Approximation of functions by Gaussians

[Orbb]

Hey everyone,

in doi:10.1016/0375-9601(82)90182-7 I found the following claim:

"Any vector of L^2(\mathbb{R}^3) can be arbitrarily well approximated by a finite sum of gaussian vectors."

Is this actually true? I lack the insight on how to prove this, but it would be a useful argument I could use in some other context. If true, I guess it would also generalize to arbitrary dimensions? Thanks in advance for any insights.

Edit: Okay, from what I've read in some papers, it seems the Gaussians form an overcomplete basis in the space of square integrable functions.

 

[Simon_Tyler]

Hi Orbb,

It seems like you've already worked it out, but just for completeness...
For the harmonic oscillator, the Gaussian functions are called "coherent states". There's quite a large literature on them (and their various generalisations) and their uses as an over-complete basis.

Simon