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

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