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 @b contained in the famous Gaussian function e^-^@b^|^x^|^^^2,[email protected]?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 @b has been unknown. People conversant with RBFs know that @b 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 @b which shows the influence of @b in the error estimate of Gaussian interpolation and with which the optimal @b can always be found.