The shape parameter in the Gaussian function II

This is a continuation of our former study, Luh [1], of the shape parameter β contained in Gaussian e − β | x | 2 , x ∈ R n . Instead of using the error bound presented by Madych and Nelson [2], here we adopt an improved error bound constructed by Luh to evaluate the influence of β on error estimates. This results in a new set of criteria for the optimal choice of β and much sharper error estimates for Gaussian interpolation. What is important is that the notorious ill-conditioning of Gaussian interpolation can be greatly relieved because in this approach the fill distance need not be very small.