Quadrature formula and zeros of para-orthogonal polynomials on the unit circle
type: Item
instance of: scholarly article
title: Quadrature formula and zeros of para-orthogonal polynomials on the unit circle
zbMATH Open document ID: 1017.42014
DOI: 10.1023/A:1019765002077
published in: Acta Mathematica Hungarica
publication date: 2003-08-06T00:00:00Z
review text: Let \(\mu\) be a probability measure supported on the unit circle \(\Gamma=\{z\in\mathbb{C}\colon |z|=1\}\) and let \((\varphi_n)_n\) be the corresponding orthonormal polynomial sequence. Let \[ B_n(z,w)= \varphi^*_{n+1}(z)\overline{\varphi^*_{n+1}(w)}- \varphi_{n+1}(z)\overline{\varphi_{n+1}(w)}, \] where \( \varphi^*_{n}(z)=z^n \overline{ \varphi_{n}(1/\overline{z})}\) are the reversed polynomials. The polynomials \(B_{n+1}\) are usually called para-orthogonal polynomials. In this paper, the author studies the properties of the zeros of these para-orthogonal polynomials. In particular, he shows that zeros of \(B_{n}\) and \(B_{n+1}\), that lie on \(\Gamma\), alternate (i.e., similar to the real line case property). Moreover, if the arc \(\gamma\in\Gamma\) is a gap in the support of \(\mu\), then for each \(n\), \(B_n\) has at most one zero in \(\overline{\gamma}\). Going further, the author gives several bounds for the distance of two consecutive zeros of \(B_n\). Finally, the attracting property of the support of \(\mu\) is discussed.
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 1960298
zbMATH Keywords: measures on the unit circle
zbMATH Keywords: para-orthogonal polynomials
zbMATH Keywords: trigonometric moment problem
zbMATH Keywords: Szegő quadrature problem
author: Leonid Golinskii
reviewed by: Renato Álvarez-Nodarse
MaRDI profile type: Publication
full work available at URL: a:1019765002077
OpenAlex ID: W2139253673
description: scientific article; zbMATH DE number 1960298