Approximation power of RBFs and their associated SBFs: A connection
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.
Fourier-legendre coefficients, Interpolation, Native space, Radial basis functions, Spherical basis functions
Narcowich, Francis J., Xinping Sun, and Joseph D. Ward. "Approximation power of RBFs and their associated SBFs: a connection." Advances in Computational Mathematics 27, no. 1 (2007): 107-124.
Advances in Computational Mathematics