See: https://smg.quora.com/Are-there-any-functions-whose-Fourier-...

Look under the section “Surprise”. This gives a construction that produces a new function g, whose transform is itself, that is a simple modification of any function ”f” that you supply.

In essence, the class of “functions that transform into themselves” is surprisingly broad and non-specific for those of us who key in on that fact about the Gaussian.