Functions which are Fourier-Stieltjes transforms

Abstract:

Let $G$ be a locally compact abelian group, $\hat G$ the dual group, $M(G)$ the algebra of regular bounded Borel measures on $G$, and $M{(G)^\wedge }$ the algebra of Fourier-Stieltjes transforms. The purpose of this paper is to characterize those continuous functions on $\hat G$ which belongs to $M(X)^\wedge$, where $X$ is a closed subset of $G$ and $M(X) = \{ \mu \in M(G)$: the support of $\mu$ is contained in $X\}$.