We study Helson matrices (also known as multiplicative Hankel matrices), i.e. infinite matrices of the form $M(alpha) = {alpha(nm)}_{n,m=1}^infty$, where $alpha$ is a sequence of complex numbers. Helson matrices are considered as linear operators on $ell^2(mathbb{N})$. By interpreting Helson matrices as Hankel matrices in countably many variables we use the theory of multivariate moment problems to show that $M(alpha)$ is non-negative if and only if $alpha$ is the moment sequence of a measure $mu$ on $mathbb{R}^infty$, assuming that $alpha$ does not grow too fast. We then characterize the non-negative bounded Helson matrices $M(alpha)$ as those where the corresponding moment measures $mu$ are Carleson measures for the Hardy space of countably many variables. Finally, we give a complete description of the Helson matrices of finite rank, in parallel with the classical Kronecker theorem on Hankel matrices.
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