Analogs of the \(m\)-function in the theory of orthogonal polynomials on the unit circle
type: Item
instance of: scholarly article
title: Analogs of the \(m\)-function in the theory of orthogonal polynomials on the unit circle
zbMATH Open document ID: 1055.33010
author: B. Simon
published in: Journal of Computational and Applied Mathematics
publication date: 2004-08-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0311050
review text: A key leit-motif which passes through the whole paper is a variety of different functions in the theory of orthogonal polynomials on the unit circle (OPUC) growing from the multitude of applications of the Weyl-Titchmarsh \(m\)-functions associated to the Schrödinger operators on the halfline and infinite Jacobi matrices. For instance, the integral representation \[ m(z,J)=\int \frac{d\mu(t)}{t-z} \] leads to the Carathéodory function \[ F(z)=\int \frac{e^{i\theta}+z}{e^{i\theta-z}}\,d\mu(\theta) \] as a natural ``\(m\)-function'' for OPUC. Yet another reasonable choice for an \(m\)-function for OPUC is the Schur function \(f\) with the Schur parameters \(\{\gamma_0, \gamma_1,\ldots\}\) due to the linear fractional relation between \(f\) and \(f_1\) with the shifted parameters \(\{\gamma_1, \gamma_2,\ldots\}\). The latter is close to the relation between \(m(z,J)\) and \(m(z,J^{(1)})\), where \(J^{(1)}\) is the Jacobi matrix with ``shifted'' entries.
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 2085603
zbMATH Keywords: Weyl-Titchmarsh \(m\)-function
zbMATH Keywords: orthogonal polynomials on the unit circle
zbMATH Keywords: Verblunsky coefficients
zbMATH Keywords: Carathéodory and Schur functions
zbMATH Keywords: Szegő function
reviewed by: Leonid Golinskii
MaRDI profile type: Publication
OpenAlex ID: W2044340537
arXiv ID: math/0311050
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DOI: 10.1016/J.CAM.2004.01.022
description: scientific article