Linear spectral transformation and Laurent polynomials
type: Item
instance of: scholarly article
title: Linear spectral transformation and Laurent polynomials
zbMATH Open document ID: 1189.42011
author: Francisco Marcellán
author: Luis E. Garza
published in: Mediterranean Journal of Mathematics
publication date: 2009-11-18T00:00:00Z
review text: Let \(\mathcal{L}\) be Hermitian linear functional on the space of Laurent polynomials \(\Lambda=\,\)span\(\{z^n\}_{n\in\mathbb{Z}}\) and let \(P_n(z)\) a complex monic polynomial. Let define the new functional \(\mathcal{L}_I: \, \langle \mathcal{L}_I,q\rangle=\langle \mathcal{L},(P_n(z)-\overline{P}_n(1/z))/(2i)\,q\rangle\) and \(\mathcal{L}_R: \, \langle \mathcal{L}_R,q\rangle=\langle \mathcal{L},(P_n(z)+\overline{P}_n(1/z))/2\,q\rangle\). In this paper the authors study the relation between the Hessenberg matrices corresponding to the functional \(\mathcal{L}\) and \(\mathcal{L}_R\) as well as the relation between the Verblunsky coefficients for the polynomials associated with \(\mathcal{L}_R\) and \(\mathcal{L}\). Finally some representative examples are studied in details.
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 5635332
zbMATH Keywords: orthogonal polynomials on the unit circle
zbMATH Keywords: Laurent polynomials
zbMATH Keywords: linear spectral transformations
reviewed by: Renato Álvarez-Nodarse
MaRDI profile type: Publication
full work available at URL: s00009-009-0008-5
OpenAlex ID: W1964555090
cites work: Darboux transformation and perturbation of linear functionals
cites work: Polynomial perturbations of bilinear functionals and Hessenberg matrices
cites work: Spectral transformations for Hermitian Toeplitz matrices
cites work: Verblunsky parameters and linear spectral transformations
cites work: Q3351682
cites work: Orthogonal Polynomials and Rational Modifications of Measures
cites work: Q3752135
cites work: Relation between polynomials orthogonal on the unit circle with respect to different weights
cites work: Moment Theory, Orthogonal Polynomials, Quadrature, and Continued Fractions Associated with the unit Circle
cites work: Q3664266
cites work: Christoffel transforms and Hermitian linear functionals
cites work: Q3159369
cites work: Generalized bounded variation and inserting point masses
cites work: DARBOUX TRANSFORMS AND ORTHOGONAL POLYNOMIALS
DOI: 10.1007/S00009-009-0008-5
description: scientific article; zbMATH DE number 5635332