Optimal Recovery of a Function Analytic in a Disk from Its Approximately Given Values on a Part of the Boundary

Abstract

We study three related extremal problems in the space H of functions analytic in the unit disk such that their boundary values on a part γ1 of the unit circle Γ belong to the space $$L_{{\psi _1}}^\infty ({\gamma _1})$$ of functions essentially bounded on γ1 with weight ψ1 and their boundary values on the set γ0 = Γ γ1 belong to the space $$L_{{\psi _0}}^\infty ({\gamma _0})$$ with weight ψ0. More exactly, on the class Q of functions from H such that the $$L_{{\psi _0}}^\infty ({\gamma _0})$$ -norm of their boundary values on γ0 does not exceed 1, we solve the problem of optimal recovery of an analytic function on a subset of the unit disk from its boundary values on γ1 specified approximately with respect to the norm of $$L_{{\psi _1}}^\infty ({\gamma _1})$$ . We also study the problem of the optimal choice of the set γ1 for 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 bounded linear operators is investigated.