Higher-order Szegő theorems with two singular points
type: Item
instance of: scholarly article
title: Higher-order Szegő theorems with two singular points
zbMATH Open document ID: 1067.30075
author: B. Simon
author: Andrej Zlatoš
published in: Journal of Approximation Theory
publication date: 2005-06-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/math-ph/0409065
full work available at URL: CaltechAUTHORS:20170512-073745126
review text: This paper is a contribution to the theory of orthogonal polynomials on the unit circle, \(\partial\mathbb D\subset\mathbb C\). Let \(d\mu\) be a non-trivial probability measure on \(\partial\mathbb D\) of the form \(d\mu=w(\theta)\frac{d\theta}{2\pi} + d\mu_s\) with Verblunsky coefficients, \(\{\alpha_j\}_{j=0}^\infty\), where \(d\mu_s\) is singular with respect to Lebesgue measure \(d\theta\) on \(\partial\mathbb D\). The goal in this paper is to analyze two singularities or a single double singularity. The authors prove for \(\theta_1\neq\theta_2\) in \([0,2\pi)\) that \[ \int[1 - \cos(\theta -\theta_1)][1 - \cos(\theta -\theta_2)] \log\left(w(\theta)\right) \frac{d\theta}{2\pi} > - \infty \] \[ \Longleftrightarrow\sum_{j=0}^\infty\left| \left\{ (\delta-e^{-i\theta_2})(\delta-e^{-i\theta_1})\alpha \right\}_j\right| ^2+| \alpha| ^4<\infty, \] where \(\delta\) is the left shift operator \((\delta\beta)_j =\beta_{j+1}\). The authors also prove a result for \(\theta_1=\theta_2\): \[ \int(1 - \cos\theta)^2\log\left(w(\theta)\right)\frac{d\theta}{2\pi} > - \infty \] \[ \Longleftrightarrow\sum_{j=o}^\infty| \alpha_{j+2} - 2\alpha_{j+1} +\alpha_j| ^2 + | \alpha_j| ^6 < \infty. \] One can replace \(\cos(\theta)\) by \(\cos(\theta-\theta_1)\) if \(\alpha_{j+2} - 2\alpha_{j+1} +\alpha_j\) is replaced by \(\left\{(\delta-e^{-i\theta_1})^2\alpha\right\}_j\).
reviewed by: Eleonora A. Storozhenko
Mathematics Subject Classification ID: 30E10
zbMATH DE Number: 2174971
zbMATH Keywords: orthogonal polynomials on the unit circle
zbMATH Keywords: relative Szegö function
zbMATH Keywords: Carathéodory and Schur functions
zbMATH Keywords: singularity at antipodal points
zbMATH Keywords: two singularities
zbMATH Keywords: single double singularity
zbMATH Keywords: Szegö theorem
MaRDI profile type: Publication
OpenAlex ID: W1993698147
arXiv ID: math-ph/0409065
cites work: On the absolutely continuous spectrum of one-dimensional Schrödinger operators with square summable potentials
cites work: On the coexistence of absolutely continuous and singular continuous components of the spectral measure for some Sturm--Liouville operators with square summable potential
cites work: Asymptotics of the orthogonal polynomials for the Szegő class with a polynomial weight
cites work: Proprietà di alcune classi di funzioni in più variabili
cites work: Q5835454
cites work: Q3270024
cites work: Sum rules for Jacobi matrices and their applications to spectral theory
cites work: On sum rules of special form for Jacobi matrices.
cites work: On a spectral property of Jacobi matrices
cites work: On new relations between spectral properties of Jacobi matrices and their coefficients
cites work: Q4673257
cites work: Q3260749
cites work: Q4748414
cites work: A canonical factorization for meromorphic Herglotz functions on the unit disk and sum rules for Jacobi matrices
cites work: Q3159369
cites work: Q3159370
cites work: Sum rules and the Szegő condition for orthogonal polynomials on the real line
cites work: Sum rules for Jacobi matrices and divergent Lieb--Thirring sums
DOI: 10.1016/J.JAT.2005.02.003
description: scientific article