Description of Helson-Szegő Measures in Terms of the Schur Parameter Sequences of Associated Schur Functions

Abstract

Let be a probability measure on the Borelian -algebra of the unit circle. Then we associate a Schur function in the unit disk with and give characterizations of the case that is a Helson-Szegő measure in terms of the sequence of Schur parameters of . Furthermore, we state some connections of these characterizations with the backward shift. Mathematics Subject Classification (2000). Primary 30E05, 47A57.

The S-recurrence of Schur Parameters of Non-inner Rational Schur Functions

“…Note that this approach is based on the description in terms of the Schur parameters of the relative position of the largest shift and the largest coshift in a completely nonunitary contraction (see [19,Section 3]). It should be mentioned that these results received a further development in [8,[21][22][23][24].…”

“…This approach is based on the description in terms of the Schur parameters of the relative position of the largest shift and the largest coshift in a completely nonunitary contraction. It should be mentioned that these results received a further development in [8,[21][22][23][24].This paper is aimed to give a survey about essential results on this direction. The main object in the approach is based on considering a Schur function as characteristic function of a contraction (see Section 1.2).…”

Scattering Theory for CMV Matrices: Uniqueness, Helson–Szegő and Strong Szegő Theorems

“…Theorem 6.6 is contained in a preliminary version of the paper (see [16,Theorem 4.5,Proposition 4.7]). It was recently observed in [11,Theorem 6.3], that operator L has a multiplicative structure. This observation gives hope that the boundedness condition on L −1 may be restated as a constructive condition on the Verblunsky coefficients via convergence of infinite products (series).…”