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 Trudy Inst. Mat. i Mekh. UrO RAN, 2018, Volume 24, Number 4, Pages 19–33 (Mi timm1572)

Optimal recovery of a function analytic in a half-plane from approximately given values on a part of the straight-line boundary

R. R. Akopyanab

a Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg
b Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg

Abstract: 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 $\gamma=ż : \arg z=\varphi_0\}$ 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.

 Funding Agency Grant Number Russian Foundation for Basic Research 18-01-00336 Ministry of Education and Science of the Russian Federation 02.A03.21.0006 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).

DOI: https://doi.org/10.21538/0134-4889-2018-24-4-19-33

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Bibliographic databases:

UDC: 517.977
MSC: 30A10, 30C80, 30C85, 30E10
Revised: 14.11.2018
Accepted:19.11.2018

Citation: 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”, Trudy Inst. Mat. i Mekh. UrO RAN, 24, no. 4, 2018, 19–33

Citation in format AMSBIB
\Bibitem{Ako18} \by R.~R.~Akopyan \paper Optimal recovery of a function analytic in a half-plane from approximately given values on a part of the straight-line boundary \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2018 \vol 24 \issue 4 \pages 19--33 \mathnet{http://mi.mathnet.ru/timm1572} \crossref{https://doi.org/10.21538/0134-4889-2018-24-4-19-33} \elib{https://elibrary.ru/item.asp?id=36517696}