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Incompleteness and minimality of complex exponential system

Abstract: A necessary and sufficient condition is obtained for the incompleteness of a complex exponential system E(A, M) in C,, where C is the weighted Banach space consisting of all complex continuous functions f on the real axis R with f (t) exp(-alpha(t)) vanishing at infinity, in the uniform norm parallel to f parallel to alpha = sup{vertical bar f(t)e(-alpha(t)) vertical bar : t is an element of R} with respect to the weight alpha(t). If the incompleteness holds, then the complex exponential system E (A, M) is minimal and each function in the closure of the linear span of complex exponential system E(A, M) can be extended to an entire function represented by a Taylor- Dirichlet series.