On the multilinear Hausdorff problem of moments

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.