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Constructive Approximation

, Volume 24, Issue 2, pp 175–186 | Cite as

Sobolev Error Estimates and a Bernstein Inequality for Scattered Data Interpolation via Radial Basis Functions

  • Francis J. Narcowich
  • Joseph D. Ward
  • Holger Wendland
Article

Abstract

Error estimates for scattered-data interpolation via radial basis functions (RBFs) for target functions in the associated reproducing kernel Hilbert space (RKHS) have been known for a long time. Recently, these estimates have been extended to apply to certain classes of target functions generating the data which are outside the associated RKHS. However, these classes of functions still were not "large" enough to be applicable to a number of practical situations. In this paper we obtain Sobolev-type error estimates on compact regions of Rn when the RBFs have Fourier transforms that decay algebraically. In addition, we derive a Bernstein inequality for spaces of finite shifts of an RBF in terms of the minimal separation parameter.

Radial basis functions Band-limited functions Error estimates Scattered data Bernstein inequality 

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Copyright information

© Springer 2006

Authors and Affiliations

  • Francis J. Narcowich
    • 1
  • Joseph D. Ward
    • 1
  • Holger Wendland
    • 2
  1. 1.Department of Mathematics, Texas A&M University, College Station, TX 77843USA
  2. 2.Universitat Gottingen, Lotzestrasse 16-18, D-37083, GottingenGermany

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