The Closed Span of some Exponential System in Weighted Banach Spaces on the Real Line and a Moment Problem

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{tkeλnt:k=0,1,2,3,..,μn−1}n=1∞ Assuming the density condition limt→∞∑λn≤tμnt=d<∞ and some other restrictions, we prove that every function in the closure of the linear span of E_Λ 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∞(∑k=0μn−1cn,kzk)eλnz,cn,k∈C We also consider a problem in a weighted L²(ℝ) Hilbert space as well as a moment problem on the real line.