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

For a real multiplicity sequence ΛΛ={λn,μn}n=1∞, that is, a sequence where {λn}{λn} are distinct positive real numbers satisfying 0<λn<λn+1↦∞0<λn<λn+1↦∞ as n↦∞n↦∞ and where each λnλn appears μnμn times, we associate the exponential systemEΛ={tkeλnt:k=0,1,2,…,μn-1}n=1∞.For a certain class of multiplicity sequences, we give necessary and sufficient conditions in order for EΛEΛ to be complete in some weighted Banach space of continuous functions on RR, and in some weighted Lp(-∞,∞)Lp(-∞,∞) spaces of measurable functions, with p∈[1,∞)p∈[1,∞). We also prove that if EΛEΛ is incomplete in the weighted spaces, then every function in the closure of the linear span of EΛ*EΛ*, where EΛ*={tμn-1eλnt}n=1∞, can be extended to an entire function represented by a Taylor–Dirichlet seriesg(z)=∑n=1∞cnzμn-1eλnz,cn∈C.Furthermore, we prove that EΛEΛ is minimal in the weighted spaces if and only if it is incomplete.