On the generalization of the theorem of Helson and Szegö (Q1064492)

Let \({\mathcal P}\) be the set of trigonometric polynomials, \({\mathcal P}_+\) be the set of analytic polynomials and \({\mathcal P}^ n_-=\{p\in {\mathcal P}:\hat p(k)=0\) for \(k>n\}\). Let \(P_+\) be the projection of \({\mathcal P}\) onto \({\mathcal P}_+\) and \(P_-=Id-P_+\). For a positive Borel measure \(\mu\) on the unit circle T and complex valued functions \(\alpha\),\(\beta\) let \(R^ n_{\mu}(\alpha,\beta)\) be the set of positive Borel measure \(\nu\) such that \(\int_{T}| (\alpha P_++\beta P_-)f|^ 2d\nu \leq \int_{T}| f|^ 2d\mu.\) The first purpose of this paper is to apply a method of a proof of Helson-Szegö theorem on a weighted norm inequality of Hilbert transform to get some inequalities given by \textit{R. Arocena}, \textit{M. Cotlar} and \textit{C. Sadosky} [see e.g. Adv. Math., Suppl. Stud. 7A, 95-128 (1981; Zbl 0494.46051)]. The author gives a characterization of the set \(R^ n_{\mu}(\alpha,\beta)\) and some applications to weight problems.