TITLE: Approximating with translated Gaussians and low-frequency trig functions
QUESTION [8 upvotes]: Defining the translated Gaussians by $f_t(x)=\exp(-(x-t)^2)$  for $t,x\in\Bbb{R}$, we showed that the linear span of $\{f_t \mid 0 \le t < \epsilon\}$ is dense in $L^2(\Bbb{R})$, for any $\epsilon>0$. As a consequence, low-frequency trigonometric functions are dense in $L^2([a,b])$. The proof of the first result uses Hermite functions, and the second one follows by taking the Fourier transform of the first. (Arxiv link.)
Two reviewers told us that these results "must be known" but didn't provide references. 
Two questions:

Are these results folklore?
What's a good place to look for similar/related results?

REPLY [2 votes]: Wiener's approximation theorem says that
Given a function $h: \mathbb{R} \to  \mathbb{R}$, the set $\{\sum a_i h(\cdot - x_i): a_i, x_i \in \mathbb{R}\}$ is dense in $L^2(\mathbb{R})$ if and only if zeros of the Fourier transform of $h$ has zero Lebesgue measure.
See Wiener's book "The Fourier Integral and Certain of Its Applications" or Chandrasekharan's "Classical Fourier Transforms".
Further question when the translating parameters are restricted to a smaller set are considered by a series of authors. See this paper for instance and the reference therein.