On the multilinear Hausdorff problem of moments

TL;DR

This paper establishes necessary and sufficient conditions for the existence of measures corresponding to multi-index moment sequences, linking multilinear Hausdorff problems with multivariate cases and applications to stochastic processes.

Contribution

It introduces a comprehensive framework for the multilinear Hausdorff moment problem, connecting weak and strong formulations and relating them to multivariate and stochastic process contexts.

Findings

Characterization of measures for weak multilinear Hausdorff moments

Conditions for the existence of Radon measures solving the strong problem

Application to weakly harmonizable stochastic processes

Abstract

Given a multi-index sequence $\mu_{\mathbf{k}}$, $\mathbf{k} = (k_1,..., k_n) \in \mathbb{N}_0^n$, necessary and sufficient conditions are given for the existence of a regular Borel polymeasure $\gamma$ on the unit interval $I= [0,1]$ such that $\mu_{\mathbf{k}} = \int_{I^n} t_1^{k_1}\otimes ... \otimes t_n^{k_n} \gamma$. This problem will be called the weak multilinear Hausdorff problem of moments for $\mu_{\mathbf{k}}$. Comparison with classical results will allow us to relate the weak multilinear Hausdorff problem with the multivariate Hausdorff problem. A solution to the strong multilinear Hausdorff problem of moments will be provided by exhibiting necessary and sufficient conditions for the existence of a Radon measure $\mu$ on $[0,1]$ such that $L_\mu(f_1,..., f_n) = \int_{I} f_1(t) ... f_n(t) \mu (dt)$ where $L_\mu$ is the $n$-linear moment functional on the space of continuous functions on the unit interval defined by the sequence $\mu_{\mathbf{k}}$. Finally the previous results will be used to provide a characterization of a class of weakly harmonizable stochastic processes with bimeasures supported on compact sets.