Integral equation, Fourier transform

Question

Find all functions $ f : \mathbb{R} \rightarrow\mathbb{R} $, that solve

$\int_{-\infty}^{\infty} f(t-x)f(x) dx =e^{-t^2}$, $ t\in \mathbb{R}$

How do I solve this?

I know that the left part is the convolution $(f\ast f)(t)$.

Answer 1

Hint: Taking the Fourier Transform of both sides, we get $$ \hat{f}^2(\tau)=\sqrt{\pi}e^{-\pi^2\tau^2} $$