The shape parameter in the Gaussian function

This is the fifth of our series of works about the shape parameter. We now explore the parameter ββ contained in the famous Gaussian function e−β|x|2,x∈Rn. In the theory of radial basis functions (RBFs), the Gaussian is frequently used in virtue of its good error bound and numerical tractability. However, the optimal choice of ββ has been unknown. People conversant with RBFs know that ββ is very influential, but do not have a reliable criterion of its choice. The purpose of this paper is to uncover its mystery. In particular, we have greatly improved the result of Madych (1992) in [15], and we present a concrete function of ββ which shows the influence of ββ in the error estimate of Gaussian interpolation and with which the optimal ββ can always be found.