Preasymptotics and asymptotics of approximation numbers of anisotropic Sobolev embeddings

TL;DR

This paper analyzes the approximation numbers of anisotropic Sobolev embeddings, revealing their asymptotic behavior, strong equivalences, and intractability, with implications for high-dimensional approximation problems.

Contribution

It provides the first detailed preasymptotic and asymptotic analysis of approximation numbers for anisotropic Sobolev embeddings, including limit spaces, showing intractability and absence of curse of dimensionality.

Findings

Approximation numbers exhibit specific asymptotic behaviors.

Embeddings are intractable but free from curse of dimensionality.

Preasymptotic behavior characterized for limit spaces.

Abstract

In this paper, we obtain the preasymptotic and asymptotic behavior and strong equivalences of the approximation numbers of the embeddings from the anisotropic Sobolev spaces $W_2^{\bf R}(\Bbb T^d)$ to $L_2(\Bbb T^d)$. We also get the preasymptotic behavior of the approximation numbers of the embeddings from the limit spaces $W_2^{\infty}(\Bbb T^d)$ of the anisotropic Sobolev spaces $W_2^{\bf R}(\Bbb T^d)$ to $L_2(\Bbb T^d)$. We show that both the above embedding problems are intractable and do not suffer from the curse of dimensionality.