A characterization of generalized exponential polynomials in terms of decomposable functions

Abstract

Let $G$ be a topological commutative semigroup with unit. We prove that a continuous function $f\colon G\to \cc$ is a generalized exponential polynomial if and only if there is an $n\ge 2$ such that $f(x_1 +\ldots +x_n )$ is decomposable; that is, if $f(x_1 +\ldots +x_n )=\sumik u_i \cd v_i$, where the function $u_i$ only depends on the variables belonging to a set $\emp \ne E_i \subsetneq \{ x_1 \stb x_n \}$, and $v_i$ only depends on the variables belonging to $\{ x_1 \stb x_n \} \se E_i$ $(i=1\stb k)$.