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The Fourier transform $F$ is an operator on a suitable space of functions. The statement that $F^4$ is the identity operator is essentially the content of the Fourieer inversion Formula. This shows that the eigenspaces are the fourth roots of unity.

My question is idle curiosity: what are other natural operators $L$ on function spaces such that $L^k$ is the idenity?

I could come up with some examples, like precomposing functions with group actions, but the Fourier transform feels of a different nature. Or there other natural examples?

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  • $\begingroup$ The Hilbert transform satisfies $H^2=-1$. $\endgroup$ Commented Feb 12 at 16:21
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    $\begingroup$ The $\mathbb{Z}/4$-action of the Fourier transform actually extends all the way to an action of $SL_2(\mathbb{C})$, see: en.wikipedia.org/wiki/Linear_canonical_transformation $\endgroup$ Commented Feb 12 at 18:19

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