Determination of all coherent pairs
type: Item
instance of: scholarly article
title: Determination of all coherent pairs
zbMATH Open document ID: 0880.42012
author: O. Diekmann
published in: Journal of Approximation Theory
publication date: 1998-01-06T00:00:00Z
review text: In the study of Sobolev orthogonal polynomials, corresponding to the inner product \[ \langle f, g \rangle_S=\int_a^b f g d\Psi_0 + \lambda \int_a^b f' g' d\Psi_1 , \] where \(\Psi_0\) and \(\Psi_1\) are distribution functions and \(\lambda \geq 0\), an essential role has been played by the concept of coherence. If \(\{P_n\}\) and \(\{T_n\}\) are monic orthogonal polynomial sequences corresponding to \(\Psi_0\) and \(\Psi_1\), respectively, then \((\Psi_0,\Psi_1)\) is a coherent pair if \[ T_n = \frac{P_{n+1}'}{n+1} - \sigma_n \frac{P_{n}'}{n} , \quad \sigma_n=\text{const. } \neq 0 . \tag{*} \] This concept can be extended to quasi-definite linear functionals \(u_0\) and \(u_1\). When \(u_0\) and \(u_1\) are coherent, the Sobolev polynomial sequence has a structure suitable for further study. Nevertheless, it is important to know all the coherent pairs precisely, and this is the aim of this paper. The author proves that if \((u_0, u_1)\) is a coherent pair, then at least one of the functionals is classical (Hermite, Laguerre, Jacobi or Bessel). Moreover, all the coherent pairs are given explicitly. Analogous results are established for symmetric coherent pairs, when the subindices involved in the r.h.s.\ of (*) are \(n+1\) and \(n-1\).
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C45
zbMATH DE Number: 1033755
zbMATH Keywords: Sobolev orthogonal polynomials
zbMATH Keywords: distribution functions
zbMATH Keywords: functionals
zbMATH Keywords: coherence
reviewed by: Andrei Martínez Finkelshtein
MaRDI profile type: Publication
full work available at URL: jath.1996.3062
OpenAlex ID: W2095409933
cites work: Q3661894
cites work: Q4843468
cites work: Q4170993
cites work: On polynomials orthogonal with respect to certain Sobolev inner products
cites work: Orthogonal polynomials on Sobolev spaces: Old and new directions
cites work: Q4843460
cites work: Orthogonal polynomials and coherent pairs: The classical case
cites work: What is beyond coherent pairs of orthogonal polynomials?
cites work: Q4853625
cites work: Coherent pairs and zeros of Sobolev-type orthogonal polynomials
cites work: Q4892493
DOI: 10.1006/JATH.1996.3062
description: scientific article; zbMATH DE number 1033755