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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Functions which operate in the Fourier algebra of a compact group


Author: Daniel Rider
Journal: Proc. Amer. Math. Soc. 28 (1971), 525-530
MSC: Primary 46.80; Secondary 42.00
MathSciNet review: 0276792
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Abstract: Let $ A(G)$ be the Fourier algebra of a compact group $ G$. It is shown that a function defined on a closed convex subset of the plane operates in $ A(G)$ if and only if it is real analytic. This was shown by Helson, Kahane, Katznelson and Rudin when $ G$ is locally compact and abelian and by Dunkl when $ G$ is compact and contains an infinite abelian subgroup. A direct proof is given of the following lemma which is all that is needed in order to apply the proof of Helson, Kahane, Katznelson and Rudin ($ \vert\vert\;\vert\vert$ is the Fourier algebra norm).

Lemma. Let $ r > 0$ and $ {S_r}$ be the set of $ f \in A(G)$ such that $ f$ is real and $ \vert\vert f\vert\vert = r$. Then

$\displaystyle \mathop {\sup }\limits_{f \in {S_r}} \vert\vert{e^{if}}\vert\vert = {e^r}.$


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1971-0276792-3
PII: S 0002-9939(1971)0276792-3
Keywords: Fourier algebra, compact group, functions which operate, real analytic
Article copyright: © Copyright 1971 American Mathematical Society