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There are many ways to choose eigenbasis for the Discrete Fourier Transform matrix since it has only $4$ distinct eigenvalues taken from $\{\pm 1,\pm i\}$.

Has there been any refereed work that provides a sparse eigenvector basis for the Discrete Fourier Transform matrix?

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FYI, I addressed this question in a non-refereed article available on ArXiv: Sparse Eigenvectors of the Discrete Fourier Transform. –  Bill Bradley Mar 12 at 14:18
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