The Penalized Lebesgue Constant for Surface Spline Interpolation. (arXiv:0911.1815v1 [math.CA])

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 those
recently employed by Hangelbroek, Narcowich and Ward to demonstrate the
stability of interpolation from quasi-uniform data on manifolds, we show that
surface spline interpolation on 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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Original source : http://arxiv.org/abs/0911.1815...