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.