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

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 $w_1,…,w_m$ 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.
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