General Stieltjes moment problems for rapidly decreasing smooth functions

We give (necessary and sufficient) conditions over a sequence $\left\{ f_{n}\right\} _{n=0}^{\infty}$ of functions under which every generalized Stieltjes moment problem $$\int_{0}^{\infty} f_{n}(x)\phi(x)\mathrm{d} x=a_{n}, \ \ \ n\in\mathbb{N},$$ has solutions $\phi\in\mathcal{S}(\mathbb{R})$ with $\operatorname*{supp} \phi\subseteq[0,\infty)$. Furthermore, we consider more general problems of this kind for measure or distribution sequences $\left\{ f_{n}\right\} _{n=0}^{\infty}$. We also study vector moment problems with values in Frechet spaces and multidimensional moment problems.