The Levi-Civita equation in function classes

Abstract

Abstract Let G be an Abelian topological semigroup with unit. By a classical result (called Theorem A), if V is a finite dimensional translation invariant linear space of complex valued continuous functions defined on G, then every element of V is an exponential polynomial. More precisely, every element of V is of the form $$\sum _{i=1}^np_i \cdot m_i$$ ∑ i = 1 n p i · m i , where $$m_1 ,\ldots ,m_n$$ m 1 , … , m n are exponentials belonging to V, and $$p_1 ,\ldots ,p_n$$ p 1 , … , p n are polynomials of continuous additive functions. We generalize this statement by replacing the set of continuous functions by any algebra $${{\mathcal {A}}}$$ A of complex valued functions such that whenever an exponential m belongs to $${{\mathcal {A}}}$$ A , then $$m^{-1}\in {{\mathcal {A}}}$$ m - 1 ∈ A . As special cases we find that Theorem A remains valid even if the topology on G is not compatible with the operation on G, or if the set of continuous functions is replaced by the set of measurable functions with respect to an arbitrary $$\sigma $$ σ -algebra. We give two proofs of the result. The first is based on Theorem A. The second proof is independent, and seems to be more elementary than the existing proofs of Theorem A.