Local asymptotics for orthonormal polynomials on the unit circle via universality

2022

J. Spectr. Theory

Self Cite

Let ¹p n º denote the orthonormal polynomials associated with a measure with compact support on the real line. Let be regular in the sense of Stahl, Totik, and Ullmann, and I be a subinterval of the support in which is absolutely continuous, while 0 is positive and continuous there. We show that boundedness of the ¹p n º in that subinterval is closely related to the spacing of zeros of p n and p n 1 in that interval. One ingredient is proving that "local limits" imply universality limits.

Section: Resultsmentioning

confidence: 99%

“…A secondary goal of this paper is to prove a converse of Theorem A, namely to show that local limits such as (1.12) imply a universality limit like (1.11). For measures on the unit circle this was undertaken in [10] -however the results necessarily take a quite different form.…”

Section: Resultsmentioning

confidence: 99%

Bounds on orthogonal polynomials and separation of their zeros

Levin

2022

J. Spectr. Theory

Self Cite

“…has at least a simple zero at v = 0. However, this contradicts the universality limit, which shows (1).…”

mentioning

confidence: 94%

“…

…”

mentioning

confidence: 99%

Correction to “Local asymptotics for orthonormal polynomials on the unit circle via universality”

2021

JAMA

Self Cite

There is a mistake in the proof of Lemma 4.2(a) in [1], namely there t jn z jn = 1, so that the denominator in K n (z jn , t jn ) is 0. This causes a gap in the proofs of Lemma 4.3(d), and Theorems 1.1 and 1.2 (but not Theorems 1.3 and 1.4). Below we give replacements for Lemmas 4.2(a) and 4.3(d) and revised proofs of Theorems 1.1 and 1.2. Note that the rest of those lemmas, including the hypotheses, remain the same.Revised Lemma 4.2(a) 1. Let ρ > 0 andThere exist C 0 , n 0 such that for n ≥ n 0 and z jn , z kn ∈ L n with j = k, we haveIn particular, all zeros of ϕ n in L n are simple.Proof. If the result is false, we can find a subsequence of integers S and for n ∈ S, z jn ,From the Christoffel-Darboux formula (2.1) and the uniform universality limit (1.1), 0 = K n (z jn , 1 z kn ) K n (τ n , τ n ) = e ιπ(α n −β n (1+o(1))) S(α n − β n (1 + o(1))) + o(1) = 1 + o(1).

“…(iii) The main ideas underlying the results of this paper are universality limits for reproducing kernels (see [10], [12] and [27]) and local limits for ratios of orthogonal polynomials (see [13]).…”

Section: § 1 Introductionmentioning

confidence: 99%

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

2022

Sb. Math.

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.