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 beta contained in the famous Gaussian function e(-beta vertical bar x vertical bar 2), x is an element of 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 beta has been unknown. People conversant with RBFs know that beta 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 beta which shows the influence of beta in the error estimate of Gaussian interpolation and with which the optimal beta can always be found. (C) 2011 Elsevier Ltd. All rights reserved.