Completeness of an exponential system in weighted Banach spaces and closure of its linear span

For a real multiplicity sequence Λ={ λ_n, μ_n }_{n = 1}^∞, that is, a sequence where { λ_n } are distinct positive real numbers satisfying 0 < λ_n < λ_{n + 1} {mapping} ∞ as n {mapping} ∞ and where each λ_n appears μ_n times, we associate the exponential systemE_Λ = { t^k e^{λn t} : k = 0, 1, 2, ..., μ_n - 1 }_{n = 1}^∞ .For a certain class of multiplicity sequences, we give necessary and sufficient conditions in order for E_Λ to be complete in some weighted Banach space of continuous functions on R, and in some weighted L^p (- ∞, ∞) spaces of measurable functions, with p ∈ [1, ∞). We also prove that if E_Λ is incomplete in the weighted spaces, then every function in the closure of the linear span of E_Λ*, where E_{Λ *} = { t^{μn - 1} e^{λn t} }_{n = 1}^∞, can be extended to an entire function represented by a Taylor-Dirichlet seriesg (z) = underover(∑, n = 1, ∞) c_n z^{μn - 1} e^{λn z}, c_n ∈ C .Furthermore, we prove that E_Λ is minimal in the weighted spaces if and only if it is incomplete.