Closure of the Linear Span of an Exponential System in a Weighted Banach Space

For a certain class of sequences with repeated terms, {λn,μn}n=1 := {λ1,λ1, . . . ,λ1 } {{ } μ1 times ,λ2 ,λ2, . . . ,λ2 } {{ } μ2 times , . . . ,λk ,λk , . . . ,λk } {{ } μk times , . . .}, we prove that every function belonging to the closed span of the exponential system {xkeλnx : n ∈ N, k = 0,1,2, . . . ,μn −1}, in some weighted Banach spaces on the real line, extends analytically as an entire function by admitting a series representation of the form ∞ ∑ n=1 ( μn−1 ∑ k=0 cn,kz k ) eλnz, cn,k ∈ C, ∀ z ∈ C. Mathematics subject classification (2010): 30B50, 30B60, 46E15, 46E20.