Closure of an exponential system in some Hardy–Smirnov spaces

Let Ω be an open, simply connected, bounded subset of the complex plane C with a rectifiable boundary ∂Ω. We investigate the relation between the Hardy–Smirnov space H²(Ω) and the closed span of the exponential system {enz}n=1∞ with respect to the Hardy–Smirnov norm ‖ · ‖ _Ω. Depending on the “height” of Ω , the two spaces may coincide or not. In the latter case we characterize the closure of the system by proving that any element f extends analytically in some half-plane as a Dirichlet series f(z)=∑n=1∞cnenz. Finally we consider the converse problem: assuming that such a Dirichlet series is in H²(Ω) , we provide some sufficient conditions for f to be in the closed linear span of the system with respect to the Hardy–Smirnov norm ‖ · ‖ _Ω.