Linear projections which implement balayage in Fourier transforms

Abstract:

Let $\Lambda$ be a closed and discrete or compact subset of a second countable ${\text {LCA}}$ group $G$ and $E$ a subset of the dual group. Balayage is said to be possible for $(\Lambda ,\;E)$ if for every finite measure $\mu$ on $G$ there is some measure $\nu$ on $\Lambda$ whose Fourier transform, $\hat \nu$, agrees on $E$ with $\hat \mu$. If balayage is assumed possible just when $\mu$ is a point measure (with the norms of all the measures $\nu$ bounded by some constant), then there is a bounded linear projection, ${B_\Lambda }$, from the measures on $G$ onto those on $\Lambda$ with ${({B_\Lambda }\mu )^ \wedge } = \hat \mu$ on $E$. An application is made to balayage in product groups.