On the High-Level Error Bound for Gaussian Interpolation

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.