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Since I've studied the Fourier transform extension to the Hilbert space $L^2$, I wondered if there is a complete study relative to its eigenvalues. I know that its adjoint operator is the inverse transform, which means that I can't use the theory of self-adjoint operators to state something about the eigenvalues. Could someone of you tell me something about them? Are they countable? Is there any "algorithm" to determine them? (just like the one working for compact self-adjoint operators). If you have references to books, they are welcome! Thank you! (sorry for a possibly repeated or stupid question)

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up vote 5 down vote accepted

The eigenvalues of the Fourier transform are $\pm 1$ and $\pm i$. Note that if $\cal F$ is the Fourier transform operator, $({\cal F})^4 = I$.

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Such a nice and easy result! Thank you very much! – Martino Jul 12 '12 at 21:17

There is a simple demonstration in wikipedia of the fact that the physicists' Hermite functions are a complete set of eigenfunctions for the unitary angular frequency version of the Fourier transform.

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