In Andy's answer to the question "What are fixed points of the Fourier Transform" on Math Overflow, he shows that the Fourier Transform has eigenvalues $\{+1, +i, -1, -i \}$ and that the projections of any function onto the corresponding four eigenspaces may be found through some simple linear algebra.
I would like to get a better feeling for these four eigenspaces of the fourier transform.
- How can I find some interesting members of each of these eigenspaces?
- How can I show that Hermite-Gaussians are in one (or more?) of the eigenspaces?
- How can one define usable projection operators onto these eigenspaces?
- The wikipedia article on the Fourier Transform mentions that Wiener defined the Fourier Transform via these projections. What exactly was Wiener's approach?