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

摘要

We catalogue the numerical properties of approximations using two novel types of radial basis functions ?(r). The QG species is a basis of exponentials of quartic argument: f(x) ≈ f~(RBF)(x; a, h) = ∑_(j=1)~N a_j exp(-[α/h]~4(x - x_j)~4) where the x_j are the RBF centers and also the interpolation points. We show that Quartic Gaussian RBFs fail at many discrete values of the shape parameter α. We show through a detailed analysis that these singularities are directly related to zeros of Q(k), the Fourier Transform of exp(-x~4). If we reverse the roles and take Q(x) as the RBF, all difficulties disappear because these IQG RBFs have a Fourier transform which is nonnegative for all real k. We explain that although the Quartic-Gaussian exp(-x~4) 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.