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; Proc. Steklov Inst. Math. (Suppl.), 308, suppl. 1 (2020), S9

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Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2018, Volume 24, Number 4, Pages 34–56
DOI: https://doi.org/10.21538/0134-4889-2018-24-4-34-56 (Mi timm1573)

This article is cited in 10 scientific papers (total in 10 papers)

Best Uniform Approximation of the Differentiation Operator by Operators Bounded in the Space $L_2$

V. V. Arestov^ab

^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

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DOI: https://doi.org/10.21538/0134-4889-2018-24-4-34-56

Abstract: We give a solution of the problem on the best uniform approximation on the number 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).

Received: 01.09.2018
Revised: 08.11.2018
Accepted: 12.11.2018

English version:
Proceedings of the Steklov Institute of Mathematics (Supplementary issues), 2020, Volume 308, Issue 1, Pages S9–S30
DOI: https://doi.org/10.1134/S0081543820020029

Bibliographic databases:

Document Type: Article

UDC: 517.518+517.983

MSC: 26D10, 47A58

Language: Russian

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; Proc. Steklov Inst. Math. (Suppl.), 308, suppl. 1 (2020), S9–S30

Citation in format AMSBIB

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\jour Proc. Steklov Inst. Math. (Suppl.)

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Linking options:

https://www.mathnet.ru/eng/timm1573

https://www.mathnet.ru/eng/timm/v24/i4/p34

This publication is cited in the following 10 articles:

K. Yu. Osipenko, “On the construction of families of optimal recovery methods for linear operators”, Izv. Math., 88:1 (2024), 92–113
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
V. V. Arestov, “Predual Spaces for the Space of (p, q)-Multipliers and Their Application in Stechkin's Problem on Approximation of Differentiation Operators”, Anal Math, 49:1 (2023), 43
K. Yu. Osipenko, “Optimal recovery in weighted spaces with homogeneous weights”, Sb. Math., 213:3 (2022), 385–411
I. V. Boykov, N. P. Krivulin, “An Approximate Method for Recovering Input Signals of Measurement Transducers”, Meas Tech, 64:12 (2022), 943
P. G. Surkov, “Approximate calculation of the Caputo-type fractional derivative from inaccurate data. Dynamical approach”, Fract. Calc. Appl. Anal., 24:3 (2021), 895–922
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
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
V. Arestov, “Uniform approximation of differentiation operators by bounded linear operators in the spacel(r)”, Anal. Math., 46:3 (2020), 425–445
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

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles

Trudy Instituta Matematiki i Mekhaniki UrO RAN

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