On second kind polynomials associated with rational transformations of linear functionals
type: Item
instance of: scholarly article
title: On second kind polynomials associated with rational transformations of linear functionals
zbMATH Open document ID: 1170.42013
author: J. Blot
published in: Journal of Mathematical Analysis and Applications
publication date: 2009-07-09T00:00:00Z
review text: This paper is concerned with the study of rational transformations of linear functionals. This is a very well written paper which describes the following situation. For a regular and hermitian functional \(u\), it is considered a new linear functional \(\mathcal{L}\) defined by means of the following relations with \(u\): \[ \lambda (z-\beta)\mathcal{L}=(z-\alpha)u, \] with \(\alpha, \beta, \lambda \in \mathbb{C}\) and \(\lambda \neq 0.\) A necessary and sufficient condition for the regularity of \(\mathcal{L}\) is obtained and the positive definite character of \( \mathcal{L}\) is also studied. Some linear relations linking the sequences of orthogonal polynomials related to \(\mathcal{L}\) and \(u\) are obtained as well as an explicit representation for the orthogonal polynomial sequences of second kind related to \(\mathcal{L}\). Finally, the particular case \(\alpha=\beta\) is analyzed. In this last situation \(\mathcal{L}\) becomes in the modification of \(u\) by addition of a Dirac delta.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33C47
zbMATH DE Number: 5577101
zbMATH Keywords: orthogonal polynomial
zbMATH Keywords: unit circle
zbMATH Keywords: Bernstein-Szegő polynomials
zbMATH Keywords: Carathéodory function
zbMATH Keywords: measure modification
MaRDI profile type: Publication
full work available at URL: j.jmaa.2009.04.058
OpenAlex ID: W2056471168
cites work: On linearly related orthogonal polynomials and their functionals
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cites work: Prolégomènes à l'étude des polynômes orthogonaux semi- classiques. (Preliminary remarks for the study of semi-classical orthogonal polynomials)
cites work: A special class of polynomials orthogonal on the unit circle including the associated polynomials
DOI: 10.1016/J.JMAA.2009.04.058
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description: scientific article; zbMATH DE number 5577101