struck S2 ⊂ ℝ3. More precisely, we consider radial basis function interpolation based on data on a (global or local) point set X ⊂ double-struck S2 for functions in the Sobolev space H s(double-struck S2) with norm ∥·∥ s, 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 Hs(double-struck S2). 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 f ∈ Hs(double-struck S2) satisfies supx∈S(z;r) |f (x) - ΛXf(x)|≤ ch(s-1)/2 ∥f∥s, 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. © 2008 Australian Mathematical Society.