Abstract: In this paper, the author obtains a necessary and sufficient condition
for the incompleteness of complex exponential polynomials in the Banach space H_α consisting of all continuous functions f on the closed right half strip I_α={z=x+iy: x≥ 0, |y|≤α} (α > 0 ), analytic in its interior and such that f(z) vanishing uniformly at infinity. The author also proves that, if some conditions of incompleteness hold, then each function in the closure of the set of complex exponential polynomials can be extended to an analytic function represented by a Dirichlet series.

Key words: Incompleteness, Closure, Exponential polynomials