R.R. Akopyan. Optimal recovery of a function analytic in a half-plane from approximately given values on a part of the straight-line boundary... P. 19-33

Let $\mathcal{H}^p(\Pi_+,\phi)$ be the class of functions analytic in the upper half-plane $\Pi_+$ and belonging to the universal Hardy class $N_*$ with boundary values from $L^p_\phi(\mathbb{R})$ with a weight $\phi$, and let $Q^p(\Pi_+,\mathbb{I},\phi)$ be the class of function $f\in \mathcal{H}^p(\Pi_+,\phi)$ such that $\|f\|_{L^p_\phi(\mathbb{R}\setminus\mathbb{I})}\le 1$, where $\mathbb{I}$ is a finite open interval or a half-line from $\mathbb{R}$ and $1\le p\le\infty.$ On the class $Q^p(\Pi_+,\mathbb{I},\phi)$, we consider the problem of optimal recovery of the value of a function at a point $z_0\in\Pi_+$ from its approximately given limit boundary values on $\mathbb{I}$ in the norm $L^p_\phi(\mathbb{I})$ and the related problem of the best approximation of a functional by linear bounded functionals. Explicit solutions of these problems are written: an extremal function, optimal recovery method, and best approximation functional. On the class $Q^p(\Pi_+,\mathbb{R}_+,\psi)$, $\psi(z)=1/|z|$, we solve the problem of optimal recovery of a function on a ray $\gamma=\{z\,:\,\arg z=\varphi_0\}$ with respect to the norm $L^p_\psi(\gamma)$ from its approximately given limit boundary values on~$\mathbb{R}_+$ in the norm $L^p_\psi(\mathbb{R}_+)$ and the related problem of the best approximation of an operator by linear bounded operators. For $f\in\mathcal{H}^p(\Pi_+,\psi)$, we obtain the exact inequality $$ \|f\|_{L^p_{\psi}(\gamma)}\le \|f\|_{L^{p}_{\psi}(-\infty, 0)}^{{\varphi_0}/{\pi}}\, \|f\|_{L_{\psi}^{p}(0, +\infty)}^{1-{\varphi_0}/{\pi}}. $$

Keywords: optimal recovery of an operator, best approximation of an unbounded operator by bounded operators, analytic function

Received August 12, 2018

Revised November 14, 2018

Accepted November 19, 2018

Funding Agency: This work was supported by the Russian Foundation for Basic Research (project no. 18-01-00336) and by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University).

Roman Razmikovich Akopyan, Ural Federal University, Yekaterinburg, 620000 Russia; Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620990 Russia, e-mail: RRAkopyan@mephi.ru


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