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 $$\Vert \cdot \Vert _\Omega $$.

中文翻译：

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一些 Hardy-Smirnov 空间中指数系统的闭包

令 $$\Omega $$ 是复平面 $$\mathbb {C}$$ 的一个开放的、简单连通的、有界子集，具有可修正的边界 $${\partial {\Omega }}$$。我们研究了 Hardy–Smirnov 空间 $$H^2(\Omega )$$ 与指数系统的闭跨度 $$\{e^{nz}\}_{n=1}^{\infty 之间的关系}$$ 相对于 Hardy–Smirnov 范数 $$\Vert \cdot \Vert _\Omega $$。根据 $$\Omega $$ 的“高度”，两个空间可能重合或不重合。在后一种情况下，我们通过证明任何元素 f 在某个半平面中解析扩展为狄利克雷级数 $$f(z)=\sum _{n=1}^{\infty } c_n e 来表征系统的闭包^{nz}$$。最后我们考虑逆向问题：假设这样的狄利克雷级数在 $$H^2(\Omega )$$ 中，