Uniformly bounded Lebesgue constants for scaled cardinal interpolation with Matérn kernels

Abstract

For $h>0$ and positive integers $m$, $d$, such that $m>d/2$, we study non-stationary interpolation at the points of the scaled grid $h{\mathbb{Z}}^d$ via the Matérn kernel ${\Phi }_{m,d}$---the fundamental solution of $(1-\Delta {)}^m$ in ${\mathbb{R}}^d$. We prove that the Lebesgue constants of the corresponding interpolation operators are uniformly bounded as $h\to 0$ and deduce the convergence rate $O(h^{2m})$ for the scaled interpolation scheme. We also provide convergence results for approximation with Matérn and related compactly supported polyharmonic kernels.

16 pages