Given a function $f$ and its Fourier transform $\hat{f}$, then the function
$$F=f^2+\hat{f}\star\hat{f},$$
with $\star$ the convolution, is its own Fourier transform. If we require that $f$ is a probability density, then any $f$ with ${\cal F}f=f$ is of this form, see Self-characteristic distributions.