散乱数据的径向函数插值是多元数据拟合中常用的方法。最常用的径向函数之一称为平移表面样条，由Dyn、Levin和Rippa在文献\cite{Dy1}中针对$R^{2}$引入，随后推广到$n\geq 1$的$R^{n}$。许多文章研究了其性质，见\cite{Bu,Du,Dy2,Po,Ri,Yo1,Yo2,Yo3,Yo4}。处理该函数时，最常用的误差界是Wu和Schaback在\cite{WS}中提出的，以及Madych和Nelson在\cite{MN2}中提出的。两者均为$O(d^{l})$（当$d o 0$时），其中$l$为正整数，$d$为填充距离。本文提出一个改进的误差界，当$d o 0$时为$O(ω^{1/d})$，其中$0<ω<1$是一个可精确计算的常数。
英文摘要：
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 $n\geq 1$. 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 $O(ω^{1/d})$ as $d\to 0$, where $0<ω<1$ is a constant which can be accurately calculated.
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