Approximation in the Zygmund Class

TL;DR

This paper investigates the approximation properties within the Zygmund class, characterizing the closure of certain subspaces related to BMO and Sobolev functions, extending results to higher dimensions.

Contribution

It provides a detailed description of the closure of I(BMO) in the Zygmund class and generalizes these results to Zygmund measures on  spaces, also linking to Sobolev spaces.

Findings

Characterization of the closure of I(BMO) in the Zygmund seminorm.

Extension of results to Zygmund measures on  spaces.

Identification of the closure of Zygmund functions in Sobolev spaces.

Abstract

We study the distance in the Zygmund class $\Lambda_{\ast}$ to the subspace $\operatorname{I}(\operatorname{BMO})$ of functions with distributional derivative with bounded mean oscillation. In particular, we describe the closure of $\operatorname{I}(\operatorname{BMO})$ in the Zygmund seminorm. We also generalise this result to Zygmund measures on $\mathbb{R}^d.$ Finally, we apply the techniques developed in the article to characterise the closure of the subspace of functions in $\Lambda_{\ast}$ that are also in the classical Sobolev space $W^{1,p},$ for $1 < p < \infty.$