The Szego class with a polynomial weight

Abstract

Let p be a trigonometric polynomial, nonnegative on the unit circle $\mathbb{T}$. We say that a measure $\sigma$ on $\mathbb{T}$ belongs to the polynomial Szego class, if $d\sigma =sigm{a}_{ac}^{\prime }d\theta +d{\sigma }_s$, ${\sigma }_s$ is singular, and $p\ln {\sigma }_{ac}^{\prime }$ is summable on $\mathbb{T}$. For the associated orthogonal polynomials, we obtain pointwise asymptotics inside the unit disc. Then, we show that this asymptotics holds in the $L^2$ sense on the unit circle. As a corollary, we get existense of certain modified wave operators.

preliminary version