Sharp inverse statements for kernel approximation: Superconvergence and saturation

TL;DR

This paper develops sharp inverse and saturation results for kernel approximation with finitely smooth Sobolev kernels, revealing a precise link between function smoothness and approximation rates, especially in the superconvergence regime.

Contribution

It provides a unified theory connecting function smoothness and approximation rates, extending beyond native spaces for kernel-based methods.

Findings

Established sharp inverse and saturation estimates.

Characterized the full range of smoothness spaces for approximation.

Extended superconvergence results beyond native spaces.

Abstract

This article establishes sharp inverse and saturation statements for kernel-based approximation using finitely smooth Sobolev kernels on bounded Lipschitz regions. The analysis focuses on the superconvergence regime, for which direct statements have only recently been obtained. The resulting theory yields a one-to-one correspondence between the smoothness of a target function - quantified in terms of power spaces - and the achievable approximation rates by kernel-based approximation. In this way, we extend existing results beyond the escaping-the-native-space regime and provide a unified characterization covering the full scale of admissible smoothness spaces.