Approximation in the Zygmund class

We study the distance in the Zygmund class ?* to the subspace I( BMO ) of functions with distributional derivative with bounded mean oscillation. In particular, we describe the closure of I( BMO ) in the Zygmund seminorm. We also generalise this result to Zygmund measures on Rd. Finally, we apply the techniques developed in the article to characterise the closure of the subspace of functions in ?* that are also in the classical Sobolev space W1,p, for 1.