The Szego class with a polynomial weight

2004/03/26 by Sergey A. Denisov, S. Denisov, Denisov, S. +2
Mathematics · #Analytic and geometric function theory #Holomorphic and Operator Theory #Mathematical functions and polynomials #math.CA #math.CV #msc:42C05 #msc:47B36

paper · pdf · doi:10.48550/arxiv.math/0403472

preliminary version

arxiv created 2004/03/26 · arxiv updated 2009/12/01

Abstract

Let p be a trigonometric polynomial, nonnegative on the unit circle \mathbbT. We say that a measure σ on \mathbbT belongs to the polynomial Szego class, if dσ=sigma'_acdθ+dσ_s, σ_s is singular, and pln σ'_ac is summable on \mathbbT. For the associated orthogonal polynomials, we obtain pointwise asymptotics inside the unit disc. Then, we show that this asymptotics holds in the L² sense on the unit circle. As a corollary, we get existense of certain modified wave operators.

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