\(N\)-extremal matrices of measures for an indeterminate matrix moment problem
Given \(\mu =(\mu_{i,j})_{i,j=1}^N\), a positive definite matrix of measures on \(\mathbb R\) with finite matrix moments \(S_k=\int_{\mathbb R} t^k d\mu \), \(k \geq 0\), denote by \(V_\mu \) the set of positive definite matrices of measures on \(\mathbb R\) having the same \(S_k\), \(k \geq 0\). The (matrix) moment problem is determinate if \(V_\mu =\{\mu \}\), and indeterminate otherwise. An indeterminate moment problem can have the degree of indetermination varying from 1 to \(N\), when it is called completely indeterminate. In analogy with the case \(N=1\), studied by M.\ Riesz, and for a completely indeterminate moment problem, the authors consider the \(N\)-extremal matrices \(W \in V_\mu \), i.e., such that polynomials are dense in \(L^2(W)\). Several properties of \(W\)'s are established, showing similarities but also differences with the classical scalar case (\(N=1\)).
scientific article; zbMATH DE number 1496043
scientific article