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Seminar on Complex Analysis (Gonchar Seminar)
June 3, 2013 18:00, Moscow, Steklov Mathematical Institute, Room 411 (8 Gubkina)

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Approximation of functions by polynomial solutions of elliptic equations

K. Yu. Fedorovskiy

N. E. Bauman Moscow State Technical University

Abstract: It is planned to consider several recently obtained results in problems on approximability of functions by polynomial solutions of homogeneous elliptic partial differential equations with constant complex coefficients on planar compact sets in norms of the spaces $C^m$. In particular it is planned to discuss problems on approximability of functions by polyanalytic polynomials, that is by polynomials of the form
$$\overline{z}^np_n(z)+\cdots+\overline{z}p_1(z)+p_0(0),$$
where $p_n,…,p_0$ — are polynomials in the complex variable $z$.
The spacial attention will be given to the problem of uniform approximability of functions by polyanalytic polynomials as well as to closely related problems about properties of Nevanlinna domains (one recalls that bounded simply connected domain $\Omega$ in $\mathbb C$ is called a Nevanlinna domain if there exist two functions $u,v\in H^\infty(\Omega)$, $v\not\equiv0$, such that the equality $\overline{z}=u(z)/v(z)$ holds almost everywhere on $\partial\Omega$ in the sense of conformal mappings). Problems about properties of Nevanlinna domain, in turn, are closely related with problems about existence and boundary behavior of bounded univalent functions belonging to model spaces (that is subspaces of the Hardy space $H^2$ that are invariant under the backward shift operator). These problems will be also discussed.