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

Fedorovskiy, K. Yu

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.

Reviewer: Konstantin Malyutin (Sumy)

MSC:

30E10	Approximation in the complex plane
30G20	Generalizations of Bers and Vekua type (pseudoanalytic, \(p\)-analytic, etc.)
41A10	Approximation by polynomials

Keywords:

Hardy space; Nevanlinna domain; \(d\)-Nevanlinna domain; pseudo-continuation; polyanalytic polynomial; uniform approximation; \(L_p\)-approximation