## Abstract

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 H^{2}(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 H^{2}(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}.

Original language | British English |
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Pages (from-to) | 69-81 |

Number of pages | 13 |

Journal | Periodica Mathematica Hungarica |

Volume | 82 |

Issue number | 1 |

DOIs | |

State | Published - Mar 2021 |

## Keywords

- Dirichlet series
- Exponential system
- Hardy–Smirnov space