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

Given a one-parameter family of continuous linear operators $ T(t):L_{2}(𝕉^{d})\to L_{2}(𝕉^{d}) $ , with $ 0\leq t<\infty $ , we consider the optimal recovery of the values of $ T(\tau) $ on the whole space by approximate information on the values of $ T(t) $ , where $ t $ runs over a compact set $ K\subset 𝕉_{+} $ and $ \tau\notin K $ . We find a family of optimal methods for recovering the values of $ T(\tau) $ . Each of these methods uses approximate measurements at no more than two points in $ K $ 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 $ T(\tau) $ 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 $ \sigma $ -algebra of Lebesgue measurable sets in $ 𝕉^{d} $ . 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.