Multivariate Gaussian cubature formulae
type: Item
instance of: scholarly article
title: Multivariate Gaussian cubature formulae
zbMATH Open document ID: 0814.41022
DOI: 10.1007/BF01193547
author: Yuan Xu
author: Hubert Berens
author: H. J. Schmid
published in: Archiv der Mathematik
publication date: 1995-03-19T00:00:00Z
review text: Gaussian quadrature can be briefly described as follows: There is a one- parameter family of minimal formulae of degree \(2n-2\) the nodes of which are the roots of \(p_ n+ \rho p_{n-1}\), \(\rho\neq 0\), where \(\{p_ n\}_{n=0}^ \infty\), denotes the system of orthogonal polynomials w.r.t. the integral under consideration; for \(\rho=0\) one even obtains a formula of degree \(2n-1\). In the multi-dimensional case, the structure of minimal cubature formulae is more complicated. For integrals on \(\mathbb{R}^ d\), \(d\geq 1\), \(\dim \prod_{[m/2]}^ d\) is a lower bound for the number of nodes of a formula of degree \(m\), where \(\prod_ m^ d\) is the linear space of all polynomials of total degree less than or equal to \(m\). As stated above, the bound is attained for \(d=1\) for arbitrary integrals; it is not, however, for classical integrals defined on the plane as has been observed by J. Radon in the late 40s. I. P. Mysovskikh and his students studied this problem intensively and came up with partial answers. Here we construct two classes of integrals on \(\mathbb{R}^ d\) allowing Gaussian cubature.
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 41A63
zbMATH DE Number: 734520
zbMATH Keywords: multivariate orthogonal polynomials
zbMATH Keywords: Gaussian cubature
MaRDI profile type: Publication
cites work: Q4170993
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full work available at URL: https://doi.org/10.1007/bf01193547
OpenAlex ID: W2008054699
description: scientific article; zbMATH DE number 734520