Orthogonal Polynomials

Szegö, Gábor

doi:10.1016/c2013-0-02379-1

Cited by 2,491 publications

(2 citation statements)

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“…(iii) The asymptotics of orthogonal polynomials on the unit circle have been studied for at least a century, and there is an extensive literature. If log µ ′ is integrable over the unit circle, then there is an L 2 asymptotic for φ n (see [8], Ch. V, [9], [23], p. 132, and [26], Ch.…”

Section: § 1 Introductionmentioning

confidence: 99%

“…It was Rakhmanov who resolved the conjecture (see [18], [20]), with definitive later contributions by Ambroladze [1], [2], and Aptekarev, Denisov and Tulyakov [3], [4]. There have been many who have contributed in a major way to the broader issue of bounds -for example, Badkov [5], Freud [8], Geronimus [9], Korous and Nevai (see [16]). Again, this is a very incomplete list.…”

Section: § 1 Introductionmentioning

confidence: 99%

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On zeros, bounds, and asymptotics for orthogonal polynomials on the unit circle

Lubinsky

2022

Sb. Math.

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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 $z_{jn}\}$ 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.

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Section: § 1 Introductionmentioning

confidence: 99%