Approximation theorems for Banach-valued almost periodic and semi-almost periodic holomorphic functions

TL;DR

This paper extends the Bohr approximation theorem to Banach-valued almost periodic holomorphic functions on a polydisk, providing new approximation results for semi-almost periodic functions with minimal boundary discontinuities.

Contribution

It introduces an extension of the classical Bohr approximation theorem to Banach-valued functions on a polydisk, addressing semi-almost periodic holomorphic functions with weak boundary discontinuities.

Findings

Extended Bohr approximation theorem to Banach-valued functions

Established approximation results for semi-almost periodic functions

Analyzed boundary behavior of holomorphic functions

Abstract

The paper studies semi-almost periodic holomorphic functions on a polydisk which have, in a sense, the weakest possible discontinuities on the boundary torus. The basic result used in the proofs is an extension of the classical Bohr approximation theorem for almost periodic holomorphic functions on a strip to the case of Banach-valued almost periodic holomorphic functions.