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Optimal Recovery of a Family of Operators from Inaccurate Measurements on a Compact
Sivkova, E. O.
Vladikavkaz Mathematical Journal 2023. Vol. 25. Issue 2.
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
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