An analogue of the two-constants theorem and optimal recovery of analytic functions

Abstract

Several related extremal problems for analytic functions in a simply connected domain with rectifiable Jordan boundary are treated. The sharp inequality is established between a value of an analytic function in the domain and the weighted integral norms of the restrictions of its boundary values to two measurable subsets and of the boundary of the domain. It is an analogue of the F. and R. Nevanlinna two-constants theorem. The corresponding problems of optimal recovery of a function from its approximate boundary values on and of the best approximation to the functional of analytic extension of a function from the part of the boundary into the domain are solved. Bibliography: 35 titles.