## 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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### References:

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