Local Accuracy for Radial Basis Function Interpolation on Finite Uniform Grids

AbstractWe consider interpolation on a finite uniform grid by means of one of the radial basis functions (RBF) φ(r)=rγ for γ>0, γ∉2N or φ(r)=rγlnr for γ∈2N+. For each positive integer N, let h=N−1 and let {xi:i =1, 2, …, (N+1)d} be the set of vertices of the uniform grid of mesh-size h on the unit d-dimensional cube [0, 1]d. Given f:[0, 1]d→R, let sh be its unique RBF interpolant at the grid vertices: sh(xi)=f(xi), i=1, 2, …, (N+1)d. For h→0, we show that the uniform norm of the error f−sh on a compact subset K of the interior of [0, 1]d enjoys the same rate of convergence to zero as the error of RBF interpolation on the infinite uniform grid hZd, provided that f is a data function whose partial derivatives in the interior of [0, 1]d up to a certain order can be extended to Lipschitz functions on [0, 1]d