On the multilinear Hausdorff problem of moments

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 $I= [0,1]$suchthat$\mu{$\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 $L_{\mu }(f_1,\mathrm{.}\mathrm{.}\mathrm{.},f_n)={\int }_I{f}_1(t)\mathrm{.}\mathrm{.}\mathrm{.}{f}_n(t)\mu (dt)$ 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.