On Power Functions and Error Estimates for Radial Basis Function Interpolation

  title={On Power Functions and Error Estimates for Radial Basis Function Interpolation},
  author={W. A. Light and Henry Wayne},
  journal={Journal of Approximation Theory},
This paper discusses approximation errors for interpolation in a variational setting which may be obtained from the analysis given by Golomb and Weinberger. We show how this analysis may be used to derive the power function estimate of the error as introduced by Schaback and Powell. A simple error tool for the power function is presented, which is similar to one appearing in the work of Madych and Nelson. It is then shown that this tool is adequate to reproducing the original error analysis… 

An error analysis for radial basis function interpolation

An error analysis is developed which works well when the Fourier transform of ϕ has a pole of order 2m at the origin and a zero at ∞ of order2κ.

Spaces of distributions, interpolation by translates of a basis function and error estimates

This paper focuses on developing a wide variety of spaces for which a variational theory is available and shows how the theory leads to efficient ways of calculating the interpolant and to new error estimates.

Error estimates for interpolation of rough and smooth functions using radial basis functions

In this thesis we are concerned with the approximation of functions by radial basis function interpolants. There is a plethora of results about the asymptotic behaviour of the error between

Spaces of Distributions and Interpolation by Translates of a Basis Function

Interpolation with translates of a basis function is a common process in approximation theory. One starts with a single function (the basis function) and a set of interpolation points. The most

Lp-error estimates for "shifted'' surface spline interpolation on Sobolev space

The purpose of this study is to discuss the Lp-approximation order (1 ≤ p ≤ ∞) of interpolation to functions in the Sobolev space Wpk(Ω) with k > max(0,d/2 - d/p).

Inverse and saturation theorems for radial basis function interpolation

It is shown that a function that can be approximated sufficiently fast must belong to the native space of the basis function in use, and certain saturation theorems are given in case of thin plate spline interpolation.

Optimal Approximation Orders in L P for Radial Basis Functions Holger Wendland

Error estimates for radial basis function interpolation are usually based on the concept of native Hilbert spaces. We investigate how good the well known L p-error estimates are by giving lower

Radial basis functions under tension

Refined Error Estimates for Radial Basis Function Interpolation

We discuss new and refined error estimates for radial-function scattered-data interpolants and their derivatives. These estimates hold on Rd, the d-torus, and the 2-sphere. We employ a new technique,

Numerical differentiation by radial basis functions approximation

A new com-putational algorithm for numerical differentiation under an a priori and an a posteriori choice rules for the regularization parameter is developed based on radial basis functions approximation.



Comparison of Radial Basis Function Interpolants

This paper compares radial basis function interpolants on diier-ent spaces. The spaces are generated by other radial basis functions, and comparison is done via an explicit representation of the norm

Multivariate interpolation at arbitrary points made simple

The concrete method of ‘surface spline interpolation’ is closely connected with the classical problem of minimizing a Sobolev seminorm under interpolatory constraints; the intrinsic structure of

Multivariate interpolation and condi-tionally positive definite functions

We continue an earlier study of certain spaces that provide a variational framework for multivariate interpolation. Using the Fourier transform to analyze these spaces, we obtain error estimates of

The uniform convergence of thin plate spline interpolation in two dimensions

Summary. Let $f$ be a function from ${\cal R}^2$ to ${\cal R}$ that has square integrable second derivatives and let $s$ be the thin plate spline interpolant to $f$ at the points $\{ \underline

Mathematical Analysis

Optimal approximation and error bounds, in On numerical approximation

  • Optimal approximation and error bounds, in On numerical approximation
  • 1959

Multivariate interpolation and conditionally positive deenite functions II

  • Math. Comp
  • 1990