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

✍ Scribed by Lev Aizenberg; Aydin Aytuna; Plamen Djakov

Publisher: Elsevier Science
Year: 2001
Tongue: English
Weight: 137 KB
Volume: 258
Category: Article
ISSN: 0022-247X
DOI: 10.1006/jmaa.2000.7355

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✦ Synopsis

In the first part, we generalize the classical result of Bohr by proving that an m Ž analogous phenomenon occurs whenever D is an open domain in ‫ރ‬ or, more .

Ž . ϱ generally, a complex manifold and is a basis in the space of holomorphic n ns0

Ž . Ž . functions H D such that s 1 and z s 0, n G 1, for some z g D.

Namely, then there exists a neighborhood U of the point z such that, whenever a 0 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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