Zeros of polynomials orthogonal over regular \(N\)-gons.
type: Item
instance of: scholarly article
title: Zeros of polynomials orthogonal over regular \(N\)-gons.
zbMATH Open document ID: 1036.30026
DOI: 10.1016/S0021-9045(03)00039-X
published in: Journal of Approximation Theory
publication date: 2003-06-09T00:00:00Z
review text: The authors investigate the location of zeros of Bergman polynomials \(Q_n(z;G_N)\) (orthogonal polynomials with respect to area measure) for regular \(N\)-gons \(G_N\) in the plane. Two conjectures posed by Eiermann and Stahl are proved: (I) For \(N = 3, 4\), the zeros of all the \(Q_n\)'s are located exactly on the `diagonals' \(\Gamma_{k,N}\) of \(G_N\). However, for \(N\geqslant5\) there are zeros that are not on the \(\Gamma_{k,N}\)'s. (II) For \(N = 3, 4\) and fixed \(j\in\{0,\dots, N- 1\}\), the real zeros of the \(Q_{Nl+j}\)'s, \(l=0,1,\dots\), interlace on \((0,1)\). Some consequences regarding the asymptotic behavior of Bergman polynomials are given.
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 1924736
zbMATH Keywords: location of zeros of Bergman polynomials
zbMATH Keywords: Bergman kernel
zbMATH Keywords: interlacing property of zeros
zbMATH Keywords: asymptotic behavior of Bergman polynomials
zbMATH Keywords: Schwarz reflection principle
zbMATH Keywords: exterior Riemann mapping function
zbMATH Keywords: Markov system
author: V. V. Maymeskul
author: E. B. Saff
reviewed by: Eleonora A. Storozhenko
MaRDI profile type: Publication
cites work: Erdős-Turán type theorems on quasiconformal curves and arcs
cites work: On zeros of polynomials orthogonal over a convex domain
cites work: Q3752617
cites work: Q5534404
cites work: Zero distribution of Bergman orthogonal polynomials for certain planar domains
cites work: Asymptotic behaviour of zeros of Bieberbach polynomials
cites work: Q4381539
cites work: Q3473617
cites work: Q4360405
cites work: Q5289656
description: scientific article; zbMATH DE number 1924736