This paper considers
some characterizations of exponential polynomials in C(G), the set of all
continuous complex valued functions on a σ-compact locally compact Abelian
group G. For f ∈ C(G), Uf will denote the subspace of C(G) obtained by
taking finite linear combinations of translates of f. It is known that f is an
exponential polynomial if and only if Uf is of finite dimension. Our main result is
to show that f is an exponential polynomial when Uf is closed in C(G) if
C(G) is given the topology of convergence uniform on all compact subsets of
G.
Further characterizations of exponential polynomials are given when G is real
Euclidean n-space, Rn.
