O.V. Akopyan, R.R. Akopyan. Optimal recovery on classes of functions analytic in a annulus ... P. 7-23

Let $C_{r,R}$ be a annulus with boundary circles $\gamma_r$ and $\gamma_R$ centered at zero; its inner and outer radii are $r$ and $R$, respectively. On the class of functions analytic in the annulus $C_{r,R}$ with finite $L^2$-norms of the angular limits on the circle $\gamma_r$ and of the $n$th derivatives (of the functions themselves for $n=0$) on the circle $\gamma_R$, we study interconnected extremal problems for the operator $\psi_{\rho}^m$ that takes the boundary values of a function on $\gamma_r$ to its restriction (for $m=0$) or the restriction of its $m$th derivative (for $m>0$) to an intermediate circle $\gamma_\rho$, $r<\rho<R$. The problem of the best approximation of $\psi_{\rho}^m$ by linear bounded operators from $L^2(\gamma_r)$ to $C(\gamma_\rho)$ is solved. A method for the optimal recovery of the $m$th derivative on a intermediate circle $\gamma_\rho$ from $L^2$-approximately given values of the function on the boundary circle $\gamma_r$ is proposed and its error is found. The Hadamard—Kolmogorov exact inequality, which estimates the uniform norm of the $m$th derivative on an intermediate circle $\gamma_\rho$ in terms of the $L^2$-norms of the limit boundary values of the function and the $n$th derivative on the circles $\gamma_r$ and $\gamma_R$, is derived.

Keywords: analytic functions, Hadamard three-circle theorem, Kolmogorov's inequality, optimal recovery

Received February 10, 2023

Revised February 27, 2023

Accepted February 27, 2023

Olga Vladimirovna Akopyan, Ural Federal University, Yekaterinburg, 620000 Russia, e-mail: olga_akopjan@rambler.ru

Roman Razmikovich Akopyan, Dr. Phys.-Math. Sci., Krasovskii Institute of Mathematics and Mechanics of the Ural Branch of the Russian Academy of Sciences, Yekaterinburg, 620108 Russia, e-mail: RRAkopyan@mephi.ru

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Cite this article as: O.V. Akopyan, R.R. Akopyan. Optimal recovery on classes of functions analytic in a annulus, Trudy Instituta Matematiki i Mekhaniki UrO RAN, 2023, vol. 29, no. 1, pp. 7–23; Proceedings of the Steklov Institute of Mathematics  (Suppl.), 2023, Vol. 321, Suppl. 1, pp. S4–S19.