The penalized Lebesgue constant for surface spline interpolation

Hangelbroek, Thomas

doi:10.1090/S0002-9939-2011-10870-0

The penalized Lebesgue constant for surface spline interpolation
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by Thomas Hangelbroek

Proc. Amer. Math. Soc. 140 (2012), 173-187

DOI: https://doi.org/10.1090/S0002-9939-2011-10870-0

Published electronically: May 5, 2011

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Abstract:

Problems involving approximation from scattered data where data is arranged quasi-uniformly have been treated by RBF methods for decades. Treating data with spatially varying density has not been investigated with the same intensity and is far less well understood. In this article we consider the stability of surface spline interpolation (a popular type of RBF interpolation) for data with nonuniform arrangements. Using techniques similar to Hangelbroek, Narcowich and Ward, who discuss interpolation from quasi-uniform data on manifolds, we show that surface spline interpolation on $\mathbb {R}^d$ is stable, but in a stronger, local sense. We also obtain pointwise estimates showing that the Lagrange function decays very rapidly, and at a rate determined by the local spacing of datasites. These results, in conjunction with a Lebesgue lemma, show that surface spline interpolation enjoys the same rates of convergence as those of the local approximation schemes recently developed by DeVore and Ron.

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