The Szegő Class with a Polynomial Weight

Let p be a trigonometric polynomial, nonnegative 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 this asymptotics holds in L 2-sense on the unit circle. As a corollary, we get existence of certain modified wave operators.