Zeros of Gegenbauer-Sobolev orthogonal polynomials: beyond coherent pairs
type: Item
instance of: scholarly article
title: Zeros of Gegenbauer-Sobolev orthogonal polynomials: beyond coherent pairs
zbMATH Open document ID: 1168.30004
author: A. Sri Ranga
author: Eliana X. L. de Andrade
author: Cleonice F. Bracciali
published in: Acta Applicandae Mathematicae
publication date: 2009-04-29T00:00:00Z
review text: Symmetrically coherent pairs of measures date back to their introduction by \textit{A. Iserles, P. E. Koch, S. P. Nørsett} and \textit{J. M. Sanz-Serna} [J. Approx. Theory 65, No. 2, 151--175 (1991; Zbl 0734.42016)] and have been studied extensively later on. The introduction of the paper under review contains a nice overview of the development relevant to the case of Gegenbauer-Sobolev orthogonality. In a paper by \textit{W. G. M. Groenevelt} [J. Approx. Theory 114, No. 1, 115--140 (2002; Zbl 1008.33006)] all possible types of coherent pairs in the Gegenbauer case have been studied and interlacing properties of zeros have been derived. In the paper under review, the situation is addressed where the pair of measures is not coherent anymore, but the relation is relaxed to the situation of the existence of a recurrence relation connecting the ordinary Gegenbauer polynomials to the Gegenbauer-Sobolev orthogonal polynomials. The following inner product is studied \[ \begin{multlined} \langle f,g\rangle = \int_{-1}^1\,f(x)g(x)(1-x^2)^{\lambda-1/2}dx\\ +\int_{-1}^1\,f'(x)g'(x)\left(\kappa_1+ {\kappa_2\over 1+qx^2}\right)(1-x^2)^{\lambda+1/2}dx \\ +\kappa_2M_q\left[f'\left({-1\over\sqrt{-q}}\right)g'\left({-1\over\sqrt{-q}}\right)+ f'\left({1\over\sqrt{-q}}\right)g'\left({-1\over\sqrt{-q}}\right)\right],\end{multlined}\tag{1} \] where \[ \lambda>-{1\over 2},\;q\geq -1;\;M_q\geq 0\text{ for }-1<q<0\text{ and }M_q=0\text{ for }q\geq 0 \] and \(q,\kappa_1,\kappa_2\) are chosen such that the integral forms above are positive definite inner products when studied separately. When \(\kappa_1\not= 0\), the pair of measures (one consisting of the first integral and the other of the second integral together with the \(M_q\)-term) is no longer coherent, but nevertheless---triggered by numerical evidence---the authors succeeded in proving interlacing properties when the parameters satisfy \[ \lambda\geq 0,\;q\geq 1\text{ and }\kappa_2\geq\left[{2\lambda +3\over 2\lambda +2}+2(1+q)\rig…
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 30C15
Mathematics Subject Classification ID: 33C45
Mathematics Subject Classification ID: 33C47
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 5549569
zbMATH Keywords: Gegenbauer polynomials
zbMATH Keywords: zeros of Gegenbauer-Sobolev orthogonal polynomials
zbMATH Keywords: symmetrically coherent pairs
zbMATH Keywords: interlacing properties of zeros
MaRDI profile type: Publication
full work available at URL: s10440-008-9265-8
OpenAlex ID: W2022639352
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DOI: 10.1007/S10440-008-9265-8
description: scientific article; zbMATH DE number 5549569