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The eigenvalues of the discrete fourier transform are $\{1, -1, i, -i\}$ in approximately equal proportions.

https://en.wikipedia.org/wiki/Discrete_Fourier_transform#Eigenvalues_and_eigenvectors

Is there a generalization of this transform where the eigenvalues are other roots of unity?

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The Fractional Fourier transform (and see here for details in the non-discrete case).

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  • $\begingroup$ From what I can understand, the fractional fourier transform is just a power of the usual fourier transform. How could it have more than 4 distinct eigenvalues? $\endgroup$ – Yi Liu Mar 2 '16 at 22:56

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