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

AbstractFor a real multiplicity sequence Λ={λn,μn}n=1∞, that is, a sequence where {λn} are distinct positive real numbers satisfying 0<λn<λn+1↦∞ as n↦∞ and where each λn appears μn 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 EΛ to be complete in some weighted Banach space of continuous functions on R, and in some weighted Lp(-∞,∞) spaces of measurable functions, with p∈[1,∞). We also prove that if EΛ is incomplete in the weighted spaces, then every function in the closure of the linear span of 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 EΛ is minimal in the weighted spaces if and only if it is incomplete