Closure of an exponential system in some Hardy–Smirnov spaces

Abstract

Let $\Omega$ be an open, simply connected, bounded subset of the complex plane $\mathbb {C}$ with a rectifiable boundary ${\partial {\Omega }}$. We investigate the relation between the Hardy–Smirnov space $H^2(\Omega )$ and the closed span of the exponential system $\{e^{nz}\}_{n=1}^{\infty }$ with respect to the Hardy–Smirnov norm $\Vert \cdot \Vert _\Omega$. Depending on the “height” of $\Omega$, 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 $f(z)=\sum _{n=1}^{\infty } c_n e^{nz}$. Finally we consider the converse problem: assuming that such a Dirichlet series is in $H^2(\Omega )$, we provide some sufficient conditions for f to be in the closed linear span of the system with respect to the Hardy–Smirnov norm $\Vert \cdot \Vert _\Omega$.