Counting via entropy: new preasymptotics for the approximation numbers of Sobolev embeddings

TL;DR

This paper establishes a new link between approximation and entropy numbers for Sobolev spaces, providing preasymptotic bounds and insights into the behavior of Gevrey spaces, with applications in Galerkin methods.

Contribution

It introduces a novel connection between approximation and entropy numbers, leading to new preasymptotic bounds for various Sobolev and Gevrey spaces.

Findings

Preasymptotic error bounds for Sobolev spaces

Approximation numbers of Gevrey spaces behave like mixed smoothness spaces

Application to Galerkin methods for Schrödinger equation

Abstract

In this paper, we reveal a new connection between approximation numbers of periodic Sobolev type spaces, where the smoothness weights on the Fourier coefficients are induced by a (quasi-)norm $\|\cdot\|$ on $\mathbb{R}^d$, and entropy numbers of the embedding $\textrm{id}: \ell_{\|\cdot\|}^d \to \ell_\infty^d$. This connection yields preasymptotic error bounds for approximation numbers of isotropic Sobolev spaces, spaces of analytic functions, and spaces of Gevrey type in $L_2$ and $H^1$, which find application in the context of Galerkin methods. Moreover, we observe that approximation numbers of certain Gevrey type spaces behave preasymptotically almost identical to approximation numbers of spaces of dominating mixed smoothness. This observation can be exploited, for instance, for Galerkin schemes for the electronic Schr\"odinger equation, where mixed regularity is present.