Analysis Math., 44 (4) (2018), 605–630 DOI: 10.1007/s10476-018-0311-0 First published online June 21, 2018 THE CLOSED SPAN OF SOME EXPONENTIAL SYSTEM IN WEIGHTED BANACH SPACES ON THE REAL LINE AND A MOMENT PROBLEM E. ZIKKOS Department of Mathematics and Statistics, University of Cyprus, Nicosia, Cyprus e-mail: zik@ucy.ac.cy (Received September 30, 2016; revised December 31, 2016; accepted January 4, 2017) Abstract. Let {λn} ∞ n=1 be a strictly increasing sequence of positive real numbers diverging to infinity and let {μn} ∞ n=1 be a sequence of positive integers. Consider the exponential system EΛ = {t k e λnt : k =0, 1, 2,...,μn - 1} ∞ n=1 . Assuming the density condition lim t→∞ ∑ λn≤t μn t = d< ∞ and some other restrictions, we prove that every function in the closure of the linear span of EΛ in some weighted Banach spaces on the real line R is extended to an entire function represented by a Taylor–Dirichlet series g(z)= ∞  n=1  μn-1  k=0 c n,k z k  e λnz , c n,k ∈ C. We also consider a problem in a weighted L 2 (R) Hilbert space as well as a moment problem on the real line. 1. Introduction Let {λ n } ∞ n=1 be a strictly increasing sequence of positive real numbers diverging to infinity and let {μ n } ∞ n=1 be a sequence of positive integers, not necessarily bounded. Then the set with multiple terms {λ 1 ,λ 1 ,...,λ 1    μ1-times ,λ 2 ,λ 2 ,...,λ 2    μ2-times ,...,λ k ,λ k ,...,λ k    μk-times ,...} Key words and phrases: completeness, closure, weighted Banach space, Taylor–Dirichlet se- ries. Mathematics Subject Classification: 30B60, 30B50, 46E15. 0133-3852 c  2018 Akad´ emiai Kiad´o, Budapest