Turán measures
type: Item
instance of: scholarly article
title: Turán measures
zbMATH Open document ID: 1032.42027
DOI: 10.1016/S0021-9045(03)00042-X
author: Yanyan Li
published in: Journal of Approximation Theory
publication date: 2003-06-09T00:00:00Z
review text: In 1988 \textit{M. Alfaro} and \textit{L. Vigil} [J. Approximation Theory 53, 195-197 (1988; Zbl 0641.33014)] proved that there is a probability measure \(\mu\) on the unit circle with the set of zeros (the set of zeros of all orthogonal with respect to \(\mu\) polynomials) dense in the unit disk (Turán's conjecture). In the present paper the author shows that the set of such measures (called Turán measures) is rather massive. Theorem 1. Given a measure \(\mu\) with parameters \(\{a_n\}\) and an arbitrary infinite set \(\Lambda\subset\mathbb N\) there exists a Turán measure \(\mu'\) with parameters \(\{a_n'\}\) satisfying \(a_n=a_n'\) for \(n\notin\Lambda\). In particular, there are Turán measures with parameters \(a_n\in\ell^p\) for all \(p>0\) as well as Turán measures with infinitely differentiable densities.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 28A12
Mathematics Subject Classification ID: 30D50
zbMATH DE Number: 1924734
zbMATH Keywords: probability measures
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: Szegő measures
zbMATH Keywords: Schur parameters
zbMATH Keywords: Schur functions
zbMATH Keywords: Turán measures
reviewed by: Leonid Golinskii
MaRDI profile type: Publication
cites work: Solution of a problem of P. Turán on zeros of orthogonal polynomials on the unit circle
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cites work: Q3889525
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cites work: Classification theorems for general orthogonal polynomials on the unit circle
cites work: Q3032769
cites work: Q4061445
full work available at URL: s0021-9045(03)00042-x
OpenAlex ID: W2911844880
description: scientific article; zbMATH DE number 1924734