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 ϕ(x;α,h)≡exp(-[α2/h2]x2)ϕ(x;α,h)≡exp(-[α2/h2]x2). The only significant numerical parameter is αα, the inverse width of the RBF functions relative to h  . In the limit α→0α→0, we demonstrate that the coefficients of the interpolant of a typical function f(x  ) grow proportionally to exp(π2/[4α2])exp(π2/[4α2]). However, we also show that the approximation to the constant f(x)≡1f(x)≡1 is a Jacobian theta function whose coefficients do not blow up as α→0α→0. The subtle interplay between the complex-plane singularities of f(x  ) (the function being approximated) and the RBF inverse width parameter αα are analyzed. For α≈1/2α≈1/2, the size of the RBF coefficients and the condition number of the interpolation matrix are both no larger than O(104)O(104) and the error saturation is smaller than machine epsilon, so this αα is the center of a “safe operating range” for Gaussian RBFs.