On the High-Level Error Bound for Gaussian Interpolation

2006/01/09 by Lin-Tian Luh, Luh, Lin-Tian
Mathematics · Computer Science · #Iterative Methods for Nonlinear Equations #Mathematical functions and polynomials #Matrix Theory and Algorithms

paper · pdf · doi:10.48550/arxiv.math/0601159

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)\Vert f\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→ 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.

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