The shape parameter in the Gaussian function II

This is a continuation of our former study, Luh [1], of the shape parameter ββ contained in Gaussian e−β|x|2e−β|x|2, x∈Rnx∈Rn. 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 ββ on error estimates. This results in a new set of criteria for the optimal choice of ββ 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.