Dirichlet polynomials and a moment problem

Abstract

Consider a linear functional L defined on the space D[s] of Dirichlet polynomials with real coefficients and the set D+[s] of nonnegative elements in D[s]. An analogue of the Riesz-Haviland theorem in this context asks: What are all D+[s]-positive linear functionals L, which are moment functionals? Since the space D[s], when considered as a subspace of C([0, ∞), 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, . . . , } and {f (log(n)} n 1 for a completely monotone function f : [0, ∞) → [0, ∞). Moreover, such an f is uniquely determined by the sequence in question.