Two point masses perturbation of regular moment functionals
type: Item
instance of: scholarly article
title: Two point masses perturbation of regular moment functionals
zbMATH Open document ID: 0877.42009
DOI: 10.1016/S0019-3577(97)83352-6
author: O. Diekmann
published in: Indagationes Mathematicae. New Series
publication date: 1997-10-08T00:00:00Z
review text: Let \(\sigma\) denote a regular moment functional. Then the authors consider \(\tau:=\sigma+\lambda_1\delta(x-a_1)+\lambda_2\delta(x-a_2)\), where \(\lambda_1,\lambda_2\in\mathbb{C}\), \(a_1,a_2\in\mathbb{R}\), and \(a_1\neq a_2\). A necessary and sufficient condition is given for \(\tau\) to be regular (or positive-definite when \(\sigma\) is positive-definite). Further the orthogonal polynomials \(\{R_n(x)\}_{n=0}^\infty\) with respect to \(\tau\) are given in terms of the orthogonal polynomials \(\{P_n(x)\}_{n=0}^\infty\) relative to \(\sigma\). When both \(\sigma\) and \(\tau\) are positive-definite, the relations between zeros of \(\{P_n(x)\}_{n=0}^\infty\) and \(\{R_n(x)\}_{n=0}^\infty\) are investigated. Finally it is shown that if \(\sigma\) is semi-classical then \(\tau\) is also semi-classical. In that case the structure relation, the second-order differential equation satisfied by the semi-classical orthogonal polynomials \(\{R_n(x)\}_{n=0}^\infty\), and the class number of \(\tau\) are given.
reviewed by: Roelof Koekoek
Mathematics Subject Classification ID: 42C05
zbMATH DE Number: 1029287
zbMATH Keywords: moment functionals
zbMATH Keywords: point masses
zbMATH Keywords: orthogonal polynomials
MaRDI profile type: Publication
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full work available at URL: s0019-3577(97)83352-6
OpenAlex ID: W2030699850
description: scientific article; zbMATH DE number 1029287