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 φ(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 φ. 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 $ψˆ$ of ψ(x) = ψ(∥x∥) in the sense $∫|fˆ|2ψˆ-1dt<∞$. 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 φ(r) = r^s for s ε R, s > 1, s ∉ 2N, and φ(r) = r^s log r for s ε 2N, which are shown to have accuracy O(h^s/2)