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Apart from Hermite functions, and the Dirac comb distribution , what other Eigenfunctions or distributions of the Fourier transform exist?

v217
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    There are a few questions already about eigenfunctions in the Schwartz space: https://math.stackexchange.com/questions/10774/how-do-i-compute-the-eigenfunctions-of-the-fourier-transform, https://math.stackexchange.com/questions/1235413/eigenvalues-of-fourier-transform-on-schwartz-functions, https://math.stackexchange.com/questions/728670/functions-that-are-their-own-fourier-transformation, https://mathoverflow.net/questions/12045/what-are-fixed-points-of-the-fourier-transform – Hans Lundmark Aug 16 '17 at 09:28
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    $\text{sech}(\pi x)$ is its own Fourier transform. The first thing to notice is that $T+\hat{T}+\hat{\hat{T}}+\hat{\hat{\hat{T}}}$ is always its own Fourier transform. Also as the Hermite functions are an orthonormal basis of $L^2$ which is dense in the distributions, any eigendistribution of the Fourier transform is a series of Hermite functions – reuns Aug 16 '17 at 09:51
  • Thank you for your comments! I will try to change my question so that it complements the already existing questions. – v217 Aug 16 '17 at 10:51

1 Answers1

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First of all, the domain of the Fourier Transform must be specifyed in this question.

The completeness of Hermite functions in $L_2$ implies that such space is divided $4$ orthogonal invariant (under the Fourier transform) spaces, which are orthogonal because the functions that composes them (The Hermite functions $\mod 4$ ) are so. Any function in such space is an eigenvector of the transform, and any funcion in the space can be expressed as linear combination of the functions, so there are no more eigenvalues. The complete reasoning can be found in

H. Dym y H.P. McKean — Fourier Series and Integrals, Academic Press (1972), pages 97-98.

Dr Potato
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