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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.
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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