Complex moment problems and recursive relations of Fibonacci type (Q1049595)

The authors study a full complex moment problem, relating it to truncated problems of the following form. The truncated \(K\)-moment problem for a doubly indexed sequence of complex numbers \[ \gamma=\gamma^{(2n)}: \gamma_{00},\gamma_{01},\gamma_{10},\dots,\gamma_{0\,2n},\dots,\gamma_{2n\,0} \] is to find a positive Borel measure \(\mu\) such that supp\((\mu)\subset \mathbb C\) and \[ \gamma_{i,j}=\int_{\mathbb C}\,\overline{z}^iz^j\,d\mu(z),\text{ for every }i,j\text{ with }0\leq i+j\leq 2n. \] The authors restrict themselves in the paper to an \(r\)-atomic representing measure, which they show to be equivalent to rows of certain moment-matrices being \(r\)-generalized Fibonacci sequences with common characteristic polynomial of degree \(r\) having distinct roots, and prove as main result for the full moment problem: Let \(M\) be an infinite matrix of finite rank \(r\), then the following assertions are equivalent: 1. \(M\) is a positive infinite moment matrix, 2. \(M(r)\geq 0\) and the family of sequences \(\{\gamma_{ij}\}_{j\geq 0}\;(i\geq 0)\) are \(r\)-generalized Fibonacci sequences with the same characteristic polynomial having distinct roots \(t_0,\ldots,t_{r-1}\in\mathbb C\). More precisely, \[ \gamma_{in}=\sum_{j=0}^{r-1}\,\rho_{i,j}t_j^n, \] with \(\rho_{i,j}=\rho_{0,j}\overline{t_j}^i\). Moreover, they give an explicit description of a constructive process to determine a flat extension of the moment matrix (for simplicity they take \(r=3\)).