This question originates an engineering application. 

There is a certain process that is presumed to be a sequence of diffusions and is usually modelled as a sum of Gaussians:

$$\Sigma\_n w\_ne^{-\frac{x^2}{\sigma\_n^2}}$$

It is known that the function $f(x)$ that is being approximated is monotone exponentially decreasing on $[0,\infty)$. The question is: can any such function be uniformly approximated with a convergent series of Gaussians?

The preferred metrics for convergence would be $L^\infty$, although any $L^p$ answer would be satisfactory.