Optimal recovery of a family of operators from inaccurate measurements on a compact

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.