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.
Bibliographical noteFunding 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