## Abstract

In the first part, we generalize the classical result of Bohr by proving that an analogous phenomenon occurs whenever D is an open domain in C^{m} (or, more generally, a complex manifold) and (φ_{n})^{∞}_{n=0} is a basis in the space of holomorphic functions H(D) such that φ_{0}=1 and φ_{n}(z_{0})=0, n≥1, for some z_{0}∈D. Namely, then there exists a neighborhood U of the point z_{0} such that, whenever a holomorphic function on D has modulus less than 1, the sum of the suprema in U of the moduli of the terms of its expansion is less than 1 too. In the second part we consider some natural Hilbert spaces of analytic functions and derive necessary and sufficient conditions for the occurrence of Bohr's phenomenon in this setting.

Original language | English |
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Pages (from-to) | 429-447 |

Number of pages | 19 |

Journal | Journal of Mathematical Analysis and Applications |

Volume | 258 |

Issue number | 2 |

DOIs | |

State | Published - 15 Jun 2001 |

### Bibliographical note

Funding Information:1Author’s research was supported by the BSF Grant 94-00113. 2Author’s research was supposed in part by NRF of Bulgaria, Grant MM-808