Asymptotics of the orthogonal polynomials for the Szego class with a polynomial weight

Let p be a trigonometric polynomial, non-negative on the unit circle T. We say that a measure σ on T belongs to the polynomial Szeg˝ o class, if dσ(e iθ) = σ ′ ac (e iθ)dθ + dσ s (e iθ), σ s is singular, and 2π 0 p(e iθ) log σ ′ ac (e iθ) dθ > −∞ For the associated orthogonal polynomials {ϕ n }, we obtain pointwise asymp-totics inside the unit disc D. Then we show that these asymptotics hold in L 2-sense on the unit circle. As a corollary, we get an existence of certain modified wave operators.