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Proceedings of the American Mathematical Society

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

 

 

Unique balayage in Fourier transforms on compact abelian groups


Author: George S. Shapiro
Journal: Proc. Amer. Math. Soc. 70 (1978), 146-150
MSC: Primary 43A05
DOI: https://doi.org/10.1090/S0002-9939-1978-0477600-4
MathSciNet review: 0477600
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Abstract: Let K be a compact subset of the compact abelian group G and let $ \Lambda $ be a subset of the dual group $ \Gamma $. Unique balayage is said to be possible for $ (K,\Lambda )$ if, for every $ \mu $ in $ M(G)$, there is a unique $ \nu $ in $ M(K)$ whose Fourier transform, $ \hat \nu $, agrees on $ \Lambda $ with $ \hat \mu $.

We prove that in order that there be any K with unique balayage possible for $ (K,\Lambda ),\Lambda $ must belong to the coset ring of $ \Gamma $. The converse of this statement is false. Some examples are given for the case where G is the circle group.


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

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Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1978-0477600-4
Keywords: Balayage in Fourier transforms, compact abelian groups, idempotent measures, coset ring
Article copyright: © Copyright 1978 American Mathematical Society