Spectral approximation orders of radial basis function interpolation on the Sobolev space.(English)Zbl 0996.41002

In this paper, spectral convergence orders, i.e. those that depend on the (variable) smoothness of the function to be approximated, are discussed for a class of radial basis functions. This includes the famous multiquadric function where the spectral convergence orders have been already looked at in various ways in the literature. Here the focus is on relating the approximation order directly to the Sobolev space the approximated function lives in. This enlarges the function classes to which the method may be applied because for multiquadrics, for instance, the so-called native spaces are much ‘smaller’ than typical Sobolev spaces due to the smoothness of the multiquadric. As a result the approximation orders obtained herein are not exponential but still faster than ordinary powers of the spacing: they are $$o(h^k)$$ for approximands from the Sobolev space of order $$k$$. Other radial basis functions to which the theory applies, e.g. ‘shifted surface splines’ are provided too.

MSC:

 41A05 Interpolation in approximation theory 41A15 Spline approximation 41A25 Rate of convergence, degree of approximation 41A30 Approximation by other special function classes 41A63 Multidimensional problems
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