The closed span of an exponential system in the Banach spaces Lp(γ,β) and C[γ,β]

Let Λ={λn,μn}n=1· be a multiplicity-sequence, that is, a sequence where the λn are complex numbers diverging to infinity, λn≠λk for n≠k, the |λn| are in an increasing order, and each λn appears μn times. We associate to Λ the exponential system EΛ={xkeλnx:k=0,1,2,...,μn-1}n=1·. In the spirit of the Müntz-Szász theorem and assuming that Λ belongs to a certain class of sequences that we denote by Uη, we investigate the closure of span(EΛ) in the Banach spaces Lp(γ,β) and C[γ,β], where -·<γ<β<· and p≥1. We prove that the closed span of EΛ in C[γ,β] is the subspace of functions that admit a Taylor-Dirichlet series representation ∑n=1·(∑j=0μn-1cn,jxj)eλnx,∀xε[γ,β). A similar result holds for the closed span of EΛ in Lp(γ,β) in an almost everywhere sense.