Uniformly bounded Lebesgue constants for scaled cardinal interpolation with Mat\'{e}rn kernels

TL;DR

This paper proves that Lebesgue constants for scaled Matérn kernel interpolation on grids remain bounded as grid spacing decreases, ensuring stable convergence with a rate of O(h^{2m}) for high-dimensional approximation.

Contribution

It establishes uniform boundedness of Lebesgue constants for Matérn kernel interpolation on scaled grids and derives convergence rates, advancing understanding of kernel-based approximation stability.

Findings

Lebesgue constants are uniformly bounded as grid spacing approaches zero.

The interpolation scheme converges at a rate of O(h^{2m}).

Convergence results extend to Matérn and related 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 Mat\'{e}rn 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 $h\to0$ and deduce the convergence rate $O(h^{2m})$ for the scaled interpolation scheme. We also provide convergence results for approximation with Mat\'{e}rn and related compactly supported polyharmonic kernels.