The Informative Power of All Possible Linear Functionals and The Mean-Square Error in the Discretization of Solutions of the Dirichlet Problem For the Laplace Equation in the Circle

Abstract

The Dirichlet problem for the Laplace equation in the case of a circle belongs to the classical ones and in various aspects has been the subject of study in various fields of mathematics. Among them are such topics as

- "Boundary properties of analytic functions", in the study of which powerful methods of function theories were created and honed,

- The Banach problem on the existence of a basis for a class of functions consisting of continuous in a closed circle and analytic in,

- Numerical methods, since this problem as a mathematical model describes many real processes.

In this article, we consider the discretization problem of solutions of the Dirichlet problem for the Laplace equation in a circle from finite numerical information obtained from the boundary function as a result of applying all possible linear functionals. The optimal order of discretization error is found and the corresponding optimal operator of discretization is constructed.

The problem of constructing probabilistic measures on functional classes is also considered. Probabilistic measures on the Korobov $E^{r} (0,2\pi)$  and Nikolsky $H_{2}^{r} (0,2\pi )$ classes are introduced. Two-sided estimates of the mean-square error of discretization the solution of the problem by operator $\left(T_{N} f\right)\left(\alpha,\, \theta \right)$ are established.