Functions which operate in the Fourier algebra of a compact group

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 ($||\;||$ 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 $||f|| = r$. Then \[ \sup \limits _{f \in {S_r}} ||{e^{if}}|| = {e^r}.\]