Direct and Inverse Sobolev Error Estimates for Scattered Data Interpolation via Spherical Basis Functions

Authors:
Francis J. Narcowich;Xingping Sun;Joseph D. Ward;Holger Wendland
Affiliations:
Department of Mathematics, Texas A&M University, College Station, TX 77843, USA;Department of Mathematics, Southwest Missouri State University, Springfield, MO 65804, USA;Department of Mathematics, Texas A&M University, College Station, TX 77843, USA;Universitat Gottingen, Lotzestrasse 16-18, D-37083, Gottingen, Germany
Venue:
Foundations of Computational Mathematics
Year:
2007

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Abstract

The purpose of this paper is to get error estimates for spherical basis function (SBF) interpolation and approximation for target functions in Sobolev spaces less smooth than the SBFs, and to show that the rates achieved are, in a sense, best possible. In addition, we establish a Bernstein-type theorem, where the smallest separation between data sites plays the role of a Nyquist frequency. We then use these Berstein-type estimates to derive inverse estimates for interpolation via SBFs.