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

Abstract

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.