Asymptotic coefficients for Gaussian radial basis function interpolants

Gaussian radial basis functions (RBFs) on an infinite interval with uniform grid pacing h are defined by @f(x;@a,h)=exp(-[@a^2/h^2]x^2). The only significant numerical parameter is @a, the inverse width of the RBF functions relative to h. In the limit @a->0, we demonstrate that the coefficients of the interpolant of a typical function f(x) grow proportionally to exp(@p^2/[4@a^2]). However, we also show that the approximation to the constant f(x)=1 is a Jacobian theta function whose coefficients do not blow up as @a->0. The subtle interplay between the complex-plane singularities of f(x) (the function being approximated) and the RBF inverse width parameter @a are analyzed. For @a~1/2, the size of the RBF coefficients and the condition number of the interpolation matrix are both no larger than O(10^4) and the error saturation is smaller than machine epsilon, so this @a is the center of a ''safe operating range'' for Gaussian RBFs.