Idempotents and moment problem for discrete measure
Let \(\mathcal{P}_d\) be the algebra of all polynomials in \(t=(t_1,\ldots, t_d)\in\mathbb R^d\), and let \(\gamma=(\gamma_\alpha)_{\alpha\in\mathbb N^d}\) be a multi-sequence of complex numbers. The linear functional \(\Lambda_\gamma:\mathcal{P}_d\mapsto\mathbb C\), given by the assignment \(t^\alpha\mapsto\gamma_\alpha\), \(\alpha\in\mathbb N^d\), is called the (associated) Riesz functional, usually also supposed to be square positive, that is, \(\Lambda_\gamma(\vert P\vert^2)\ge 0\) for all \(P\in \mathcal{P}_d\). Finding an integral representation for \(\Lambda_\gamma\) means to solve the moment problem for the sequence \(\gamma\). When the sequence \(\gamma\) is finite, we have a truncated moment problem, otherwise it is a full moment problem. In the article under review, the authors adopt the reviewer's point of view, who showed that a square positive Riesz functional \(\Lambda\), corresponding to a truncated moment problem, has a representing measure if and only if an associated Hilbert space has a an orthogonal basis consisting of idempotents with respect to \(\Lambda\). Analysing deeply the concept of idempotent, the authors investigate the full moment problem for discrete measures, giving a sufficient condition for the existence of a discrete integral representation of the Riesz functional, which happens to be also necessary in the bounded shift case, in particular, when the density of polynomials is assumed. The concept of \(\Lambda\)-multiplicative element, also introduced by the reviewer, is further studied, the authors giving criteria insuring are that such an element is the characteristic function of a singleton.
scientific article; zbMATH DE number 7390156
scientific article