On the best recovery of a family of operators on a class of functions according to their inaccurately specified spectrum

The paper considers a one-parameter family of linear continuous operators in $L_2 (\mathbb{R}^d)$ and poses the problem of optimal reconstruction of the operator for a given value of a parameter on a class of functions whose Fourier transforms are square integrable with power weight (spaces of such a structure play an important role in questions of embedding function spaces and the theory of differential equations) using the following information: about each function from this class is knows (generally speaking, approximately) its Fourier transform on some measurable subset of $\mathbb{R}^d$. A family of optimal methods for restoring operators for each parameter value is constructed. Optimal methods do not use all the available information about the Fourier transform of functions from the class, but use only information about the Fourier transform of a function in a ball centered at zero of maximum radius, which has the property that its measure is equal to the measure of its intersection with the set, where is known (exactly or approximately) Fourier transform of the function. As a consequence, the following results were obtained: a family of optimal methods for recovery the solution of the heat equation in $\mathbb{R}^d$ at a given time, provided that the initial function belongs to the specified class and its Fourier transform is known exactly or approximately on some measurable set, and also a family of optimal methods for reconstructing the solution of the Dirichlet problem for a half-space on a hyperplane from the Fourier transform of a boundary function belonging to the specified class, which is known exactly or approximately on some measurable set in $\mathbb{R}^d$.