Error Bounds for Minimal Energy Interpolatory Spherical Splines

Abstract

The convergence of the minimal energy interpolatory splines on the unit sphere is studied in this paper. An upper bound on the dierence between a sucien tly smooth function and its in- terpolatory spherical spline in the innit y norm is given. The error bound is expressed in terms of a second order spherical Sobolev-type seminorm of the original function. x1. Introduction We study the minimal energy method for scattered data interpolation over the unit sphere. Mainly, we are interested in the convergence rate of the minimal energy method based on spherical spline functions. In the planar setting, such convergence rate was recently determined in (5). It is natural to generalise the convergence result to the setting of spherical splines. This is the purpose of the paper. To be more precise about what we will study in this paper, let us introduce some necessary notation and denition. Let S denote the unit sphere in R 3 . Given two points u; v on S that are not antipodal, the shortest curve connecting them is an arc c of the the great circle through them. Given three points v1; v2 and v3 on S such that v1; v2; v3 form a basis for R 3 , a spherical triangle is a domain bounded by the arcs d v1v2, d v2v3 and d v3v1, which are called edges of the spherical triangle . The points v1; v2 and v3 are called vertices of . Given a set V of points on S we can form a triangulation : a collection of spherical triangles. We will assume that the triangulation is regular in the sense that any two triangles do not intersect each other, or share either a common vertex or a common edge and every edge of is shared by exactly two triangles. Under the regularity assumption of , the number of vertices V = #(V) and the number of triangles T := #(f 2 g) are