A characterization of generalized exponential polynomials in terms of decomposable functions (Q2322496)

Let \(X\) be a nonempty set. A function \(F\colon X^n\to\mathbb{C}\) is called decomposable, if it can be written as \(F=\sum_{i=1}^ku_iv_i\), where \(u_i\) only depends on the variables \(E_i\subset\{x_1,\dots,x_n\}\), \(v_i\) only depends on the variables \(\{x_1,\dots,x_n\}\setminus E_i\), and \(E_i\) is a nonempty and proper subset of the variables \(\{x_1,\dots,x_n\}\) for all \(i\in\{1,\dots,k\}\).\par The main motivation of the paper is a result of \textit{E. Shulman} [Aequationes Math. 79, No. 1--2, 13--21 (2010; Zbl 1216.39032)]: if \(G\) is a topological semigroup with a unit, and \(f: G\to\mathbb{C}\) is continuous such that \(f(x_1+\dots+x_n)\) is decomposable for some \(n\ge 2\), then \(f\) is contained in a finite dimensional subspace of continuous complex valued functions acting on \(G\) (equivalently, \(f\) satisfies the Levi-Civita functional equation). The author proves the following.\par Theorem. Let \(G\) be a commutative unital subsemigroup and let \(f: G\to\mathbb{C}\) be continuous. Then the next statements are equivalent:\par (i) \(f(x_1+\dots+x_n)\) is decomposable for some \(n\ge 2\);\par(ii) \(f\) is a generalized exponential polynomial.\par Under a generalized exponential polynomial we mean a finite sum of products of a generalized polynomial and an exponential function. In the commutative case, this theorem improves the above mentioned result of Shulman.