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

This article is cited in 1 scientific paper (total in 1 paper)

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

Full text: PDF file (317 kB)
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Bibliographic databases:

UDC: 517.518+517.983
MSC: 26D10, 47A58
Received: 01.09.2018
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
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    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  crossref  zmath  isi  scopus
  • Trudy Instituta Matematiki i Mekhaniki UrO RAN
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