In his fundamental paper (RAIRO Anal. Numer. 12 (1978) 325) Duchon presented a strategy for analysing the accuracy of surface spline interpolants to sufficiently smooth target functions. In the mid-1990s Duchon's strategy was revisited by Light and Wayne (J. Approx. Theory 92 (1992) 245) and Wendland (in: A. Le Méhauté, C. Rabut, L.L. Schumaker (Eds.), Surface Fitting and Multiresolution Methods, Vanderbilt Univ. Press, Nashville, 1997, pp. 337–344), who successfully used it to provide useful error estimates for radial basis function interpolation in Euclidean space. A relatively new and closely related area of interest is to investigate how well radial basis functions interpolate data which are restricted to the surface of a unit sphere. In this paper we present a modified version Duchon's strategy for the sphere; this is used in our follow up paper (Lp-error estimates for radial basis function interpolation on the sphere, preprint, 2002) to provide new Lp error estimates (p[1,∞]) for radial basis function interpolation on the sphere.
