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, where m_1 ,ldots ,m_n are exponentials belonging to V, and p_1 ,ldots ,p_n are polynomials of continuous additive functions. We generalize this statement by replacing the set of continuous functions by any algebra {{mathcal {A}}} of complex valued functions such that whenever an exponential m belongs to {{mathcal {A}}}, then m^{-1}in {{mathcal {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 -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.
Highlights
Introduction and main resultsLet G be an Abelian topological semigroup with unit, and let C(G) denote the set of complex valued continuous functions defined on G
· mi, where pi is a polynomial and mi is an exponential for every i = 1, . . . , n, are called exponential polynomials
We know that every element of V is of the form k i=1 pi mi, where p1, . . . , pk are discrete polynomials, and m1, . . . , mk are exponentials
Summary
Let G be an Abelian topological semigroup with unit, and let C(G) denote the set of complex valued continuous functions defined on G. From Theorem A it follows that if V is a finite dimensional translation invariant linear subspace of CG, every element of V is a discrete exponential polynomial. A necessary condition for such a generalization to hold is that if p is a discrete polynomial and m is an exponential, and the translates of p · m belong to A, p ∈ A. This is not true for every algebra. Bk are additive functions belonging to Af , to A This means that f is admissible w.r.t. A, proving the ‘if’ part of Theorem 2. We show that Theorems 2, 3, and some other results we present are easy consequences of Theorem 1
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