Optimal recovery of a function analytic in a disk from approximately given values on a part of the boundary

We study three related extremal problems in the space $\mathcal{H}$ of functions analytic in the unit disk such that their boundary values on a part $\gamma_1$ of the unit circle $\Gamma$ belong to the space $L^\infty_{\psi_1}(\gamma_1)$ of functions essentially bounded on $\gamma_1$ with weight $\psi_1$ and their boundary values on the set $\gamma_0=\Gamma\setminus\gamma_1$ belong to the space $L^\infty_{\psi_0}(\gamma_0)$ with weight $\psi_0$. More exactly, on the class $Q$ of functions from $\mathcal{H}$ such that the norm $L^\infty_{\psi_0}(\gamma_0)$ of their boundary values on $\gamma_0$ does not exceed one, we solve the problem of optimal recovery of an analytic function on a subset of the unit disk from its boundary values on $\gamma_1$ specified approximately with respect to the norm $L^\infty_{\psi_1}(\gamma_1)$. We also study the problem of the optimal choice of the set $\gamma_1$ under a given fixed value of its measure. The problem of the best approximation of the operator of analytic continuation from a part of the boundary by linear bounded operators is investigated.