Quartic Gaussian and Inverse-Quartic Gaussian radial basis functions: The importance of a nonnegative Fourier transform

We catalogue the numerical properties of approximations using two novel types of radial basis functions ϕ(r)ϕ(r). The QG species is a basis of exponentials of quartic argument: f(x)≈fRBF(x;α,h)≡∑j=1Najexp(−[α/h]4(x−xj)4) where the xjxj are the RBF centers and also the interpolation points. We show that Quartic Gaussian RBFs fail at many discrete values of the shape parameter αα. We show through a detailed analysis that these singularities are directly related to zeros of Q(k)Q(k), the Fourier Transform of exp(−x4)exp(−x4). If we reverse the roles and take Q(x)Q(x) as the RBF, all difficulties disappear because these IQG RBFs have a Fourier transform which is nonnegative for all real kk. We explain that although the Quartic-Gaussian exp(−x4)exp(−x4) is positive definite in the physics/dynamical systems sense of being zero-free and nonnegative, it lacks the crucial property of being positive definition in the RBF/analysis sense.