2021/07/22 by Chavan, Sameer, Sahu, Chaman Kumar
#FOS: Mathematics #Functional Analysis (math.FA)
paper · doi:10.48550/arxiv.2107.10603
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, ∞), \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, …, \ and \f(log(n)\n \geqslant 1 for a completely monotone function f : [0, ∞) → [0, ∞). Moreover, such an f is uniquely determined by the sequence in question.