Sharp inverse statements for kernel interpolation

TL;DR

This paper establishes sharp inverse theorems for kernel interpolation, revealing a precise link between a function's smoothness and its approximation rate, especially for finitely smooth kernels like Matérn and Wendland.

Contribution

It provides the first sharp inverse statements for kernel interpolation with finitely smooth kernels, clarifying the relationship between smoothness and approximation rate.

Findings

Sharp inverse statements for kernel interpolation derived

One-to-one correspondence between function smoothness and approximation rate established

Results apply to popular RBF kernels like Matérn and Wendland

Abstract

While direct statements for kernel based interpolation on regions $\Omega \subset \mathbb{R}^d$ are well researched, far less is known about corresponding inverse statements. The available inverse statements for kernel based interpolation so far are not sharp. In this paper, we derive sharp inverse statements for interpolation using finitely smooth kernels, such as popular radial basis function (RBF) kernels like the class of Mat\'ern or Wendland kernels. In particular, the results show that there is a one-to-one correspondence between the smoothness of a function and its approximation rate via kernel interpolation: If a function can be approximated with a given rate, it has a corresponding smoothness and vice versa.