Optimal Recovery of a Family of Operators from Inaccurate Measurements on a Compact Set

Given a one-parameter family of continuous linear operators , with , we consider the optimal recovery of the values of on the whole space by approximate information on the values of , where runs over a compact set and . We find a family of optimal methods for recovering the values of . Each of these...

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Vydáno v:	Siberian mathematical journal Ročník 65; číslo 2; s. 495 - 504
Hlavní autor:	Sivkova, E. O.
Médium:	Journal Article
Jazyk:	angličtina
Vydáno:	Moscow Pleiades Publishing 01.03.2024 Springer Nature B.V
Témata:	Accuracy Constraints Dirichlet problem Fourier transforms Half spaces Hyperplanes Inequality Linear operators Linear programming Mathematics Mathematics and Statistics Methods Operators (mathematics) Recovery Thermodynamics Upper bounds Vector spaces heat equation optimal method 517.9 extremal problem Fourier transform Dirichlet problem optimal recovery
ISSN:	0037-4466, 1573-9260
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Abstract	Given a one-parameter family of continuous linear operators , with , we consider the optimal recovery of the values of on the whole space by approximate information on the values of , where runs over a compact set and . We find a family of optimal methods for recovering the values of . Each of these methods uses approximate measurements at no more than two points in and depends linearly on these measurements. As a corollary, we provide some families of optimal methods for recovering the solution of the heat equation at a given moment of time from inaccurate measurements on other time intervals and for solving the Dirichlet problem for a half-space on a hyperplane by inaccurate measurements on other hyperplanes. The optimal recovery of the values of from the indicated information reduces to finding the value of an extremal problem for the maximum with continuum many inequality-type constraints, i.e., to finding the exact upper bound of the maximized functional under these constraints. This rather complicated task reduces to the infinite-dimensional problem of linear programming on the vector space of all finite real measures on the -algebra of Lebesgue measurable sets in . This problem can be solved by some generalization of the Karush–Kuhn–Tucker theorem, and its significance coincides with the significance of the original problem.
AbstractList	Given a one-parameter family of continuous linear operators , with , we consider the optimal recovery of the values of on the whole space by approximate information on the values of , where runs over a compact set and . We find a family of optimal methods for recovering the values of . Each of these methods uses approximate measurements at no more than two points in and depends linearly on these measurements. As a corollary, we provide some families of optimal methods for recovering the solution of the heat equation at a given moment of time from inaccurate measurements on other time intervals and for solving the Dirichlet problem for a half-space on a hyperplane by inaccurate measurements on other hyperplanes. The optimal recovery of the values of from the indicated information reduces to finding the value of an extremal problem for the maximum with continuum many inequality-type constraints, i.e., to finding the exact upper bound of the maximized functional under these constraints. This rather complicated task reduces to the infinite-dimensional problem of linear programming on the vector space of all finite real measures on the -algebra of Lebesgue measurable sets in . This problem can be solved by some generalization of the Karush–Kuhn–Tucker theorem, and its significance coincides with the significance of the original problem. Given a one-parameter family of continuous linear operators , with , we consider the optimal recovery of the values of on the whole space by approximate information on the values of , where runs over a compact set and . We find a family of optimal methods for recovering the values of . Each of these methods uses approximate measurements at no more than two points in and depends linearly on these measurements. As a corollary, we provide some families of optimal methods for recovering the solution of the heat equation at a given moment of time from inaccurate measurements on other time intervals and for solving the Dirichlet problem for a half-space on a hyperplane by inaccurate measurements on other hyperplanes. The optimal recovery of the values of from the indicated information reduces to finding the value of an extremal problem for the maximum with continuum many inequality-type constraints, i.e., to finding the exact upper bound of the maximized functional under these constraints. This rather complicated task reduces to the infinite-dimensional problem of linear programming on the vector space of all finite real measures on the -algebra of Lebesgue measurable sets in . This problem can be solved by some generalization of the Karush–Kuhn–Tucker theorem, and its significance coincides with the significance of the original problem.
Author	Sivkova, E. O.
Author_xml	– sequence: 1 givenname: E. O. surname: Sivkova fullname: Sivkova, E. O. email: e.o.sivkova@mail.ru, sivkova_elena@inbox.ru organization: Southern Mathematical Institute, Moscow Power Engineering Institute
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Cites_doi	10.1007/978-1-4684-2388-4_1 10.1023/A:1026084617039 10.32523/2077-9879-2019-10-4-75-84 10.1070/SM2009v200n05ABEH004014 10.1007/s10688-010-0029-7 10.1134/S0965542520100036 10.1007/BFb0075157 10.1137/0716007 10.1090/mmono/222
ContentType	Journal Article
Copyright	Pleiades Publishing, Ltd. 2024. Russian Text © The Author(s), 2023, published in Vladikavkazskii Matematicheskii Zhurnal, 2023, Vol. 25, No. 2, pp. 124–135. Pleiades Publishing, Ltd. 2024.
