Norm Estimates for Inverses of Distance Matrices

Abstract

A radial basis function approximation has the form s(x) = ▪ is some given function, (yj)n1 are real coefficients, and the centres (xj)n1 are points in ℝd. For a wide class of functions φ, it is known that the interpolation matrix ▪ is invertible (see [5]). Further, several recent papers have provided upper bounds on ‖A−1‖2, where the points (xj)n1 satisfy the condition ‖xj – xk‖2 ≥ δ, j ≠ k, for some positive constant 6. In this paper, we show that the generalized Fourier transform of φ provides a simple and elegant tool for the derivation of such estimates.