Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
MathSciNet review: 0283500
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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}$.

References [Enhancements On Off] (What's this?)

  • [1] Walter Rudin, Fourier analysis on groups, Interscience Tracts in Pure and Applied Mathematics, No. 12, Interscience Publishers (a division of John Wiley and Sons), New York-London, 1962. MR 0152834
  • [2] Sadahiro Saeki, Spectral synthesis for the Kronecker sets, J. Math. Soc. Japan 21 (1969), 549–563. MR 0254525
  • [3] Nicholas Th. Varopoulos, Sur les ensembles parfaits et les séries trigonométriques, C. R. Acad. Sci. Paris 260 (1965), 4668-4670; ibid. 260 (1965; 5165-5 168; ibid. 260 (1965), 5997–6000 (French). MR 0178309
  • [4] N. Th. Varopoulos, Tensor algebras and harmonic analysis, Acta Math. 119 (1967), 51–112. MR 0240564

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 42.58

Retrieve articles in all journals with MSC: 42.58

Additional Information

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