The Szego class with a polynomial weight
Abstract
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.
- Publication:
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arXiv Mathematics e-prints
- Pub Date:
- March 2004
- DOI:
- 10.48550/arXiv.math/0403472
- arXiv:
- arXiv:math/0403472
- Bibcode:
- 2004math......3472D
- Keywords:
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- Classical Analysis and ODEs;
- Complex Variables;
- 47B36;
- 42C05
- E-Print:
- preliminary version