Inverse inequalities for kernel-based approximation on bounded domains and Riemannian manifolds

TL;DR

This paper develops inverse inequalities for kernel-based approximation spaces on bounded domains and Riemannian manifolds, extending classical polynomial inequalities to more general kernel methods with potential applications in approximation theory.

Contribution

It extends Bernstein and Nikolskii inequalities to kernel-based spaces on bounded Lipschitz domains and Riemannian manifolds, covering all Sobolev orders and restricted kernels.

Findings

Extended Bernstein inequalities to all Sobolev orders.

Derived Nikolskii inequalities bounding $L_$ norms by $L_2$ norms.

Established inverse inequalities for restricted kernels on manifolds.

Abstract

This paper establishes inverse inequalities for kernel-based approximation spaces defined on bounded Lipschitz domains in $\mathbb{R}^d$ and compact Riemannian manifolds. While inverse inequalities are well-studied for polynomial spaces, their extension to kernel-based trial spaces poses significant challenges. For bounded Lipschitz domains, we extend prior Bernstein inequalities, which only apply to a limited range of Sobolev orders, to all orders on the lower bound and $L_2$ on the upper, and derive Nikolskii inequalities that bound $L_\infty$ norms by $L_2$ norms. Our theory achieves the desired form but may require slightly more smoothness on the kernel than the regular $>d/2$ assumption. For compact Riemannian manifolds, we focus on restricted kernels, which are defined as the restriction of positive definite kernels from the ambient Euclidean space to the manifold, and prove their counterparts.