Moment problem (Q1865681)

Let \(({\mathbb S},+,*)\) be a (commutative) \(*\)-semi-group, and let \({\mathbb S}^*\) be the set of all semi-characters on \(\mathbb S\). A moment function \(\phi:{\mathbb S}\to {\mathbb C}\) is a map such that \(\phi(s)=\int_{{\mathbb S}^*}\rho(s)d\nu(s)\), where \(\nu\) is a positive (Radon) measure on \({\mathbb S}^*\). The aim of this paper is to characterize the moment functions, defined on \(\mathbb S\), which are not necessarily of finite type. The main results generalize certain assertions proved by \textit{M. Putinar} and \textit{F.-H. Vasilescu} [Ann. Math. (2) 149, No. 3, 1087--1107 (1999; Zbl 0939.44003)], stated on \({\mathbb R}^n\). As an application, the author characterizes some nondefinite sequences having a singularity at the point \((1,\ldots,1)\in{\mathbb C}^p\) of order at least \(2\kappa\), where \(\kappa\) is a positive integer.