On the Best Recovery of a Family of Operators on the Manifold $\mathbb R^n\times \mathbb T^m$

Given a one-parameter family of operators on the manifold $\mathbb R^n\times \mathbb T^m$, we solve the problem of the best recovery of an operator for a given value of the parameter from inaccurate data on the operators for other values of the parameter from a certain compact set. We construct a family of best recovery methods. As a consequence, we obtain families of best recovery methods for the solutions of the heat equation and the Dirichlet problem for a half-space.