The shape parameter in the Gaussian function

Abstract

This is the fifth of our series of works about the shape parameter. We now explore the parameter β contained in the famous Gaussian function e − β | x | 2 , x ∈ R n . 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 β has been unknown. People conversant with RBFs know that β 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 β which shows the influence of β in the error estimate of Gaussian interpolation and with which the optimal β can always be found.