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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

A characterization of $ {\rm SH}$-sets


Author: Sadahiro Saeki
Journal: Proc. Amer. Math. Soc. 30 (1971), 497-503
MSC: Primary 42.58
DOI: https://doi.org/10.1090/S0002-9939-1971-0283500-9
MathSciNet review: 0283500
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Abstract: Let G be a locally compact abelian group, and $ A(G)$ the Fourier algebra on G. A Helson set in G is called an SH-set if it is also an S-set for the algebra $ A(G)$. In this article we prove that a compact subset K of G is an SH-set if and only if there exists a positive constant b such that: For any disjoint closed subsets $ {K_0}$ and $ {K_1}$ of K, we can find a function u in $ A(G)$ such that $ \left\Vert u \right\Vert < b,u = 1$ on some neighborhood of $ {K_0}$, and $ u = 0$ on some neighborhood of $ {K_1}$.


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DOI: https://doi.org/10.1090/S0002-9939-1971-0283500-9
Keywords: Locally compact abelian group, Helson set, S-set, SH-set, quasi-Kronecker set, $ {K_p}$-set, character, pseudomeasure
Article copyright: © Copyright 1971 American Mathematical Society