Optimal recovery of functions from anisotropic Sobolev classes on a power – logarithmic scale

Abstract

In this paper, within the framework of the C (N) D - formulation of the recovery problem, the problem of optimal recovery of functions from anisotropic Sobolev classes in a power-logarithmic scale in the metric $L^{q} \, (2\le q\le \infty )$ is solved. Namely, in the case when the values $l_{N}^{\eqref{GrindEQ__1_}} (f),...,l_{N}^{(N)} (f)$ of linear functionals $l_{N}^{\eqref{GrindEQ__1_}} ,...,l_{N}^{(N)} $ defined on the considered functional class are used as numerical information about a function, firstly, the exact order of the recovery error is established, and secondly, a specific computing unit $\bar{\varphi }_{N} \left(\bar{l}_{N}^{(1)} (f),...,\bar{l}_{N}^{(N)} (f);\, \cdot \right)$ is indicated that implements the established exact order.