Fourier transform invariant functions other than the bell curve? Are there any functions that are their own Fourier transforms other than $e^{-\pi x^2} $?

There is a complete characterization of the probability densities that are (modulo a constant factor) their own Fourier transforms (aka characteristic functions) in a paper of K. Schladitz and H.J. Engelbert: 
"On probability density functions which are their own characteristic functions", 
Theory Probab. Appl., vol. 40 (1995) pp. 577–581. The class is surprisingly large.
