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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.

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# Approximation of functions by Gaussians

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