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.

给定一族连续线性算子 \( T(t):L_{2}(𝕉^{d})\to L_{2}(𝕉^{d}) \)，其中 \( 0\leq t<\infty \) ，我们通过\( T(t) \)值的近似信息来考虑整个空间上 \( T(\tau) \)值的最优恢复 ，其中\( t \ )运行在一个紧集 \( K\subset 𝕉_{+} \)和\( \tau\notin K \)上。我们找到了一系列用于恢复T(tau)值的最佳方法。这些方法中的每一种都使用\( K \)中不超过两个点的近似测量值，并且线性依赖于这些测量值。作为推论，我们提供了一些最优方法族，用于从其他时间间隔上的不准确测量中恢复给定时刻的热方程的解，以及通过对其他超平面。从指示的信息中最佳恢复 T(tau)的值简化为寻找具有连续体许多不等式约束的最大值的极值问题，即找到在这些限制下实现功能最大化。这个相当复杂的任务简化为在\( \sigma \)中勒贝格可测集的代数上所有有限实测度的向量空间上线性规划的无限维问题\( 𝕉^{d} \)。这个问题可以通过卡鲁什-库恩-塔克定理的某种推广来解决，其意义与原问题的意义是一致的。