Dirichlet polynomials and a moment problem

Abstract

Consider a linear functional $L$ defined on the space $\mathcal{D}[s]$ of Dirichlet polynomials with real coefficients and the set ${\mathcal{D}}_{+}[s]$ of non-negative elements in $\mathcal{D}[s].$ An analogue of the Riesz-Haviland theorem in this context asks: What are all ${\mathcal{D}}_{+}[s]$-positive linear functionals $L,$ which are moment functionals? Since the space $\mathcal{D}[s],$ when considered as a subspace of $C([0,\infty ),\mathbb{R}),$ fails to be an adapted space in the sense of Choquet, the general form of Riesz-Haviland theorem is not applicable in this situation. In an attempt to answer the forgoing question, we arrive at the notion of a moment sequence, which we call the Hausdorff log-moment sequence. Apart from an analogue of the Riesz-Haviland theorem, we show that any Hausdorff log-moment sequence is a linear combination of $\{1,0,\dots ,\}$ and $\{f(\log (n){\}}_{n⩾1}$ for a completely monotone function $f:[0,\infty )\to [0,\infty ).$ Moreover, such an $f$ is uniquely determined by the sequence in question.

20 pages