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 optimalrecovery of the values of $ T(\tau) $ on the whole space by approximate informationon 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 thevalues of $ T(\tau) $ .Each of these methods uses approximate measurementsat no more than two points in $ K $ anddepends linearly on these measurements.As a corollary, we provide some families of optimal methodsfor recovering the solution of the heat equationat a given moment of time frominaccurate measurements on other time intervals and forsolving the Dirichlet problem fora half-space on a hyperplane by inaccuratemeasurements on other hyperplanes.The optimal recovery of the values of $ T(\tau) $ from the indicatedinformation reduces to finding the value ofan extremal problem for the maximum withcontinuum many inequality-type constraints, i.e.,to finding the exact upper bound of themaximized functional under these constraints.This rather complicated task reducesto the infinite-dimensional problem of linearprogramming on the vector space of allfinite real measures on the $ \sigma $ -algebra ofLebesgue measurable sets in $ ^{d} $ .This problem can be solved by some generalization ofthe Karush–Kuhn–Tucker theorem,and its significance coincides with the significanceof the original problem.
