The Penalized Lebesgue Constant for Surface Spline Interpolation
For researchers in scattered data approximation, this provides a theoretical foundation for stable surface spline interpolation with nonuniform data, extending prior quasi-uniform results.
The paper proves that surface spline interpolation on R^d is stable in a local sense for nonuniform data, with Lagrange function decay determined by local data spacing, achieving convergence rates comparable to local approximation schemes.
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.