Regular Sobolev type orthogonal polynomials: The Bessel case
type: Item
instance of: scholarly article
title: Regular Sobolev type orthogonal polynomials: The Bessel case
zbMATH Open document ID: 0858.42016
author: Miguel A. Piñar
author: Francisco Marcellán
author: Teresa E. Pérez
published in: Rocky Mountain Journal of Mathematics
publication date: 1997-03-19T00:00:00Z
review text: We are dealing with a nice paper, studying Sobolev-type orthogonal polynomials arising from a regular linear functional on the linear space of polynomials with real coefficients in the following manner: \[ \varphi(p,q):=\langle u,pq\rangle+ \lambda p'(c)q'(c)\quad (\lambda, c\in\mathbb{R}). \] The ordinary multiplication operator is not selfadjoint for \(\varphi\), but luckily the multiplication operator \((x-c)^2\) is! Now using the functional \(u_1=(x-c)^2 u\), the authors derive Christoffel-Darboux formulae, 5-term recurrence relations and a recursive algorithm to calculate its coefficients for the orthogonal polynomials connected with \(u_1\). Finally, the general results are used to treat the linear functional \(u\) that gives rise to the Bessel polynomials, with the choice \(c=0\). The authors specialize their general results on this case, giving explicit formulae. Moreover, they give a Rodrigues type formula and the form of a second-order differential equation satisfied by the Sobolev orthogonal polynomials (instead of polynomial coefficients of degrees 2, 1 and 0, resp., the degrees are now 4, 3 and 2; no proof, however, is given that these coefficients do not have a common factor).
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 894660
zbMATH Keywords: regular functionals
zbMATH Keywords: Sobolev-type orthogonal polynomials
zbMATH Keywords: Christoffel-Darboux formulae
zbMATH Keywords: Bessel polynomials
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1216/rmjm/1181072155
OpenAlex ID: W1969508349
cites work: Q3331506
cites work: On Orthogonal Polynomials of Sobolev Type: Algebraic Properties and Zeros
cites work: On orthogonal polynomials with respect to an inner product involving derivatives: Zeros and recurrence relations
cites work: Q4170993
cites work: On Recurrence Relations for Sobolev Orthogonal Polynomials
cites work: Bessel polynomials
cites work: Q3677993
cites work: A New Class of Orthogonal Polynomials: The Bessel Polynomials
cites work: On a class of polynomials orthogonal with respect to a discrete Sobolev inner product
DOI: 10.1216/RMJM/1181072155
description: scientific article; zbMATH DE number 894660