Incompleteness and closure of a linear span of exponential system in a weighted Banach space

Abstract

In this paper, we obtain a necessary and sufficient condition for the incompleteness of complex exponential polynomials in C α , where C α is a weighted Banach space of complex continuous functions f on the real axis R with f(t) exp(−α(t)) vanishing at infinity, in the uniform norm with respect to the weight α( t). We also prove that, if the above condition of incompleteness holds, then each function in the closure of complex exponential polynomials can be extended to an entire function represented by a Dirichlet series.