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. |
Keywords
Hardy spaces, Riemann surface,Muckenhoupt condition, Toeplitz operators, Fredholm operators
Math Review Classification
Primary 46J15, 30D55, 30F99, 47B25
; Secondary 30F15, 30F30, 14K20, 46E22, 30H05
Last Updated
8 September 1998
Length
22 pages
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