Optimal recovery of a function holomorphic in a polydisc from its approximate values on a part of the skeleton

R. R. Akopyan

Matematičeskie trudy, Tome 26 (2023) no. 2, pp. 3-29

Résumé

We consider a series of related extremal problems for holomorphic functions in a polydisc $\mathbb{D}^m$, $m\in\mathbb{N}$. The sharp inequality $|f(z)|\le\mathscr{C}\|f\|^{\alpha_1}_{L_{\phi_1}^{p_1}(G_1)}\|f\|^{\alpha_0}_{L_{\phi_0}^{p_0}(G_0)}$, with $0{p_0}$, $p_1\le\infty$ is established between the value of a function holomorphic in $\mathbb{D}^m$ and the norms of its limit values on measurable sets $G_1$ and $G_0$, where $G_0=\mathbb{S}^m\setminus G_1$ and $\mathbb{S}^m$ is the skeleton (the Shilov boundary) of $\mathbb{D}^m$. This result is an analog of the two-constant theorem by the Nevanlinna brothers. We study conditions under which the above inequality provides us with the value of the modulus of continuity of the functional for holomorphic extension of a function on $G_1$ at a prescribed point of the polydisc. In these cases, a solution was obtained of the problem of optimal recovery of a function from approximately given values on a part of the skeleton $G_1$ and the related problem of the best approximation of the functional of the continuation of a function into a polydisk from $G_1$.