Editing Fourier transform (section)

=== Inversion and periodicity ===
{{Further|Fourier inversion theorem|Fractional Fourier transform}}

Under suitable conditions on the function <math>f</math>, it can be recovered from its Fourier transform <math>\hat{f}</math>. Indeed, denoting the Fourier transform operator by <math>\mathcal{F}</math>, so <math>\mathcal{F} f := \hat{f}</math>, then for suitable functions, applying the Fourier transform twice simply flips the function: <math>\left(\mathcal{F}^2 f\right)(x) = f(-x)</math>, which can be interpreted as "reversing time". Since reversing time is two-periodic, applying this twice yields <math>\mathcal{F}^4(f) = f</math>, so the Fourier transform operator is four-periodic, and similarly the inverse Fourier transform can be obtained by applying the Fourier transform three times: <math>\mathcal{F}^3\left(\hat{f}\right) = f</math>. In particular the Fourier transform is invertible (under suitable conditions).

More precisely, defining the ''parity operator'' <math>\mathcal{P}</math> such that <math>(\mathcal{P} f)(x) = f(-x)</math>, we have:
<math display="block">\begin{align}
  \mathcal{F}^0 &= \mathrm{id}, \\
  \mathcal{F}^1 &= \mathcal{F}, \\
  \mathcal{F}^2 &= \mathcal{P}, \\
  \mathcal{F}^3 &= \mathcal{F}^{-1} = \mathcal{P} \circ \mathcal{F} = \mathcal{F} \circ \mathcal{P}, \\
  \mathcal{F}^4 &= \mathrm{id}
\end{align}</math>
These equalities of operators require careful definition of the space of functions in question, defining equality of functions (equality at every point? equality [[almost everywhere]]?) and defining equality of operators – that is, defining the topology on the function space and operator space in question. These are not true for all functions, but are true under various conditions, which are the content of the various forms of the [[Fourier inversion theorem]].

This fourfold periodicity of the Fourier transform is similar to a rotation of the plane by 90°, particularly as the two-fold iteration yields a reversal, and in fact this analogy can be made precise. While the Fourier transform can simply be interpreted as switching the time domain and the frequency domain, with the inverse Fourier transform switching them back, more geometrically it can be interpreted as a rotation by 90° in the [[time–frequency domain]] (considering time as the {{mvar|x}}-axis and frequency as the {{mvar|y}}-axis), and the Fourier transform can be generalized to the [[fractional Fourier transform]], which involves rotations by other angles. This can be further generalized to [[linear canonical transformation]]s, which can be visualized as the action of the [[special linear group]] {{math|[[SL2(R)|SL<sub>2</sub>('''R''')]]}} on the time–frequency plane, with the preserved symplectic form corresponding to the [[#Uncertainty principle|uncertainty principle]], below. This approach is particularly studied in [[signal processing]], under [[time–frequency analysis]].