In Stein's *Fourier Analysis*, there's the following exercise:

The function $e^{-\pi x^2}$ is its own Fourier transform. Generate other functions [presumably in the Schwartz space $S(\mathbb{R})$] that, up to a constant multiple, are their own FTs. What must the constant multiples be? To decide this, prove that $F^4 = I$ where $F$ is the FT operator.

My problem is that I don't really understand what the question is asking. Is it asking us to find the class of all functions such that $F(f) = cf$ for constant c and prove the above identity, or is it asking us to just find other examples? Or, is the $F^4 = I$ identity for any $f \in S(\mathbb{R})$?

Finding every function would mean finding all $f$ such that $$f(\xi) = \int^{\infty}_{-\infty} f(x)e^{-2\pi i x \xi}dx$$ which seems rather difficult.

I can come up with a specific example: if I can find a function such that $\hat{f}(\xi) = 1$, I can just take $g = f \ast K_{\delta}(x)$ where $K_{\delta}$ is the family of Gaussian functions as approximations of the identity.