Approximation power of RBFs and their associated SBFs: a connection

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 R n 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 R n+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 (R n+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.

Order-preserving derivative approximation with periodic radial basis functions

“…Estimates on k β (n) were carried out in [17] to show that k β (n) ∼ n −2β+1 for large n. We repeat some of those arguments here to show that this kernel satisfies Proposition 2 with s = −2β + 1. First, for t > 1 the following expansion is valid:…”

“…3 Below we consider classes of these functions, and show that the associated zonal PBFs satisfy (14). Important to us will be the interplay between the Fourier transform of the kernel Φ in R d and Fourier coefficients of the associated PBF φ on the circle (see [17,26]). …”

“…When β / ∈ 2N, the power kernel Φ β = C β r β is conditionally positive definite of order q = β 2 on R d , and in the other case we get the thin plate spline Φ β (r) = C β r β log(r), which is CPD of order q = β/2 + 1 [22,Corollary 8.18]. Asymptotic formulas for the spherical Fourier coefficients of these kernels were worked out in [17]. In particular, when one of these RBFs is restricted from R 2 to the circle, using [17,Equation (4.12)] (with the substitutions n = 2, 2s − 2 = β) we immediately get…”

“…Asymptotic formulas for the spherical Fourier coefficients of these kernels were worked out in [17]. In particular, when one of these RBFs is restricted from R 2 to the circle, using [17,Equation (4.12)] (with the substitutions n = 2, 2s − 2 = β) we immediately get…”

Stability and Error Estimates for Vector Field Interpolation and Decomposition on the Sphere with RBFs

“…We claim that this ensures that our native spaces are Sobolev. Indeed, if ψ is an SBF obtained by restricting φ to the sphere, by [26,Proposition 4…”