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Mat. Sb., 2019, Volume 210, Number 10, Pages 3–16 (Mi msb8952)  

An analogue of the two-constants theorem and optimal recovery of analytic functions

R. R. Akopyanab

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

Abstract: Several related extremal problems for analytic functions in a simply connected domain $G$ with rectifiable Jordan boundary $\Gamma$ are treated. The sharp inequality
$$ |f(z)|\le\mathscr C^{r,q}(z;\gamma_0,\varphi_0;\gamma_1,\varphi_1)\|f\|^\alpha_{L^q_{\varphi_1}(\gamma_1)}\|f\|^{1-\alpha}_{L^r_{\varphi_0}(\gamma_0)} $$
is established between a value of an analytic function in the domain and the weighted integral norms of the restrictions of its boundary values to two measurable subsets $\gamma_1$ and $\gamma_0=\Gamma\setminus\gamma_1$ of the boundary of the domain. It is an analogue of the F. and R. Nevanlinna two-constants theorem. The corresponding problems of optimal recovery of a function from its approximate boundary values on $\gamma_1$ and of the best approximation to the functional of analytic extension of a function from the part of the boundary $\gamma_1$ into the domain are solved.
Bibliography: 35 titles.

Keywords: analytic functions, F. and R. Nevanlinna two-constants theorem, optimal recovery of a functional, best approximation of an unbounded functional by bounded functionals, harmonic measure.

Funding Agency Grant Number
Russian Foundation for Basic Research 15-01-02705-а
Ministry of Education and Science of the Russian Federation 02.A03.21.0006
НШ-9356.2016.1
This research was carried out with the support of the Russian Foundation for Basic Research (grant no. 15-01-02705-a), by the Russian Academic Excellence Project ‘5-100’ (grant no. 02.A03.21.0006) and by the programme of the President of the Russian Federation for state support of leading scientific schools (grant no. НШ-9356.2016.1).


DOI: https://doi.org/10.4213/sm8952

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English version:
Sbornik: Mathematics, 2019, 210:10, 1348–1360

Bibliographic databases:

UDC: 517.538.3+517.544
MSC: Primary 30C85, 65E05; Secondary 30H99
Received: 02.04.2017 and 24.05.2019

Citation: R. R. Akopyan, “An analogue of the two-constants theorem and optimal recovery of analytic functions”, Mat. Sb., 210:10 (2019), 3–16; Sb. Math., 210:10 (2019), 1348–1360

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