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 Trudy Inst. Mat. i Mekh. UrO RAN, 2016, Volume 22, Number 4, Pages 29–42 (Mi timm1351)

Optimal recovery of a function analytic in a disk from approximately given values on a part of the boundary

R. R. Akopyanab

a Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
b Institute of Mathematics and Computer Science, Ural Federal University, Ekaterinburg

Abstract: We study three related extremal problems in the space $\mathcal{H}$ of functions analytic in the unit disk such that their boundary values on a part $\gamma_1$ of the unit circle $\Gamma$ belong to the space $L^\infty_{\psi_1}(\gamma_1)$ of functions essentially bounded on $\gamma_1$ with weight $\psi_1$ and their boundary values on the set $\gamma_0=\Gamma\setminus\gamma_1$ belong to the space $L^\infty_{\psi_0}(\gamma_0)$ with weight $\psi_0$. More exactly, on the class $Q$ of functions from $\mathcal{H}$ such that the norm $L^\infty_{\psi_0}(\gamma_0)$ of their boundary values on $\gamma_0$ does not exceed one, we solve the problem of optimal recovery of an analytic function on a subset of the unit disk from its boundary values on $\gamma_1$ specified approximately with respect to the norm $L^\infty_{\psi_1}(\gamma_1)$. We also study the problem of the optimal choice of the set $\gamma_1$ under a given fixed value of its measure. The problem of the best approximation of the operator of analytic continuation from a part of the boundary by linear bounded operators is investigated.

Keywords: optimal recovery of analytic functions, best approximation of unbounded operators, Szegő function.

 Funding Agency Grant Number Russian Foundation for Basic Research 15-01-02705 Ministry of Education and Science of the Russian Federation ÍØ-9356.2016.102.A03.21.0006

DOI: https://doi.org/10.21538/0134-4889-2016-22-4-29-42

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English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2018, 300, suppl. 1, 25–37

Bibliographic databases:

UDC: 517.5
MSC: 30E10, 30E25, 30C85, 41A35

Citation: R. R. Akopyan, “Optimal recovery of a function analytic in a disk from approximately given values on a part of the boundary”, Trudy Inst. Mat. i Mekh. UrO RAN, 22, no. 4, 2016, 29–42; Proc. Steklov Inst. Math. (Suppl.), 300, suppl. 1 (2018), 25–37

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. R. R. Akopyan, “An analogue of the two-constants theorem and optimal recovery of analytic functions”, Sb. Math., 210:10 (2019), 1348–1360
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