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On Helson matrices: moment problems, non-negativity, boundedness, and finite rank

Perfekt, K.-M. and Pushnitski, A. (2018) On Helson matrices: moment problems, non-negativity, boundedness, and finite rank. Proceedings of the London Mathematical Society, 116 (1). pp. 101-134. ISSN 1460-244X

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To link to this item DOI: 10.1112/plms.12068


We study Helson matrices (also known as multiplicative Hankel matrices), that is, infinite matrices of the form M(α)={α(nm)}∞n,m=1M(α)={α(nm)}n,m=1∞, where α is a sequence of complex numbers. Helson matrices are considered as linear operators on ℓ2(N)ℓ2(N). By interpreting Helson matrices as Hankel matrices in countably many variables we use the theory of multivariate moment problems to show that M(α) is non-negative if and only if αα is the moment sequence of a measure μ on R∞, assuming that α does not grow too fast. We then characterize the non-negative bounded Helson matrices M(α) as those where the corresponding moment measures μ 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.

Item Type:Article
Divisions:Science > School of Mathematical, Physical and Computational Sciences > Department of Mathematics and Statistics
ID Code:72120

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