Universality Type Limits for Bergman Orthogonal Polynomials

Lubinsky, D. S.

doi:10.1007/bf03321759

Cited by 8 publications

(26 citation statements)

References 30 publications

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“…Recently, the subject has experienced a new surge, with many new interesting results in a variety of topics such as the asymptotic behavior and zero distribution of the orthogonal polynomials [4,6,7,8,10,11,12,14,16], universality and Christoffel functions [9,14,20], and the existence of recurrence relations [1,18,19]. In particular, [6] and [20] consider orthogonality over several domains, while [14] considers orthogonality with respect to certain potential theoretic varying weights.…”

Section: Introduction and New Resultsmentioning

confidence: 99%

Asymptotics of Carleman Polynomials for Level Curves of the Inverse of a Shifted Zhukovsky Transformation

Dragnev

Miña-Díaz

Northington

2013

Comput. Methods Funct. Theory

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This paper complements the recent investigation of [4] on the asymptotic behavior of polynomials orthogonal over the interior of an analytic Jordan curve L. We study the specific case of L = {z = w−1+(w−1) −1 , |w| = R}, for some R > 2, providing an example that exhibits the new features discovered in [4], and for which the asymptotic behavior of the orthogonal polynomials is established over the entire domain of orthogonality. Surprisingly, this variation of the classical example of the ellipse turns out to be quite sophisticated. After properly normalizing the corresponding orthonormal polynomials pn, n = 0, 1, . . ., and on certain critical subregion of the orthogonality domain, a subsequence {pn k } converges if and only if log µ 4 (n k ) converges modulo 1 (µ being an important quantity associated to L). As a consequence, the limiting points of the sequence {pn} form a one parameter family of functions, the parameter's range being the interval [0, 1). The polynomials pn are much influenced by a certain integrand function, the explained behavior being the result of this integrand having a nonisolated singularity that is a cluster point of poles. The nature of this singularity sparks purely from geometric considerations, as opposed to the more common situation where the critical singularities come from the orthogonality weight.1991 Mathematics Subject Classification. 42C05, 30E10, 30E15.

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Section: Introduction and New Resultsmentioning

confidence: 99%

Asymptotics of Carleman Polynomials for Level Curves of the Inverse of a Shifted Zhukovsky Transformation

Dragnev

Miña-Díaz

Northington

2013

Comput. Methods Funct. Theory

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“…A Model Case. As in [24], we begin by proving Theorem 1.4 in one specific case. We will consider area measure on the region G r,m , which we denote by µ 0 .…”

Section: Proof Of Theorem 14mentioning

confidence: 99%

“…Indeed, many of the results of [24,Sections 3 & 4] do not depend on the simple connectivity of the support of the measure, only on the fact that z 0 is a peak polynomial point (as defined in [24]) and a model case for comparison. Rather than duplicate the proof in [24], we provide here only a sketch of the necessary arguments. The first step in proving the general case is to establish estimates on the Christoffel functions, specifically on expressions of the form lim n→∞ n 2 λ n (u n ; µ), where u n ∈ ∂G r,m and |u n −u| = O(n −1 ) as n → ∞ for some u ∈ ∂G r,m ∩D.…”

Section: Recall Thatmentioning

confidence: 99%

“…In recent years there has also been interest in proving universality results for measures supported on sets other than the unit circle or real line. Such results include measures supported on arcs of the unit circle (see [26]), smooth Jordan regions (see [24,34]), and smooth Jordan curves (see [17]). Our second main result is a comparable result for a collection of polynomial lemniscates.…”

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confidence: 99%

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Two universality results for polynomial reproducing kernels

Simanek

2017

Journal of Approximation Theory

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“…If J ⊂ ∂G, a ∂G neighborhood of J means a relatively open subset J 1 of ∂G containing J. In [58], I proved:…”

Section: Bergman Polynomialsmentioning

confidence: 99%

An Update on Local Universality Limits for Correlation Functions Generated by Unitary Ensembles

Lubinsky¹

2016

SIGMA

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Abstract. We survey the current status of universality limits for m-point correlation functions in the bulk and at the edge for unitary ensembles, primarily when the limiting kernels are Airy, Bessel, or Sine kernels. In particular, we consider underlying measures on compact intervals, and fixed and varying exponential weights, as well as universality limits for a variety of orthogonal systems. The scope of the survey is quite narrow: we do not consider β ensembles for β = 2, nor general Hermitian matrices with independent entries, let alone more general settings. We include some open problems.

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Universality Type Limits for Bergman Orthogonal Polynomials

Abstract: Abstract. We establish universality type limits for Bergman orthogonal polynomials on simply connected regions in the complex plane with smooth boundary.

Cited by 8 publications

References 30 publications

Asymptotics of Carleman Polynomials for Level Curves of the Inverse of a Shifted Zhukovsky Transformation

Asymptotics of Carleman Polynomials for Level Curves of the Inverse of a Shifted Zhukovsky Transformation

Two universality results for polynomial reproducing kernels

An Update on Local Universality Limits for Correlation Functions Generated by Unitary Ensembles

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