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On the density of certain modules of polyanalytic type in spaces of integrable functions on the boundaries of simply connected domains. (English) Zbl 1345.30041
Sb. Math. 207, No. 1, 140-154 (2016); translation from Mat. Sb. 207, No. 1, 151-166 (2016).
Let $$L^p$$, $$1\leq p\leq\infty$$, be the Lebesgue space on the unit circle $$\mathbb{T}=\{z\in\mathbb{C}:|z| =1\}$$ with respect to the normalized Lebesgue measure on $$\mathbb{T}$$, and let $$H^p(\mathbb{T})$$ be the Hardy space on $$\mathbb{T}$$. In this paper, the question of the density in the space $$L^p$$ of the subspaces $$M^p(w_1,\dots,w_m):=H^p+\sum_{k=1}^mw_kH^p$$, where $$w_1,\dots,w_m$$ are given functions in the class $$L^\infty$$, is considered. 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. Let $$\mathbb{D}$$ be the unit disk. The space $$M^p(w_1,\dots,w_m)$$ is not dense in $$L^p$$, and the space $$M^\infty(w_1,\dots,w_m)$$ is not weak-star dense in $$L^\infty$$ if and only if each of the functions $$w_k$$, $$k=1,\dots,m$$, admits a pseudocontinuation to $$\mathbb{D}$$. The question of the approximation by polyanalytic polynomials is also considered.
##### MSC:
 30E10 Approximation in the complex plane 30G20 Generalizations of Bers and Vekua type (pseudoanalytic, $$p$$-analytic, etc.) 41A10 Approximation by polynomials
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