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Alexandre Eremenko
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There is a whole chapter in Titchmarsh, Fourier transform, describing all sigenfunctions in great detail. He calls them "self-reciprocal" functions. A more difficult question is the about eigenvectors of discrete Fourier transform, but this you do not ask.

Alexandre Eremenko
  • 91.8k
  • 9
  • 259
  • 429