The near-equivalence of five species of spectrally-accurate radial basis functions (RBFs): Asymptotic approximations to the RBF cardinal functions on a uniform, unbounded grid

These results generalize to higher dimensions. The Gaussian RBF cardinal functions in any number of dimensions d are, without approximation, the tensor product of one dimensional Gaussian cardinal functions: Cd(x1,x2…,xd)=∏j=1dC(xj). For other RBF species, we show that the two-dimensional cardinal functions are well approximated by the products of one-dimensional cardinal functions; again the error goes to zero as α → 0. The near-identity of the cardinal functions implies that all five species of RBF interpolants are (almost) the same, despite the great differences in the RBF ϕ's themselves.