  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 $d\sigma=sigma'_{ac}d\theta+d\sigma_s$, $\sigma_s$
is singular, and $p\ln \sigma'_{ac}$ 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.
