Some properties of zeros of Sobolev-type orthogonal polynomials
type: Item
instance of: scholarly article
title: Some properties of zeros of Sobolev-type orthogonal polynomials
zbMATH Open document ID: 0862.33005
DOI: 10.1016/0377-0427(95)00034-8
author: Yanyan Li
published in: Journal of Computational and Applied Mathematics
publication date: 1997-05-26T00:00:00Z
review text: The authors consider the monic polynomials \(Q_n (n=1,2, \dots)\) which are orthogonal with respect to a certain inner product involving also a discrete part (Sobolev-type product) of the form (1) \(\langle f,g \rangle = \int_Ifgd \mu+ \sum^r_{i=0} M_if^{(i)} (c)g^{(i)} (c)\), where \(\mu\) stands for finite positive Borel measure supported on an interval \(I \subset\mathbb{R}\) while \(c\notin {\overset\circ I}\) (the interior of \(I)\), \(r\geq 1\), \(M_i\geq 0\) \((i=0, \dots, r-1)\) and \(M_r>0\), and derive some properties of their zeros. The results are given in the form of two main theorems. One of them states that \(Q_n\) has at least \((n- \overline n)\) changes of sign in the interior of the convex hull of \(I\), where \(\overline n\) denotes the number of terms in the discrete part in (1) whose order of derivatives is less than \(\overline n\). The other main result concerns the interlacing properties of the zeros and gives the conditions under which \(Q_n\) and \(Q_{n+1}\) have common zeros.
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 908409
zbMATH Keywords: Sobolev-type product
reviewed by: Mithat Idemen
MaRDI profile type: Publication
full work available at URL: 0377-0427(95)00034-8
OpenAlex ID: W2077307830
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description: scientific article; zbMATH DE number 908409