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 Trudy Inst. Mat. i Mekh. UrO RAN, 2018, Volume 24, Number 4, Pages 34–56 (Mi timm1573)

Best uniform approximation of the differentiation operator by operators bounded in the space $L_2$

V. V. Arestovab

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: We give a solution of the problem on the best uniform approximation on the numerical axis of the first-order differentiation operator on the class of functions with bounded second derivative by linear operators bounded in the space $L_2$. This is one of the few cases of the exact solution of the problem on the approximation of the differentiation operator in some space with the use of approximating operators that are bounded in another space. We obtain a related exact inequality between the uniform norm of the derivative of a function, the variation of the Fourier transform of the function, and the $L_\infty$-norm of its second derivative. This inequality can be regarded as a nonclassical variant of the Hadamard-Kolmogorov inequality.

Keywords: Stechkin problem, differentiation operator, Hadamard-Kolmogorov inequality.

 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/10.21538/0134-4889-2018-24-4-34-56

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Bibliographic databases:

UDC: 517.518+517.983
MSC: 26D10, 47A58
Revised: 08.11.2018
Accepted:12.11.2018

Citation: V. V. Arestov, “Best uniform approximation of the differentiation operator by operators bounded in the space $L_2$”, Trudy Inst. Mat. i Mekh. UrO RAN, 24, no. 4, 2018, 34–56

Citation in format AMSBIB
\Bibitem{Are18}
\by V.~V.~Arestov
\paper Best uniform approximation of the differentiation operator by operators bounded in the space $L_2$
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2018
\vol 24
\issue 4
\pages 34--56
\mathnet{http://mi.mathnet.ru/timm1573}
\crossref{https://doi.org/10.21538/0134-4889-2018-24-4-34-56}
\elib{https://elibrary.ru/item.asp?id=36517697}

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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. V. V. Arestov, “Best approximation of a differentiation operator on the set of smooth functions with exactly or approximately given Fourier transform”, Mathematical Optimization Theory and Operations Research, Lecture Notes in Computer Science, 11548, eds. M. Khachay, Y. Kochetov, P. Pardalos, Springer, 2019, 434–448
2. V. V. Arestov, R. R. Akopyan, “Zadacha Stechkina o nailuchshem priblizhenii neogranichennogo operatora ogranichennymi i rodstvennye ei zadachi”, Tr. IMM UrO RAN, 26, no. 4, 2020, 7–31
3. R. R. Akopyan, “Optimal recovery of a derivative of an analytic function from values of the function given with an error on a part of the boundary. II”, Anal. Math., 46:3 (2020), 409–424
4. V. Arestov, “Uniform approximation of differentiation operators by bounded linear operators in the spacel(r)”, Anal. Math., 46:3 (2020), 425–445
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