The vector-valued Stieltjes moment problem with general exponents

We characterize the sequences of complex numbers \((z_{n})_{n \in \mathbb {N}}\) and the locally complete (DF)-spaces E such that for each \((e_{n})_{n \in \mathbb {N}} \in E^\mathbb {N}\) there exists an E-valued function \(\textbf{f}\) on \((0,\infty )\) (satisfying a mild regularity condition) such that

$$\begin{aligned} \int _{0}^{\infty } t^{z_{n}} \textbf{f}(t) dt = e_{n}, \qquad \forall n \in \mathbb {N}, \end{aligned}$$

where the integral should be understood as a Pettis integral. Moreover, in this case, we show that there always exists a solution \(\textbf{f}\) that is smooth on \((0,\infty )\) and satisfies certain optimal growth bounds near 0 and \(\infty \). The scalar-valued case \((E = \mathbb {C})\) was treated by Durán (Math Nachr 158:175–194, 1992). Our work is based upon his result.