Abstract:
This paper discusses Stechkin's problem on the best approximation of a linear unbounded operator by bounded linear operators and related extremal problems. The main attention is paid to the approximation of differentiation operators in Lebesgue spaces on the axis and to the operator of the continuation of an analytic function to a domain from a part of the boundary of the domain. This is a review paper based on the materials of the authors' lecture on September 14, 2020, at the X Internet video-conference “Day of Mathematics and Mechanics” of four institutes of the Russian Academy of Sciences: Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of RAS (Yekaterinburg), Sobolev Institute of the Siberian Branch of RAS (Novosibirsk), Steklov Mathematical Institute (Moscow), and the St. Petersburg Department of the Steklov Mathematical Institute. The lecture of the authors was dedicated to the 100th anniversary of the birth of Sergei Borisovich Stechkin. The problem of the best approximation of a linear unbounded operator by bounded ones is one of his legacies. We tried to at least partially reflect the new results, methods, and statements that appeared in this topic after the publication of the review papers (Arestov, Gabushin, 1995–1996). The material on this topic is wide; the selection of the material for the lecture and paper is the responsibility of the authors.
This work was performed as a part of the research conducted in the Ural Mathematical Center and also 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).
Citation:
V. V. Arestov, R. R. Akopyan, “Stechkin's problem on the best approximation of an unbounded operator by bounded ones and related problems”, Trudy Inst. Mat. i Mekh. UrO RAN, 26, no. 4, 2020, 7–31
\Bibitem{AreAko20}
\by V.~V.~Arestov, R.~R.~Akopyan
\paper Stechkin's problem on the best approximation of an unbounded operator by bounded ones and related problems
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2020
\vol 26
\issue 4
\pages 7--31
\mathnet{http://mi.mathnet.ru/timm1763}
\crossref{https://doi.org/10.21538/0134-4889-2020-26-4-7-31}
\elib{https://elibrary.ru/item.asp?id=44314654}
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This publication is cited in the following 7 articles:
R. R. Akopyan, V. V. Arestov, V. G. Timofeev, “Stechkin's Problem on the Approximation of the Differentiation Operator in the Uniform Norm on the Half-Line”, Math. Notes, 115:6 (2024), 853–867
V. V. Arestov, “Variant zadachi Stechkina o nailuchshem priblizhenii operatora differentsirovaniya drobnogo poryadka na osi”, Tr. IMM UrO RAN, 30, no. 4, 2024, 37–54
V. V. Arestov, “A Variant of Stechkin's Problem on the Best Approximation of a Fractional Order Differentiation Operator on the Axis”, Proc. Steklov Inst. Math., 327:S1 (2024), S10
Vitalii V. Arestov, “Approximation of differentiation operators by bounded linear operators in lebesgue spaces on the axis and related problems in the spaces of (p,q)-multipliers and their predual spaces”, Ural Math. J., 9:2 (2023), 4–27
A. S. Demidov, A. S. Kochurov, “Calculation of nth derivative with minimum error based on function’s measurement”, Comput. Math. Math. Phys., 63:9 (2023), 1571–1579
R. R. Akopyan, “Optimal recovery of a function holomorphic in a polydisc from its approximate values on a part of the skeleton”, Siberian Adv. Math., 33:4 (2023), 261–277
R. R. Akopyan, “Nailuchshee priblizhenie operatorov differentsirovaniya na klasse Soboleva analiticheskikh v polose funktsii”, Sib. elektron. matem. izv., 18:2 (2021), 1286–1298
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