Four-fold periodicity of Fourier transform

[Log]

I was looking through some examples which applied the duality principle while studying for an up and coming exam when it hit me that the transform applied 4 times gives you back the same function.

So is there some theory that uses this? perhaps some sort of operator?

I thought it interesting but couldn't find any information on it, what should I look up to learn more about this?

 

[mathman]

When you take two steps you get back the original function with a sign change of the argument, i.e. f(x)->f(-x), so after two more steps you get back to f(x).