The vector-valued Stieltjes moment problem with general exponents

Abstract

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 $\mathbf{f}$ on $(0,\infty)$ (satisfying a mild regularity condition) such that $$\int_{0}^{\infty} t^{z_{n}} \mathbf{f}(t) dt = e_{n}, \qquad \forall n \in \mathbb{N},$$ where the integral should be understood as a Pettis integral. Moreover, in this case, we show that there always exists a solution $\mathbf{f}$ 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 Dur\'an [Math. Nachr. 158 (1992), 175-194]. Our work is based upon his result.