On the High-Level Error Bound for Gaussian Interpolation

Abstract

It's well-known that there is a very powerful error bound for Gaussians put forward by Madych and Nelson in 1992. It's of the form$% | f(x)-s(x)| \leq (Cd)^{\frac{c}{d}}\left\Vert f\right\Vert_{h}$ where $C,c$ are constants, $h$ is the Gaussian function, $% s$ is the interpolating function, and d is called fill distance which, roughly speaking, measures the spacing of the points at which interpolation occurs. This error bound gets small very fast as $d\to 0$. The constants $C$ and $c$ are very sensitive. A slight change of them will result in a huge change of the error bound. The number $c$ can be calculated as shown in [9]. However, $C$ cannot be calculated, or even approximated. This is a famous question in the theory of radial basis functions. The purpose of this paper is to answer this question.