Unique balayage in Fourier transforms on compact abelian groups

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.