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