Higher-order Szegő theorems with two singular points
This paper is a contribution to the theory of orthogonal polynomials on the unit circle, \(\partial\mathbb D\subset\mathbb C\). Let \(d\mu\) be a non-trivial probability measure on \(\partial\mathbb D\) of the form \(d\mu=w(\theta)\frac{d\theta}{2\pi} + d\mu_s\) with Verblunsky coefficients, \(\{\alpha_j\}_{j=0}^\infty\), where \(d\mu_s\) is singular with respect to Lebesgue measure \(d\theta\) on \(\partial\mathbb D\). The goal in this paper is to analyze two singularities or a single double singularity. The authors prove for \(\theta_1\neq\theta_2\) in \([0,2\pi)\) that \[ \int[1 - \cos(\theta -\theta_1)][1 - \cos(\theta -\theta_2)] \log\left(w(\theta)\right) \frac{d\theta}{2\pi} > - \infty \] \[ \Longleftrightarrow\sum_{j=0}^\infty\left| \left\{ (\delta-e^{-i\theta_2})(\delta-e^{-i\theta_1})\alpha \right\}_j\right| ^2+| \alpha| ^4<\infty, \] where \(\delta\) is the left shift operator \((\delta\beta)_j =\beta_{j+1}\). The authors also prove a result for \(\theta_1=\theta_2\): \[ \int(1 - \cos\theta)^2\log\left(w(\theta)\right)\frac{d\theta}{2\pi} > - \infty \] \[ \Longleftrightarrow\sum_{j=o}^\infty| \alpha_{j+2} - 2\alpha_{j+1} +\alpha_j| ^2 + | \alpha_j| ^6 < \infty. \] One can replace \(\cos(\theta)\) by \(\cos(\theta-\theta_1)\) if \(\alpha_{j+2} - 2\alpha_{j+1} +\alpha_j\) is replaced by \(\left\{(\delta-e^{-i\theta_1})^2\alpha\right\}_j\).
orthogonal polynomials on the unit circle
scientific article