# real valued functions that are invariant under convolution

Consider the function $$f(x) = e^{-x^2}$$, notice that $$f$$ convoluted with itself is of the form $$a\cdot f(bx+c)$$ for reals $$a, b$$ and $$c$$. Another way of saying this is that the shape of the function $$f(x)$$ does not change when convoluted with itself. It only gets stretch and shifted.

Here is another such function: $$f(x) = \frac1{1+x^2}$$

I am interested in a categorization of all such functions. Is such a categorization known? I am interested in any resources discussing this topic. If you are unable to do either, I would still be interested in any example of such a function other than the two I listed.

• Use the FT to transform the convolution into a product, you'll find the condition $F(r \xi) =a(r) F(\xi)^{b(r)} e^{2 i\pi c(r)\xi}$. – reuns May 13 at 23:59