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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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