# On Power Functions and Error Estimates for Radial Basis Function Interpolation

@article{Light1998OnPF, title={On Power Functions and Error Estimates for Radial Basis Function Interpolation}, author={W. A. Light and Henry Wayne}, journal={Journal of Approximation Theory}, year={1998}, volume={92}, pages={245-266} }

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…

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## References

SHOWING 1-10 OF 11 REFERENCES

### Comparison of Radial Basis Function Interpolants

- Mathematics
- 1993

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

- Mathematics
- 1979

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

- Mathematics
- 1988

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

- Mathematics
- 1994

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…

### Sur l’erreur d’interpolation des fonctions de plusieurs variables par les $D^m$-splines

- Mathematics
- 1978

### 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