Bohr property of bases in the space of entire functions and its generalizations

Abstract

We prove that if $(\varphi_n)_{n=0}^\infty, \; \varphi_0 \equiv 1, $ is a basis in the space of entire functions of $d$ complex variables, $d\geq 1,$ then for every compact $K\subset \mathbb{C}^d$ there is a compact $K_1 \supset K$ such that for every entire function $f= \sum_{n=0}^\infty f_n \varphi_n$ we have $\sum_{n=0}^\infty |f_n|\, \sup_{K}|\varphi_n| \leq \sup_{K_1} |f|.$ A similar assertion holds for bases in the space of global analytic functions on a Stein manifold with the Liouville Property.