Laguerre-Sobolev orthogonal polynomials: Asymptotics for coherent pairs of type II
type: Item
instance of: scholarly article
title: Laguerre-Sobolev orthogonal polynomials: Asymptotics for coherent pairs of type II
zbMATH Open document ID: 1031.42024
DOI: 10.1016/S0021-9045(03)00034-0
published in: Journal of Approximation Theory
publication date: 2003-06-09T00:00:00Z
review text: A Laguerre-Sobolev orthogonal polynomial \(S_n\) of degree \(n\) corresponding to a coherent pair of type II satisfies \((S_n, q)_S=0\) for polynomials \(q\) of degree \(\leq n-1\), where \[ (p, q)_S=\int_0^{+\infty} p q d\mu_0+ \int_0^{+\infty} p' q' d\mu_1 , \] \(\mu_0\) is standard Laguerre measure on \((0, +\infty)\) and \(\mu_1\) is its rational modification plus eventually a single mass point on the negative semiaxis. The authors establish strong asymptotics for the sequence \(S_n(x)\) for \(x \notin (0,+\infty)\), bounds on \((0,+\infty)\), along with Mehler-Heine (for \(S_n(x/n)\) on \(\mathbb C\)) and Plancherel-Rotach (for \(S_n(nx)\) on \(\mathbb C \setminus [0, 4]\)) formulas. As a corollary, asymptotic behavior of the zeros of \(S_n\) is obtained, illustrated by several numerical experiments. As an auxiliary result, several properties of polynomials orthogonal with respect to \(\mu_1\) are studied.
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 1924732
zbMATH Keywords: Sobolev orthogonal polynomials
zbMATH Keywords: Laguerre polynomials
zbMATH Keywords: asymptotics
zbMATH Keywords: zeros
author: M. Alfaro
author: J. J. Moreno-Balcázar
author: M. L. Rezola
reviewed by: Andrei Martínez Finkelshtein
MaRDI profile type: Publication
cites work: Q3331506
cites work: Q5690368
cites work: Zeros and critical points of Sobolev orthogonal polynomials
cites work: On polynomials orthogonal with respect to certain Sobolev inner products
cites work: An upper bound for the Laguerre polynomials
cites work: Strong and Plancherel-Rotach asymptotics of non-diagonal Laguerre-Sobolev orthogonal polynomials
cites work: Bernstein-Szegő's theorem for Sobolev orthogonal polynomials
cites work: Asymptotics of Sobolev orthogonal polynomials for a Jacobi weight
cites work: Determination of all coherent pairs
cites work: Zeros of Sobolev orthogonal polynomials following from coherent pairs
cites work: Asymptotics of Sobolev orthogonal polynomials for coherent pairs of Laguerre type
cites work: On Sobolev orthogonal polynomials with coherent pairs: The Laguerre case, type 1
cites work: Q4061445
cites work: Asymptotics for orthogonal polynomials
cites work: Q3473619
description: scientific article; zbMATH DE number 1924732