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.
Categories: math.NA cs.NA math.CA