I am wondering if there is a way to find a (or multiple) Schwartz function f such that its threefold convolution produces the same gaussian. Aka
$ f_3\left(x\right) = e^{-π(x-1)^2}$ where $f_3\left(x\right) = (f*f*f)$
Is there some sort of type of schwartz function that has this property?