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We all know Fourier transform has invariants such as $e^{-x^2}$, and another MSE post has shown the non-existence of invariant function under Hilbert transform using Fourier transform. I am wondering if there is any general method to find the invariant functions (or prove the non-existence) given an integral transform?

(You can impose any constrains you deem necessary on the integral transform. Or if you know anything interesting about a specific transform, please also share with us.)

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