Laurent polynomial perturbations of linear functionals. An inverse problem
type: Item
instance of: scholarly article
title: Laurent polynomial perturbations of linear functionals. An inverse problem
zbMATH Open document ID: 1191.42012
author: Kenier Castillo
author: Luis E. Garza
author: Francisco Marcellán
publication date: 2010-04-14T00:00:00Z
full work available at URL: https://eudml.org/doc/230524
full work available at URL: pp83-98.pdf
review text: The authors continue their study of linear functionals \(L\) on the space of Laurent polynomials, generated by the sequence \(\{c_n\}_{n\in\mathbb{Z}}\) of moments \[ L(q)=\sum_{k=-n}^m c_kq_k, \qquad q(z)=\sum_{k=-n}^m q_kz^k, \] and corresponding Carathéodory functions \(F(z)=1+2\sum_{n\geq1} c_{-n}z^n\). In a previous paper they considered a perturbation \(L_R\) of \(L\) given by \[ L_R(q):=L\left(\frac{z+z^{-1}-\alpha-\bar\alpha}2\,q\right), \qquad \alpha\in\mathbb{C}. \] The main object under consideration in the present paper is the inverse transformation \(L_{R^{-1}}\), defined in a natural way. The authors obtain conditions for \(L_{R^{-1}}\) to be quasi-definite, as well as the relation between the corresponding Carathéodory functions. Several illuminating examples are analyzed.
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 5692978
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: linear functionals
zbMATH Keywords: Laurent polynomials
zbMATH Keywords: linear spectral transformations
zbMATH Keywords: inverse transformations
zbMATH Keywords: Carathéodory functions
reviewed by: Leonid Golinskii
MaRDI profile type: Publication
published in: ETNA - Electronic Transactions on Numerical Analysis
description: scientific article; zbMATH DE number 5692978