Dirichlet polynomials and a moment problem

TL;DR

This paper explores the characterization of positive linear functionals on Dirichlet polynomials, introducing Hausdorff log-moment sequences and extending classical moment problem results to this setting.

Contribution

It introduces Hausdorff log-moment sequences and establishes a Riesz-Haviland type theorem for Dirichlet polynomials, linking moment sequences to completely monotone functions.

Findings

Any Hausdorff log-moment sequence is a linear combination of specific sequences.

Such sequences are uniquely determined by an associated completely monotone function.

The paper extends classical moment problem results to Dirichlet polynomial spaces.

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 $\{1, 0, \ldots, \}$ and $\{f(\log(n)\}_{n \geqslant 1}$ for a completely monotone function $f : [0, \infty) \rightarrow [0, \infty).$ Moreover, such an $f$ is uniquely determined by the sequence in question.