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 <A HREF="http://www.utdallas.edu/~aria/papers/jfi98.pdf">Self-characteristic distributions.</A>