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.

令G为具有单位的Abelian拓扑半群。根据经典结果（称为定理A），如果V是在G上定义的复值连续函数的有限维平移不变线性空间，则V的每个元素都是指数多项式。更准确地说，V的每个元素的形式为\（\ sum _ {i = 1} ^ np_i \ cdot m_i \），其中\（m_1，\ ldots，m_n \）是属于V的指数，而\（p_1 ，\ ldots，p_n \）是连续加法函数的多项式。我们通过用任何代数\（{{\ mathcal {A}}} \\的复杂值函数，以便每当指数m属于\（{{\ mathcal {A}}} \\）时，然后是{{\ mathcal {A}}} \中的\（m ^ {-1} \}。作为特殊情况下，我们发现，定理的证明仍然有效，即使在拓扑摹不与操作兼容摹，或者如果集的连续函数是由一套可衡量的功能所取代相对于任意\（\西格玛\ ）-代数。我们给出了两个证明。第一个基于定理A。第二个证明是独立的，并且比定理A的现有证明更基本。