On rational transformations of linear functionals: direct problem
type: Item
instance of: scholarly article
title: On rational transformations of linear functionals: direct problem
zbMATH Open document ID: 1066.33008
author: A. Peña
author: M. Alfaro
author: Francisco Marcellán
author: M. L. Rezola
published in: Journal of Mathematical Analysis and Applications
publication date: 2004-11-23T00:00:00Z
review text: In this paper the authors continue their research on linearly related orthogonal polynomials (\(P_n\) and \(Q_n\)) and their functionals (\(u\) and \(v\), respectively) [\textit{M. Alfaro, F. Marcellán, A. Pena, M. L. Rezola}, J. Math. Anal. Appl. 287, No.1, 307-319 (2003; Zbl 1029.42014)]. Here they analyze the direct problem for linear functionals acting on the linear space of polynomials with complex coefficients satisfying the functional equation \(pu=qv\), where \(p(x)=x-\tilde{a}\), \(q(x)=\lambda(x-a)\). If \(u\) is a quasi-definite functional they give necessary and sufficient conditions for \(v\) to be quasi-definite. Under these conditions they characterize when the corresponding families \(P_n\) and \(Q_n\) satisfy a relation of the form \(P_n(x) + s_n P_{n-1}(x) = Q_n(x) + t_n Q_{n-1}(x)\), \(s_nt_n\neq0\).
Mathematics Subject Classification ID: 33C47
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 2117735
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: recurrence relations
zbMATH Keywords: linear functionals
reviewed by: Renato Álvarez-Nodarse
MaRDI profile type: Publication
full work available at URL: j.jmaa.2004.04.065
OpenAlex ID: W1992622100
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DOI: 10.1016/J.JMAA.2004.04.065
description: scientific article; zbMATH DE number 2117735