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

TL;DR

This paper establishes a Bohr-type property for bases in the space of entire functions in several complex variables, linking basis coefficients and supremum norms over nested compact sets, with extensions to Stein manifolds.

Contribution

It proves a Bohr property for bases in the space of entire functions and extends the result to Stein manifolds with the Liouville Property.

Findings

The Bohr property holds for bases in the space of entire functions.

A similar property is valid for bases in Stein manifolds with Liouville Property.

Provides bounds relating basis coefficients to supremum norms over nested compact sets.

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.