The crucial constants in the exponential-type error estimates 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)| ≤ (Cd) c d f 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 → 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.

The shape parameter in the Gaussian function

“…In (5) the constant C highly depends on β. It's tempting to think that in (5) only C is influenced by β. In fact, f h also changes as β changes.…”

“…However some crucial constants in the error bound had been unknown and considered to be a hard question. Fortunately these constants are thoroughly clarified in [5]. In [5] the author presents a complete and lucid exponential-type error bound for gaussian interpolation.…”

“…Fortunately these constants are thoroughly clarified in [5]. In [5] the author presents a complete and lucid exponential-type error bound for gaussian interpolation. In order to develop useful criteria of the optimal choice of β, we need the following theorem which is cited from [5] directly.…”

“…In [5] the author presents a complete and lucid exponential-type error bound for gaussian interpolation. In order to develop useful criteria of the optimal choice of β, we need the following theorem which is cited from [5] directly. Theorem 2.2 Let h(x) = e −β|x| 2 , β > 0, be the gaussian function in R n .…”

“…This is the key to understanding this seemingly complicated theorem. In (5) the constant C highly depends on β. It's tempting to think that in (5) only C is influenced by β.…”