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For a function applying the fourier transform twice is equivalent to the parity transformation, applying it three times is the same as applying the inverse of the fourier transform, and applying four times, is the identity transformation. Wikipedia has a good description of this.

What do we get when we apply the transform multiple times to a distribution? For example applying it twice would look like this:

$ \mathscr F \mathscr F T(f) = T(\mathscr F \mathscr F f) = T( f_-) = ? $

Where $f_- = f(-x)$.

Here $\mathscr F$ is the fourier transform $T$ is a distribution, and $f$ is a function.

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Is there anything not well explained by the Wikipedia section on invertibility and periodicity of the Fourier transform? –  dmckee May 19 '13 at 16:08
    
Well, that section applies to functions. Are saying this applies to distributions? In that case applying the transform to a distribution twice would be T(-f) = -T(f) –  yalis May 19 '13 at 19:41

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