On the density of certain modules of polyanalytic type in spaces of integrable functions on the boundaries of simply connected domains

K. Yu. Fedorovskiy

Sbornik. Mathematics, Tome 207 (2016) no. 1, pp. 140-154

Résumé

We consider the question of the density in the space $L^p$, $1\leq p\leq\infty$, on the unit circle, of the subspaces $H^p+\sum_{k=1}^mw_kH^p$, where $H^p$ is the standard Hardy space and $w_1,\dots,w_m$ are given functions in the class $L^\infty$. This question is closely related to problems of uniform and $L^p$-approximations of functions by polyanalytic polynomials on the boundaries of simple connected domains in $\mathbb C$. The obtained results are formulated in terms of Nevanlinna and $d$-Nevanlinna domains, that is, in terms of special analytic characteristics of simply connected domains in $\mathbb C$, which are related to the pseudocontinuation property of bounded holomorphic functions. Bibliography: 19 titles.

Détail

Keywords: pseudocontinuation, uniform approximation, $L^p$-approximation.
Mots-clés : Nevanlinna domain, $d$-Nevanlinna domain, polyanalytic polynomial