Optimal approximation of multivariate periodic Sobolev functions in the sup-norm

TL;DR

This paper develops a general operator-theoretic framework to accurately approximate multivariate periodic Sobolev functions in the supremum norm, bridging $L_2$ and $L_ty$ approximation estimates with precise constant control.

Contribution

It introduces a novel approach using operator ideals and s-numbers to transfer approximation estimates from $L_2$ to $L_ty$, with applications to various Sobolev and Besov spaces.

Findings

Derived new bounds for $L_ty$ approximation of Sobolev functions.

Established transfer principles from $L_2$ to $L_ty$ approximation.

Applied results to isotropic and mixed smoothness Sobolev and Besov spaces.

Abstract

Using tools from the theory of operator ideals and s-numbers, we develop a general approach to transfer estimates for $L_2$ -approximation of Sobolev functions into estimates for $L_\infty$-approximation, with precise control of all involved constants. As an illustration, we derive some results for periodic isotropic Sobolev spaces $H^s ({\mathbb T}^d)$ and Sobolev spaces of dominating mixed smoothness $H^s_{\rm mix} ({\mathbb T}^d)$, always equipped with natural norms. Some results for isotropic as well as dominating mixed Besov spaces are also obtained.