Classical orthogonal polynomials in two variables: a matrix approach
type: Item
instance of: scholarly article
title: Classical orthogonal polynomials in two variables: a matrix approach
zbMATH Open document ID: 1069.42015
author: Lidia Fernández
author: Teresa E. Pérez
author: Miguel A. Piñar
published in: Numerical Algorithms
publication date: 2005-05-20T00:00:00Z
review text: In the past, several authors have tried to establish a characterization of two-variable orthogonal polynomials along the line of the Bochner-classification of classical orthogonal polynomials in 1929. In 1967, \textit{H. L. Krall} and \textit{I. M. Sheffer} [Ann. Mat Pura Appl., IV. Ser. 76, 325--376 (1967; Zbl 0186.38602)] extended the Bochner approach to the study of a second order partial differential equation for the weight function \(w(x,y)\) with polynomial coefficients of certain degrees in \(x\) and \(y\), but that classification did not include for instance the tensor product of the Jacobi polynomials in one variable as these did not satisfy a differential equation of the type studied. The authors of this paper, however, succeeded in giving an extension of the just mentioned results that does include tensor products of one variable polynomials. The main tools are 2D moment functionals taken from the topological dual of the space of polynomials in two variables and the calculus for these functionals (including distributional counterparts to the gradient and divergence operators, originally defined on the polynomials). An important role is played by the matrix equation \[ L[\mathbf{P}_n^T]=\text\textbf{P}_n^T\Lambda_n^T\;(n\geq 0), \] where \(L[p]=\text{div}(\phi\nabla p)+{\tilde{\psi}}\nabla p\) and (\textbf{P}\(_n)_n\) is the corresponding weak orthogonal polynomial system. A quite interesting paper.
reviewed by: Marcel G. de Bruin
Mathematics Subject Classification ID: 42C05
Mathematics Subject Classification ID: 33C50
zbMATH DE Number: 2170133
zbMATH Keywords: orthogonal polynomials in two variables
zbMATH Keywords: classical orthogonal polynomials
MaRDI profile type: Publication
full work available at URL: s11075-004-3625-x
OpenAlex ID: W2356207438
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DOI: 10.1007/S11075-004-3625-X
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description: scientific article; zbMATH DE number 2170133