Three term recurrence relation modulo ideal and orthogonality of polynomials of several variables
type: Item
instance of: scholarly article
title: Three term recurrence relation modulo ideal and orthogonality of polynomials of several variables
zbMATH Open document ID: 1070.42013
author: Dariusz Cichoń
author: Jan Stochel
author: Franciszek Hugon Szafraniec
published in: Journal of Approximation Theory
publication date: 2005-06-10T00:00:00Z
review text: This lengthy paper is bent on a thorough treatment of polynomials in several variables, orthogonal with respect to a positive measure supported on an algebraic set. To achieve their goal, the authors give a systematic treatment of the underlying concepts. The main difference to the study of orthogonal polynomials in one variable lies in the fact that there the equivalence between the determining measure of the linear functional and the well-known three term recurrence relation can be exploited. In the several variables case, there does not exist such a `simple' description; for a recent account on the theory (connected with the order in which the orthonormalization process is applied to `basis' functions), the reader is referred to [\textit{C. F. Dunkl, Y. Xu}, ``Orthogonal polynomials of several variables'', Encyclopedia of Mathematics and Its Applications. 81. (2001; Zbl 0964.33001)]. Apart from the problem of correctly defining a three term recurrence relation, there is also the possibility that the recurrence relation does not determine any orthogonality measure at all. A solution to these problems is found in the introduction of recurrence relations of metric type, satisfied modulo an ideal. The right type of ideals can be found using the methods and results due to \textit{J. Stochel} [Ark. Mat. 30, 133--48 (1992; Zbl 0819.47015)]. Implementing the program sketched in the introduction of the paper under review, the authors devote a substantial part of the paper to so-called quasi-orthogonality. The first four sections (page 13: Prerequisites; page 15: Monomial bases related to an ideal \(V\); page 18: \(V\)-bases; page 22: Rigid \(V\)-bases) contain basic ingredients, while in section 5 (page 24: Quasi-orthogonality: the real case) polynomials of several variables quasi-orthogonal with respect to a Hermitian linear functional are treated. The results of section 5 are looked into and discussed more deeply in the two sections thereafter (page 34: Quasi-orthogonal…
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 47B15
Mathematics Subject Classification ID: 47B25
zbMATH DE Number: 2174967
zbMATH Keywords: polynomials in several variables
zbMATH Keywords: orthogonal polynomials
zbMATH Keywords: three term recurrence relation
zbMATH Keywords: Favards theorem
zbMATH Keywords: ideal of polynomials
zbMATH Keywords: algebraic set
zbMATH Keywords: symmetric operator
zbMATH Keywords: selfadjoint operator
zbMATH Keywords: joint spectral measure
MaRDI profile type: Publication
full work available at URL: j.jat.2004.12.011
OpenAlex ID: W2007340536
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DOI: 10.1016/J.JAT.2004.12.011
description: scientific article; zbMATH DE number 2174967