Local error estimates for radial basis function interpolation of scattered data.(English)Zbl 0762.41006

Summary: Introducing a suitable variational formulation for the local error of scattered data interpolation by radial basis functions $$\varphi(r)$$, 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 $$\varphi$$. 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 $$\hat f$$ is ‘dominated’ by the nonnegative Fourier transform $$\hat\psi$$ of $$\psi(x)=\varphi(\| x\|)$$ in the sense $$\int| \hat f|^ 2\hat\psi^{-1}dt<\infty$$. 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 $$\hat f$$, 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 $$\varphi(r)=r^ s$$ for $$s\in\mathbb{R}$$, $$s>1$$, $$s\not\in2\mathbb{N}$$, and $$\varphi(r)=r^ s\log r$$ for $$s\in 2\mathbb{N}$$, which are also to have accuracy $$O(h^{s/2})$$.

MSC:

 41A05 Interpolation in approximation theory
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