Fourier Transform Eigenvalues and Eigenvectors

Question

I'm interested in calculate Fourier transform, $\mathcal{F}$, eigenvalues and eigenvectors. I've read some post here but they didn't answer my questions. I know that $\mathcal{F}^4=I$ (I've proven it via Fourier inverse transform), but I don't understand why this implies that its eigenvalues are $\pm1$ and $\pm i$.

Now, even if I know the eigenvalues, I'm not capable of finding the eigenvectors associated to them. Can you guys explain it to me o at least give a reference where this is explained with detail?

Thank you very much.

Answer 1

Suppose that $f$ is an eigenfunction of the Fourier transform $\mathcal{F}$ with eigenvalue $\lambda$, i.e. $$ \mathcal{F}(f) = \lambda f. $$ As $\mathcal{F}^4(f) = f$, we find that $$ \mathcal{F}^4(f) = \lambda \mathcal{F}^3(f) = \cdots = \lambda^4 f. $$ Thus $\lambda^4 = 1$. This implies that $\lambda$ is one of the fourth roots of unity, which are $\pm 1, \pm i$.

Even though $\mathcal{F}$ is linear, it acts on an infinite-dimensional space and there are infinitely many eigenfunctions. It's possible to find a couple very directly, as in Willie Wong's answer here. It's also possible to characterize all such eigenfunctions, but they can be very complicated.

Answer 2

I am considering the DFT in this answer.

There are different sets of eigenvectors known since there are repeated eigenvalues. They can be described in terms of character theory and are exponential sums. See the question on mathoverflow here and they also depend on $n$, the size of the $n\times n$ DFT matrix.

This nice online article heredescribes the eigenvectors at a more basic level.