Copyright_xml	– notice: Pleiades Publishing, Ltd. 2024. Russian Text © The Author(s), 2023, published in Vladikavkazskii Matematicheskii Zhurnal, 2023, Vol. 25, No. 2, pp. 124–135. – notice: Pleiades Publishing, Ltd. 2024.
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References_xml	– reference: Micchelli C.A. and Rivlin T.J., “Lectures on optimal recovery,” in: Lecture Notes in Mathematics; vol. 1129, Springer, Berlin (1985), 21–93. – reference: TraubJFWoźniakowskiHA General Theory of Optimal Algorithms1980New YorkAcademic – reference: Magaril-IlyaevGGOsipenkoKYOptimal recovery of functions and their derivatives from inaccurate information about the spectrum and inequalities for derivativesFunct. Anal. Appl.2003373203214202041410.1023/A:1026084617039 – reference: Magaril-IlyaevGGOsipenkoKYOptimal recovery of the solution of the heat equation from inaccurate dataSb. Math.20092005665682254122110.1070/SM2009v200n05ABEH004014 – reference: AbramovaEVMagaril-Il’yaevGGSivkovaEOBest recovery of the solution of the Dirichlet problem in a half-space from inaccurate dataComp. Math. Math. Phys.2020601016561665418177110.1134/S0965542520100036 – reference: Magaril-IlyaevGGTikhomirovVMConvex Analysis: Theory and Applications2003ProvidenceAmer. Math. Soc.10.1090/mmono/222 – reference: SteinEWeissGIntroduction to Harmonic Analysis on Euclidean Spaces1970PrincetonPrinceton UniversityPrinceton Mathematical Series; vol. 1 – reference: MelkmanAAMicchelliCAOptimal estimation of linear operators in Hilbert spaces from inaccurate dataSIAM J. Numer. Anal.19791618710551868610.1137/0716007 – reference: Micchelli C.A. and Rivlin T.J., “A survey of optimal recovery,” in: Optimal Estimation in Approximation Theory, Plenum, New York (1977), 1–54. – reference: Magaril-Il’yaevGGSivkovaEOOptimal recovery of the semi-group operators from inaccurate dataEurasian Math. J.20191047584406946410.32523/2077-9879-2019-10-4-75-84 – reference: SmolyakSAOn Optimal Recovery of Functions and Functionals over Them1965MoscowMoscow UniversityExtended Abstract of Cand. Sci. Dissertation; Language: Russian – reference: Magaril-IlyaevGGOsipenkoKYOn optimal harmonic synthesis from inaccurate spectral dataFunct. Anal. Appl.2010443223225276051810.1007/s10688-010-0029-7 – ident: 1424_CR4 doi: 10.1007/978-1-4684-2388-4_1 – volume: 37 start-page: 203 issue: 3 year: 2003 ident: 1424_CR8 publication-title: Funct. Anal. Appl. doi: 10.1023/A:1026084617039 – volume: 10 start-page: 75 issue: 4 year: 2019 ident: 1424_CR11 publication-title: Eurasian Math. J. doi: 10.32523/2077-9879-2019-10-4-75-84 – volume-title: On Optimal Recovery of Functions and Functionals over Them year: 1965 ident: 1424_CR3 – volume: 200 start-page: 665 issue: 5 year: 2009 ident: 1424_CR9 publication-title: Sb. Math. doi: 10.1070/SM2009v200n05ABEH004014 – volume: 44 start-page: 223 issue: 3 year: 2010 ident: 1424_CR10 publication-title: Funct. Anal. Appl. doi: 10.1007/s10688-010-0029-7 – volume: 60 start-page: 1656 issue: 10 year: 2020 ident: 1424_CR12 publication-title: Comp. Math. Math. Phys. doi: 10.1134/S0965542520100036 – ident: 1424_CR6 doi: 10.1007/BFb0075157 – volume-title: A General Theory of Optimal Algorithms year: 1980 ident: 1424_CR7 – volume: 16 start-page: 87 issue: 1 year: 1979 ident: 1424_CR5 publication-title: SIAM J. Numer. Anal. doi: 10.1137/0716007 – volume-title: Introduction to Harmonic Analysis on Euclidean Spaces year: 1970 ident: 1424_CR2 – volume-title: Convex Analysis: Theory and Applications year: 2003 ident: 1424_CR1 doi: 10.1090/mmono/222
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SubjectTerms	Accuracy Constraints Dirichlet problem Fourier transforms Half spaces Hyperplanes Inequality Linear operators Linear programming Mathematics Mathematics and Statistics Methods Operators (mathematics) Recovery Thermodynamics Upper bounds Vector spaces
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Title	Optimal Recovery of a Family of Operators from Inaccurate Measurements on a Compact Set
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