On the subspaces of analytic and antianalytic functions

Abstract

The problem of description of those positive weights on the boundary $Gamma$ of a finitely connected domain $Omega$ for which the angle in a weighted $L_2$ space on $Gamma$ between the linear space ${cal R}(Omega)$ of all rational functions on $bar{bf {C}}$ with poles outside of $Clos Omega$ and the linear space ${cal R}(Omega)_-={bar{f}vert fin {cal R}(Omega)}$ of antianalytic rational functions, is a natural analog of the problem solved in a famous Helson-Szeg"o theorem. In this paper we solve more general problem and give a complete description (in terms of necessary and sufficient conditions) of those positive weights $w$ on $Gamma$ for which the sum of the closures in $L_2(Gamma, w)$ of the subspaces ${cal R}(Omega)$ and ${cal R}(Omega)_-$ is closed and their intersection is finite dimensional. The given description is similar to that one in the Helson-Sarason Theorem, i.e. the "modified" weight should satisfy the Muckenhoupt condition.