We study the distance in the Zygmund class ${\mathrm{\Lambda }}_{*}$ to the subspace $\mathrm{I}(\mathrm{BMO})$ of functions with distributional derivative with bounded mean oscillation. In particular, we describe the closure of $\mathrm{I}(\mathrm{BMO})$ 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 ${\mathrm{\Lambda }}_{*}$ that are also in the classical Sobolev space ${\mathrm{W}}^{1,p}$, for $1<p<\infty$.

中文翻译：

---

Zygmund类中的近似

我们在Zygmund类中研究距离 ${\mathrm{\Lambda }}_{*}$ 到子空间 $\mathrm{一世}（\mathrm{BMO}）$具有有限平均振荡的分布导数的函数 特别是，我们描述了$\mathrm{一世}（\mathrm{BMO}）$在Zygmund半范式中。我们还将这个结果推广到Zygmund测度上${\mathbb{[R}}^d$。最后，我们运用本文中开发的技术来表征函数中子空间的闭合${\mathrm{\Lambda }}_{*}$ 在经典的Sobolev空间中 ${\mathrm{w\; ^}}^{\mathrm{1个}，p}$，对于 $\mathrm{1个}<p<\infty$。