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JOURNALS // Trudy Instituta Matematiki i Mekhaniki UrO RAN // Archive

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

This article is cited in 2 papers

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.

UDC: 517.5

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

Received: 28.03.2016

DOI: 10.21538/0134-4889-2016-22-4-29-42


 English version:
Proceedings of the Steklov Institute of Mathematics (Supplement Issues), 2018, 300, suppl. 1, S25–S37

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