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The penalized Lebesgue constant for surface spline interpolation

Author: Thomas Hangelbroek
Journal: Proc. Amer. Math. Soc. 140 (2012), 173-187
MSC (2010): Primary 41A05, 41A25, 46E35, 65D05
Published electronically: May 5, 2011
MathSciNet review: 2833530
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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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Additional Information

Thomas Hangelbroek
Affiliation: Department of Mathematics, Texas A&M University, College Station, Texas 77843
Address at time of publication: Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240

Keywords: Interpolation, surface splines, Lebesgue constant, radial basis function
Received by editor(s): November 9, 2009
Received by editor(s) in revised form: October 30, 2010
Published electronically: May 5, 2011
Additional Notes: The author was supported by an NSF Postdoctoral Research Fellowship.
Communicated by: Walter Van Assche
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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