I am currently reading a paper titled "Multilayer Feedforward Networks With a Nonpolynomial Activation Function Can Approximate Any Function" (https://doi.org/10.1016/S0893-6080(05)80131-5). In Remark 4, the authors mention a result from the theory of mean period functions. I have no knowledge about mean period functions and would like to find a formal statement and a proof of the theorem.
From the short discussion in the paper, it seems like the theorem can be stated as follows:
Let $\sigma(x) : \mathbb{R} \rightarrow \mathbb{R}$ be a continuous, non-polynomial function. Define $$V := \text{span}\{ \sigma(x + \theta) : \theta \in \mathbb{R}\}.$$ Then the closure $\overline{V}$ contains a function of the form $e^{\lambda x} \cos(\nu x)$, where $\lambda, \nu \in \mathbb{R}$ and one of $\lambda$ and $\nu$ is non-zero.
Although the paper doesn't explicitly mention it, I assume the closure is defined using the $L^\infty$ norm.
I am looking for a reference on this result.