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An error analysis for radial basis function interpolation. (English) Zbl 1128.41002

Summary: Radial basis function interpolation refers to a method of interpolation which writes the interpolant to some given data as a linear combination of the translates of a single function \(\varphi\) and a low degree polynomial. We develop an error analysis which works well when the Fourier transform of \(\varphi\) has a pole of order \(2m\) at the origin and a zero at \(\infty\) of order \(2\kappa\). In case \(0\leq m \leq \kappa\), we derive error estimates which fill in some gaps in the known theory; while in case \(m>\kappa\) we obtain previously unknown error estimates. In this latter case, we employ dilates of the function \(\varphi\), where the dilation factor corresponds to the fill distance between the data points and the domain.

MSC:

41A05 Interpolation in approximation theory
41A25 Rate of convergence, degree of approximation
65D05 Numerical interpolation
41A63 Multidimensional problems
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