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

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.