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 \(\{1, 0, \ldots , \}\) 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.