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On riesz subsets of abelian discrete groups

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Abstract

We study the class of the Riesz subsets of abelian discrete groups, that is, the sets for which the F. and M. Riesz theorem extends. We show that the “classical” tools of the theory — Riesz projections, localization in the Bohr sense, products — are leading to Riesz sets which are satisfying nice additional properties, e.g., the Mooney-Havin result extends to this class. We give an alternative proof of a result of A. B. Alexandrov, and we improve a construction of H. P. Rosenthal. The connection is made between this class and theM-structure theory. We show a result of convergence at the boundary for holomorphic functions on the polydisc. The Bourgain-Davis result on convergence of analytic martingales is improved.

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Authors and Affiliations

  1. Equipe d’Analyse, Université Paris VI, Tour 46-0, 4eme étage, 4, Place Jussieu, 75230, Paris Cedex 05, France

    Gilles Godefroy

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  1. Gilles Godefroy
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Godefroy, G. On riesz subsets of abelian discrete groups. Israel J. Math. 61, 301–331 (1988). https://doi.org/10.1007/BF02772575

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