Editing Fractional Fourier transform (section)

===Fractional kernel===
The FRFT is an [[integral transform]]
<math display=block>\mathcal{F}_\alpha f (u) = \int K_\alpha (u, x) f(x)\, \mathrm{d}x</math>
where the α-angle kernel is
<math display=block>K_\alpha (u, x) = \begin{cases}\sqrt{1-i\cot(\alpha)} \exp \left(i \pi (\cot(\alpha)(x^2+ u^2) -2 \csc(\alpha) u x) \right) & \mbox{if } \alpha \mbox{ is not a multiple of }\pi, \\
\delta (u - x) & \mbox{if } \alpha \mbox{ is a multiple of } 2\pi, \\
\delta (u + x) & \mbox{if } \alpha+\pi \mbox{ is a multiple of } 2\pi, \\
\end{cases}</math>

Here again the special cases are consistent with the limit behavior when  {{mvar|α}}  approaches a multiple of {{mvar|π}}.

The FRFT has the same properties as its kernels :
* symmetry: <math>K_\alpha~(u, u')=K_\alpha ~(u', u)</math>
* inverse: <math>K_\alpha^{-1} (u, u') = K_\alpha^* (u, u') = K_{-\alpha} (u', u) </math>
* additivity: <math>K_{\alpha+\beta} (u,u') = \int K_\alpha (u, u'') K_\beta (u'', u')\,\mathrm{d}u''.</math>