Generalization of a Theorem of Bohr for Bases in Spaces of Holomorphic Functions of Several Complex Variables

Lev Aizenberg; Aydin Aytuna; Plamen Djakov

doi:10.1006/jmaa.2000.7355

Generalization of a Theorem of Bohr for Bases in Spaces of Holomorphic Functions of Several Complex Variables

Lev Aizenberg, Aydin Aytuna, Plamen Djakov

Department of Mathematics

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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 φ₀=1 and φ_n(z₀)=0, n≥1, for some z₀∈D. Namely, then there exists a neighborhood U of the point z₀ 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
Pages (from-to)	429-447
Number of pages	19
Journal	Journal of Mathematical Analysis and Applications
Volume	258
Issue number	2
DOIs	https://doi.org/10.1006/jmaa.2000.7355
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

Funding

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

Funders	Funder number
NRF of Bulgaria	MM-808
United States-Israel Binational Science Foundation	94-00113

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title = "Generalization of a Theorem of Bohr for Bases in Spaces of Holomorphic Functions of Several Complex Variables",

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 Cm (or, more generally, a complex manifold) and (φn)∞n=0 is a basis in the space of holomorphic functions H(D) such that φ0=1 and φn(z0)=0, n≥1, for some z0∈D. Namely, then there exists a neighborhood U of the point z0 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.",

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Generalization of a Theorem of Bohr for Bases in Spaces of Holomorphic Functions of Several Complex Variables

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