Advances in Computational Mathematics

, Volume 27, Issue 1, pp 107–124 | Cite as

Approximation power of RBFs and their associated SBFs: a connection

  • Francis J. Narcowich
  • Xinping Sun
  • Joseph D. Ward
Article
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Accepted:

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.

Keywords

interpolation radial basis functions spherical basis functions native space Fourier–Legendre coefficients 

Mathematics subject classifications (2000)

41A2454 41A05 41A63 42C10 
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Copyright information

© Springer 2006

Authors and Affiliations

  • Francis J. Narcowich
    • 1
  • Xinping Sun
    • 2
  • Joseph D. Ward
    • 1
  1. 1.Department of MathematicsTexas A&M UniversityCollege StationUSA
  2. 2.Department of MathematicsSouthwest Missouri State UniversitySpringfieldUSA

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