Approximation power of RBFs and their associated SBFs: a connection |
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Authors: | Francis J Narcowich Xinping Sun Joseph D Ward |
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Institution: | (1) Department of Mathematics, Texas A&M University, College Station, TX 77843-3368, USA;(2) Department of Mathematics, Southwest Missouri State University, Springfield, MO 65804, USA |
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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.
Joseph D. Ward: Francis J. Narcowich: Research supported by grant DMS-0204449 from the National Science Foundation. |
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Keywords: | interpolation radial basis functions spherical basis functions native space Fourier– Legendre coefficients |
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