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.



Can a monotone exponentially decreasing function be uniformely approximated bt Gaussians?
You want the $w\_n$ to be nonnegative, right?


With that interpretation, the answer is no. Just substitute $x^2=y$, and the problem becomes one of approximating by a sum of decaying exponentials. Such functions are well studied; they are called completely monotone.


