Accuracy of radial basis function interpolation and derivative approximations on 1-D infinite grids

Radial basis function (RBF) interpolation can be very effective for scattered data in any number of dimensions. As one of their many applications, RBFs can provide highly accurate collocation-type numerical solutions to several classes of PDEs. To better understand the accuracy that can be obtained, we survey here derivative approximations based on RBFs using a similar Fourier analysis approach that has become the standard way for assessing the accuracy of finite difference schemes. We find that the accuracy is directly linked to the decay rate, at large arguments, of the ( generalized) Fourier transform of the radial function. Three different types of convergence rates can be distinguished as the node density increases - polynomial, spectral, and superspectral, as exemplified, for example, by thin plate splines, multiquadrics, and Gaussians respectively.