[sharpneli]

Inverse fourier transform of a non transformed signal gives you basically the fourier transform with some changes (I can't remember which, were the numbers conjugates or something?). Applying it the second time gives you same result as if you'd do the forward direction transform twice.

If you apply fourier transform 4 times you get your original function back. You can think of it as 90 degree rotation. Inverse transform just rotates it in the opposite direction.

The rotation analog is not even too far fetched as fractional fourier transform allows you to do an arbitrary angle rotation.

[araes]

Having never heard of this for the Fourier Transform, needed to read.

F0: original signal

F1: frequency domain signal

F2: reverse time signal

F3: inverse fourier signal

F4: original signal

Also, has further weird applications I've never heard of with "Fractional Fourier Transforms" [1] which can apparently result in smooth smears of time -> frequency domain [2].

[1] https://en.wikipedia.org/wiki/Fractional_Fourier_transform

[2] https://en.wikipedia.org/wiki/File:FracFT_Rec_by_stevencys.j...

[laszlokorte]

See [1] for my visualization of this 4-cycle and the fractional fourier transform

[1]: https://static.laszlokorte.de/frft-cube/