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

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/b9762-8415-3252-n Optimal Recovery of a Family of Operators from Inaccurate Measurements on a Compact Sivkova, E. O. Vladikavkaz Mathematical Journal 2023. Vol. 25. Issue 2. Abstract: For a one-parameter family of linear continuous operators \(T(t)\colon L_2(\mathbb R^d)\to L_2(\mathbb R^d)\), \(0\le t<\infty\), we consider the problem of optimal recovery of the values of the operator \(T ( \tau)\) on the whole space by approximate information about the values of the operators \(T(t)\), where \(t\) runs through some compact set \(K\subset \mathbb R_ + \) and \(\tau\notin K\). A family of optimal methods for recovering the values of the operator \(T(\tau)\) is found. Each of these methods uses approximate measurements at no more than two points from \(K\) and depends linearly on these measurements. As a consequence, families of optimal methods are found for restoring the solution of the heat equation at a given moment of time from its inaccurate measurements on other time intervals and for solving the Dirichlet problem for a half-space on a hyperplane from its inaccurate measurements on other hyperplanes. The problem of optimal recovery of the values of the operator \(T(\tau)\) from the indicated information is reduced to finding the value of some extremal problem for the maximum with a continuum of inequality-type constraints, i.e., to finding the least upper bound of the a functional under these constraints. This rather complicated task is reduced, in its turn, to the infinite-dimensional problem of linear programming on the vector space of all finite real measures on the \(\sigma\)-algebra of Lebesgue measurable sets in \(\mathbb R^d\). This problem can be solved using some generalization of the Karush-Kuhn-Tucker theorem, and its the value coincides with the value of the original problem. Keywords: optimal recovery, optimal method, extremal problem, Fourier transform, heat equation, Dirichlet problem. Language: Russian Download the full text For citation: Sivkova, E. O. Optimal Recovery of a Family of Operators from Inaccurate Measurements on a Compact // Vladikavkaz Math. J., 2023, vol. 25, no. 2, pp.124-135 (in Russian). DOI 10.46698/b9762-8415-3252-n + References 1. Magaril-Il'yaev, G.G. and Tikhomirov, V.M. Vypuklyy Analiz i ego Prilozheniy [Convex Analysis and its Applications], Moscow, URSS, 2020, 176p. (in Russian). 2. Stein, E. M. and Weiss, G. Introduction to Fourier Analysis on Euclidean Spaces, Princeton Mathematical Series, vol. 1, New Jersey, Princeton University Press, 1971. 3. Smolyak, S.A. Ob optimal'nom vosstanovlenii funksii i funksionalov ot nikh. Diss. [On Optimal Recovery of Functions and Functionals from them. PhD], Moscow, MSU, 1965 (in Russian). 4. Micchelli, C.A. and Rivlin, T.J. A Survey of Optimal Recovery, Optimal Estimation in Approximation Theory, C.A.Micchelli and T.J.Rivlin (eds.), New York, Plenum Press, 1977, pp.1-54. 5. Melkman, A.A. and Micchelli, C.A. Optimal Estimation of Linear Operators in Hilbert Spaces from Inaccurate Data, SIAM Journal on Numerical Analysis, 1979, vol. 16, no. 1, pp. 87-105. DOI: 10.1137/0716007. 6. Micchelli, C.A. and Rivlin,T.J. Lectures on Optimal Recovery, Numerical Analysis Lancaster, Turner P.R. (ed.), Lecture Notes in Mathematics, vol.1129, Berlin, Springer-Verlag, 1985, pp.21-93. DOI: 10.1007/BFb0075157. 7. Traub, J.F. and Wozniakowski, H. A General Theory of Optimal Algorithms, New York, Academic Press, 1980. 8. Magaril-Il'yaev, G.G. and Osipenko, K.Yu. Optimal Recovery of Functions and Their Derivatives from Inaccurate Information about the Spectrum and Inequalities for Derivatives, Functional Analysis and Its Applications, 2003, vol. 37, no. 3, pp. 203-214. DOI: 10.1023/A:1026084617039. 9. Magaril-Il'yaev, G.G. and Osipenko, K.Yu. Optimal Recovery of the Solution of the Heat Equation from Inaccurate Data, Sbornik: Mathematics, 2009, vol. 200, no. 5, pp. 665-682. DOI: 10.1070/SM2009v200n05ABEH004014. 10. Magaril-Il'yaev, G.G. and Osipenko, K.Yu. On Optimal Harmonic Synthesis from Inaccurate Spectral Data, Functional Analysis and Its Applications, 2010, vol. 44, no. 3, pp. 223-225. DOI: 10.1007/s10688-010-0029-7. 11. Magaril-Il'yaev, G.G. and Sivkova, E.O. Optimal Recovery of the Semi-Group Operators from Inaccurate Data, Eurasian Mathematical Journal, 2019, vol. 10, no. 4, pp. 75-84. DOI: 10.32523/2077-9879-2019-10-4-75-84. 12. Abramova, E.V., Magaril-Il'yaev, G.G. and Sivkova, E.O. Best Recovery of the Solution of the Dirichlet Problem in a Half-Space from Inaccurate Data, Computational Mathematics and Mathematical Physics, 2020, vol. 60, no. 10, pp. 1656-1665. DOI: 10.1134/S0965542520100036. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). The use of metadata of the scientific article, including the title, abstract, author information, references, identifiers, and other bibliographic description elements for subsequent unrestricted use, is carried out under the terms of the CC BY or CC0 open licenses. ← Contents of issue


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