Consider the Gaussian $\exp(-t^2/2)$. Is it the case that any function in $L^2(\Bbb R)$ can be written as a limit of a sum of scalings and translations of Gaussians? That is, for any $f\in L^2(\Bbb R)$, is it the case that
$$f(t) = \sum_{n=0}^{\infty} a_n\exp(-(t-t_n)^2/2),$$
where $a_n\in\Bbb C$ and $t_n\in\Bbb R$ and the convergence is understood in the $L^2$ sense? Via the Fourier transform, we can view this as asking if any function $g$ in $L^2(\Bbb R)$ can be written in the form
$$g(\omega) = \sum_{n=0}^{\infty}a_n\exp(-i\omega t_n)\exp(-\omega^2/2).$$
It seems to me that this encroaches a bit on frame theory but I haven't had much exposure to it so I can't quite see how to show this is true or not.
If this is indeed the case, could we even extend this to all Schwartz space functions since they have a lot in common with the Gaussian?
Yes, this follows from Wiener's Tauberian theorem, the $L^2$ case.
Since the answer is short, I'll keep the old version below the cut. It was based on misreading of the question (RBF allow horizontal scaling).
Yes, this is true. Key terms: radial basis function, universal approximation property.
A classical reference (with a proof of more general result) is Universal Approximation Using Radial-Basis-Function Networks by J. Park and I. W. Sandberg.