Interpolation by transforms of discrete measures

Abstract:

Let $G$ be a compact abelian group, and $\Gamma$ its character group. Given $E \subset \Gamma ,{E^a}$ denotes the set of all accumulation points of $E$ in $\overline \Gamma$, the Bohr compactification of $\Gamma$. In this paper it is shown that the inclusion ${({L^1}(G))^ \wedge }{|_E} \subset {({l^1}(G))^ \wedge }{|_E}$ obtains if and only if $E \cap {E^a} = \emptyset$ and there exists a measure $\mu \epsilon M(G)$ such that $\widehat \mu = 1$ on $E$ and $\widehat \mu = 0$ on $\Gamma \cap {E^a}$.