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](http://arxiv.org/abs/0805.3795).)


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?



Approximating with translated Gaussians and low-frequency trig functions
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](https://doi.org/10.1007/BF02384485 "Faxén, B. On approximation by translates and related problems in function theory. Ark. Mat. 19, 271–289 (1981)") for instance and the reference therein.


