Email this article On zeros, bounds, and asymptotics for orthogonal polynomials on the unit circle
- Authors: Lubinsky D.S.1
-
Affiliations:
- School of Mathematics, Georgia Institute of Technology
- Issue: Vol 213, No 11 (2022)
- Pages: 31-49
- Section: Articles
- URL: https://journals.rcsi.science/0368-8666/article/view/133492
- DOI: https://doi.org/10.4213/sm9569
- ID: 133492
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Abstract
Let $\mu$ be a measure on the unit circle that is regular in the sense of Stahl, Totik and Ullmann. Let $\{\varphi_{n}\}$ be the orthonormal polynomials for $\mu$ and $ż_{jn}\}$ their zeros. Let $\mu$ be absolutely continuous in an arc $\Delta$ of the unit circle, with $\mu'$ positive and continuous there. We show that uniform boundedness of the orthonormal polynomials in subarcs $\Gamma$ of $\Delta$ is equivalent to certain asymptotic behaviour of their zeros inside sectors that rest on $\Gamma$. Similarly the uniform limit $\lim_{n\to \infty}|\varphi_{n}(z)|^{2}\mu'(z)=1$ is equivalent to related asymptotics for the zeros in such sectors. Bibliography: 27 titles.
About the authors
Doron Shaul Lubinsky
School of Mathematics, Georgia Institute of Technology
Email: lubinsky@math.gatech.edu
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