Spectral approximation orders of radial basis function interpolation on the Sobolev space

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Abstract

In this study, we are mainly interested in error estimates of interpolation, using smooth radial basis functions such as multiquadrics. The current theories of radial basis function interpolation provide optimal error bounds when the basis function φ is smooth and the approximand f is in a certain reproducing kernel Hilbert space Fφ. However, since the space Fφ is very small when the function φ is smooth, the major concern of this paper is to prove approximation orders of interpolation to functions in the Sobolev space. For instance, when φ is a multiquadric, we will observe the error bound o(hk) if the function to be approximated is in the Sobolev space of smoothness order k.

Original languageEnglish
Pages (from-to)946-958
Number of pages13
JournalSIAM Journal on Mathematical Analysis
Volume33
Issue number4
DOIs
StatePublished - 2001

Keywords

  • "Shifted" surface spline
  • Interpolation
  • Multiquadric
  • Positive definite function
  • Radial basis function
  • Sobolev space

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