Quadrature formulas for matrix measures --- a geometric approach
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.
scientific article; zbMATH DE number 1903576
scientific article