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Optimal recovery of a family of operators from inaccurate measurements on a compact
E. O. Sivkovaab a Southern Mathematical Institute VSC RAS,
53 Vatutina St., Vladikavkaz 362025, Russia
b NRU “Moscow Power Engineering Institute”,
14 Krasnokazarmennaya St., Moscow 111250, Russia
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.
Key words:
optimal recovery, optimal method, extremal problem, Fourier transform, heat equation, Dirichlet problem.
UDC:
517.9
MSC: 34K29,
65K10,
90C25 Received: 15.07.2022
DOI:
10.46698/b9762-8415-3252-n
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