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

Let p   be a trigonometric polynomial, non-negative on the unit circle TT. We say that a measure σσ on TT belongs to the polynomial Szegő class, if dσ(eiθ)=σac′(eiθ)dθ+dσs(eiθ), σsσs is singular, and∫02πp(eiθ)logσac′(eiθ)dθ>-∞.For the associated orthogonal polynomials {ϕn}{ϕn}, we obtain pointwise asymptotics inside the unit disc DD. Then we show that these asymptotics hold in L2L2-sense on the unit circle. As a corollary, we get an existence of certain modified wave operators.