Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Approximation of functions by Gaussians

  1. May 10, 2010 #1
    Hey everyone,

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

    "Any vector of [tex]L^2(\mathbb{R}^3)[/tex] 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.
     
    Last edited: May 11, 2010
  2. jcsd
  3. May 16, 2010 #2
    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
     
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook