Quadrature formulas for matrix measures --- a geometric approach
type: Item
instance of: scholarly article
title: Quadrature formulas for matrix measures --- a geometric approach
zbMATH Open document ID: 1019.41020
DOI: 10.1016/S0024-3795(02)00532-3
published in: Linear Algebra and its Applications
publication date: 2003-05-04T00:00:00Z
review text: Let \(\mu=(\mu_{ij})\) be a \(p\times p\) matrix measure of finite signed measures \(\mu_{ij}\) on the Borel sets of the real line, such that \(\mu(A)\) is symmetric and nonnegative definite for all Borel sets \(A\subset \mathbb{R}\). A quadrature formula for \(\mu\), with degree of exactness \(n\), can be constructed by considering real nodes \(x_k\), \(k=1,\dots ,m\), and matrix quadrature weights \(\Lambda_k \in \mathbb{R}^{p \times p}\) such that \[ S_\nu(\mu):=\int t^\nu d\mu(t)=\sum_{k=1}^m x_k^\nu \Lambda_k, \quad \nu=0,\dots ,n.\tag{1} \] The present paper gives new proofs of the existence and uniqueness of a quadrature formula of precision \(n=2m-1\), such that the sum of ranks of the coefficients \(\Lambda_k\) is equal to \(mp\). In connection with a recent paper by \textit{A. J. Durán} and \textit{B. Polo} [Gaussian quadrature formulas for matrix weights, Linear Algebra Appl. 355, 119-146 (2002)] some new quadrature formulas are also presented for which the sum of ranks is \(mp+h\) \((0<h<p)\). Let \(M_{n+1}=\{(S_0(\eta),\dots ,S_{n-1}(\eta),S^-_{n}(\eta))\), \(\eta \in \Xi\}\), where \(\Xi\) is the set of all matrix measures with finite moments up to the order \(n\), and \(S^-_{n}(\mu)\) is a suitable modification of the \(n\)-th moment. An alternative approach is given to the construction of a matrix quadrature formula which interpolates matrix polynomials. It consists in connecting identities (1) with the fact that the degree of exactness corresponds to a representation of particular boundary points of the moment space \(M_{2m}\), in terms of a matrix measure with finite support.
Mathematics Subject Classification ID: 41A55
Mathematics Subject Classification ID: 65D32
Mathematics Subject Classification ID: 15A99
zbMATH DE Number: 1903576
zbMATH Keywords: numerical integration
zbMATH Keywords: quadrature formulas
zbMATH Keywords: interpolation quadrature formulas
zbMATH Keywords: matrix measures
zbMATH Keywords: moment spaces
zbMATH Keywords: orthogonal matrix polynomials
author: Holger Dette
author: William J. Studden
reviewed by: J. Illán
MaRDI profile type: Publication
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full work available at URL: s0024-3795(02)00532-3
OpenAlex ID: W3021213911
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description: scientific article; zbMATH DE number 1903576