Variational Theory for Interpolation on Spheres

In this paper we consider the problem of developing a variational theory for interpolation by radial basis functions on spheres. The interpolants have the property that they minimise the value of a certain semi-norm, which we construct explicitly. We then go on to investigate forms of the interpolant which are suitable for computation. Our main aim is to derive error bounds for interpolation from scattered data sets, which we do in the final section of the paper. 1 Introduction Surface splines were introduced by Duchon [2], although one should not overlook earlier work of Atteia [1]. Given a function f in C(IR d ) and a set of points a 1 ; : : : ; am 2 IR d , a surface 0 This version produced at 14:43 on August 1, 1997 1 Research partially supported by NATO grant CRG910885 2 Research partially supported by EPSRC grant GR/K79710 3 Research partially supported by EPSRC grant GR/J19481 spline is an interpolant to f on the set fa 1 ; : : : ; am g which minimises a certain (ro..