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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.

  1. How can I find some interesting members of each of these eigenspaces?
  2. How can I show that Hermite-Gaussians are in one (or more?) of the eigenspaces?
  3. How can one define usable projection operators onto these eigenspaces?
  4. The wikipedia article on the Fourier Transform mentions that Wiener defined the Fourier Transform via these projections. What exactly was Wiener's approach?

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