The high-level error bound for shifted surface spline interpolation

Abstract

Radial function interpolation of scattered data is a frequently used method for multivariate data fitting. One of the most frequently used radial functions is called shifted surface spline, introduced by Dyn, Levin and Rippa in \cite{Dy1} for $R^2$. Then it's extended to $R^n$ for $n\ge 1$. Many articles have studied its properties, as can be seen in \cite{Bu,Du,Dy2,Po,Ri,Yo1,Yo2,Yo3,Yo4}. When dealing with this function, the most commonly used error bounds are the one raised by Wu and Schaback in \cite{WS}, and the one raised by Madych and Nelson in \cite{MN2}. Both are $O(d^l)$ as $d\to 0$, where $l$ is a positive integer and $d$ is the fill-distance. In this paper we present an improved error bound which is $O({\omega }^{1/d})$ as $d\to 0$, where $0<\omega <1$ is a constant which can be accurately calculated.

14 pages, radial basis functions, approximation theory. arXiv admin note: text overlap with arXiv:math/0601158

References in corpus (1)

• On Wu and Schaback's Error Bound

Cited by in corpus (2)

• The Shape Parameter in the Shifted Surface Spline
• A New Error Bound for Shifted Surface Spline Interpolation