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 phi(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 phi. The explicit construction of locally well-behaving admissible coefficient 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 psi of psi(x) = phi(\\x\\) in the sense integral \f\2 psi-1 dt < infinity. 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 different and simpler analytic technique and allows to handle the cases of interpolation with phi(r) = r(s) for S is-an-element-of R, s > 1, s is-not-an-element-of 2N, and phi(r) = r(s) log r for S is-an-element-of 2N, which are shown to have accuracy O(h(s/2)).