Optimal Recovery on Classes of Functions Analytic in an Annulus

Let $C_{r,R}$ be an annulus with boundary circles $\gamma_r$ and $\gamma_R$ centered at zero; its inner and outer radii are $r$ and $R$, respectively, $0<r<R<\infty$. 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 bounded linear operators from $L^2(\gamma_r)$ to $C(\gamma_\rho)$ is solved. A method for the optimal recovery of the $m$th derivative on an 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.