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Approximation power of RBFs and their associated SBFs: a connection

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Abstract

Error estimates for scattered data interpolation by “shifts” of a conditionally positive definite function (CPD) for target functions in its native space, which is its associated reproducing kernel Hilbert space (RKHS), have been known for a long time. Regardless of the underlying manifold, for example ℝn or S n, these error estimates are determined by the rate of decay of the Fourier transform (or Fourier series) of the CPD. This paper deals with the restriction of radial basis functions (RBFs), which are radial CPD functions on ℝn+1, to the unit sphere S n. In the paper, we first strengthen a result derived by two of us concerning an explicit representation of the Fourier–Legendre coefficients of the restriction in terms of the Fourier transform of the RBF. In addition, for RBFs that are related to completely monotonic functions, we derive a new integral representation for these coefficients in terms of the measure generating the completely monotonic function. These representations are then utilized to show that if an RBF has a native space equivalent to a Sobolev space H s(ℝn+1), then the restriction to S n has a native space equivalent to H s−1/2(S n). In addition, they are used to recover the asymptotic behavior of such coefficients for a wide variety of RBFs. Some of these were known earlier.

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Authors and Affiliations

  1. Department of Mathematics, Texas A&M University, College Station, TX, 77843-3368, USA

    Francis J. Narcowich & Joseph D. Ward

  2. Department of Mathematics, Southwest Missouri State University, Springfield, MO, 65804, USA

    Xinping Sun

Authors
  1. Francis J. Narcowich
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  2. Xinping Sun
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  3. Joseph D. Ward
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Additional information

Communicated by R. Schaback

Joseph D. Ward: Francis J. Narcowich: Research supported by grant DMS-0204449 from the National Science Foundation.

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Narcowich, F.J., Sun, X. & Ward, J.D. Approximation power of RBFs and their associated SBFs: a connection. Adv Comput Math 27, 107–124 (2007). https://doi.org/10.1007/s10444-005-7506-1

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  • DOI: https://doi.org/10.1007/s10444-005-7506-1

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