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

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

2012-01-26

arxiv.org/abs/1201.5607v1

Mathematics

Complex Variables

Scientific paper

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.

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