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

Abstract

We prove that if $({\phi }_n{)}_{n=0}^{\infty },{\phi }_0\equiv 1,$ is a basis in the space of entire functions of $d$ complex variables, $d\ge 1,$ then for every compact $K\subset {\mathbb{C}}^d$ there is a compact $K_1\supset K$ such that for every entire function $f={\sum }_{n=0}^{\infty }{f}_n{\phi }_n$ we have ${\sum }_{n=0}^{\infty }|f_n|{\sup }_K|{\phi }_n|\le {\sup }_{K_1}|f|.$ A similar assertion holds for bases in the space of global analytic functions on a Stein manifold with the Liouville Property.

This version is accepted for publication in the Bulletin of the London Mathematical Society

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