A Duchon framework for the sphere

  • Authors:

  • Simon Hubbert;Tanya M. Morton

  • Affiliations:

  • Lehrstuhl für Numerische Mathematik, Justus-Liebig-Universität, Heinrich-Buff-Ring 44, Giessen 35392, Germany;The Mathworks Ltd, Matrix House, Cowley Park, Cambridge CB4 0HH, UK

  • Venue:
  • Journal of Approximation Theory
  • Year:
  • 2004

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Abstract

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.