Generalization of a theorem of Bohr for bases in spaces of holomorphic functions of several complex variables


Aizenberg L., Aytuna A., Djakov P.

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, vol.258, no.2, pp.429-447, 2001 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 258 Issue: 2
  • Publication Date: 2001
  • Doi Number: 10.1006/jmaa.2000.7355
  • Title of Journal : JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
  • Page Numbers: pp.429-447

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 (phi (n))(n=0)(infinity) is a basis in the space of holomorphic functions H(D) such that phi (0) = 1 and phi (n)(z(0)) = 0, n greater than or equal to 1, for some z(0) is an element of D. Namely, then there exists a neighborhood U of the point to 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. (C) 2001 Academic Press