Higher-Order Szego Theorems With Two Singular Points

TL;DR

This paper establishes higher-order Szego theorems for measures on the unit circle with two singular points, linking integrability conditions of the weight function to decay and smoothness properties of Verblunsky coefficients.

Contribution

It extends Szego theorems by deriving new conditions involving second differences and shifted coefficients for measures with two singular points.

Findings

Characterization of integrability of weighted log measures via Verblunsky coefficients.

Conditions involving second differences of Verblunsky coefficients for higher-order Szego theorems.

Results applicable to measures with multiple singularities on the unit circle.

Abstract

We consider probability measures, $d\mu=w(\theta) \f{d\theta}{2\pi} +d\mu_\s$, on the unit circle, $\partial\bbD$, with Verblunsky coefficients, $\{\alpha_j\}_{j=0}^\infty$. We prove for $\theta_1\neq\theta_2$ in $[0,2\pi)$ and $(\delta\beta)_j=\beta_{j+1}$ that \[ \int [1-\cos(\theta-\theta_1)][1-\cos(\theta-\theta_2)] \log w(\theta) \f{d\theta}{2\pi} >-\infty \] if and only if \[ \sum_{j=0}^\infty \bigl|\bigl\{(\delta -e^{-i\theta_2}) (\delta -e^{-i\theta_1}) \alpha\bigr\}_j\bigr|^2 +\abs{\alpha_j}^4 <\infty \] We also prove that \[ \int (1-\cos\theta)^2 \log w(\theta) \f{d\theta}{2\pi} >-\infty \] if and only if \[ \sum_{j=0}^\infty \abs{\alpha_{j+2}-2\alpha_{j+1} +\alpha_j}^2 + \abs{\alpha_j}^6 <\infty \]