Bochner transforms, perturbations and amoebae of holomorphic almost periodic mappings in tube domains

TL;DR

This paper introduces a new representation of the Bochner transform closure for holomorphic almost periodic mappings, defines a novel amoeba concept, and proves convexity properties of amoeba complements in tube domains.

Contribution

It presents an alternative representation of the Bochner transform closure, introduces a new amoeba notion, and establishes convexity results for amoeba complements in the context of holomorphic almost periodic mappings.

Findings

New representation of Bochner transform closure.

Amoeba notion coincides with Favorov's for regular mappings.

Amoeba complement is Henriques m-convex.

Abstract

We give an alternative representation of the closure of the Bochner transform of a holomorphic almost periodic mapping in a tube domain. For such mappings we introduce a new notion of amoeba and we show that, for mappings which are regular in the sense of Ronkin, this new notion agrees with Favorov's one. We prove that the amoeba complement of a regular holomorphic almost periodic mapping, defined on Cn and taking its values in Cm+1, is a Henriques m-convex subset of Rn. Finally, we compare some different notions of regularity.