Local Error Estimates for Radial Basis Function Interpolation of Scattered Data

Introducing a suitable variational formulation for the local error of scattered data interpolation by radial basis functions (r), the error can be bounded by a term depending on the Fourier transform of the interpolated function f and a certain \Kriging function", which allows a formulation as an integral involving the Fourier transform of. The explicit construction of locally well{behaving admissible coeecient vectors makes the Kriging function bounded by some power of the local density h of data points. This leads to error estimates for interpolation of functions f whose Fourier transform ^ f is \dominated" by the nonnegative Fourier transform ^ of (x) = (kxk) in the sense R j ^ fj 2 ^ ?1 dt < 1. Approximation orders are arbitrarily high for interpolation with Hardy multiquadrics, inverse multiquadrics and Gaussian kernels. This was also proven in recent papers by Madych and Nelson, using a reproducing kernel Hilbert space approach and requiring the same hypothesis as above on ^ f, which limits the practical applicability of the results. This work uses a diierent and simpler analytic technique and allows to handle the cases of interpolation with (r) = r s for s 2 IR; s > 1; s = 2 2IN, and (r) = r s log r for s 2 2IN, which are shown to have accuracy O(h s=2).