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.