Editing Fourier inversion theorem (section)

{{Short description|Mathematical theorem about functions}}
{{more references needed|date=May 2024}}
In [[mathematics]], the '''Fourier inversion theorem''' says that for many types of functions it is possible to recover a function from its [[Fourier transform]]. Intuitively it may be viewed as the statement that if we know all [[frequency#Frequency_of_waves|frequency]] and [[phase (waves)|phase]] information about a wave then we may reconstruct the original wave precisely.

The theorem says that if we have a function <math>f:\R \to \Complex</math> satisfying certain conditions, and we use the [[Fourier transform#Other conventions|convention for the Fourier transform]] that 

:<math>(\mathcal{F}f)(\xi):=\int_{\mathbb{R}} e^{-2\pi iy\cdot\xi} \, f(y)\,dy,</math>

then

:<math>f(x)=\int_{\mathbb{R}} e^{2\pi ix\cdot\xi} \, (\mathcal{F}f)(\xi)\,d\xi.</math>

In other words, the theorem says that 

:<math>f(x)=\iint_{\mathbb{R}^2} e^{2\pi i(x-y)\cdot\xi} \, f(y)\,dy\,d\xi.</math>

This last equation is called the '''Fourier integral theorem'''.

Another way to state the theorem is that if <math>R</math> is the flip operator i.e. <math>(Rf)(x) := f(-x)</math>, then 

:<math>\mathcal{F}^{-1}=\mathcal{F}R=R\mathcal{F}.</math>

The theorem holds if both <math>f</math> and its Fourier transform are [[absolutely integrable function|absolutely integrable]] (in the [[Lebesgue integration|Lebesgue sense]]) and <math>f</math> is continuous at the point <math>x</math>. However, even under more general conditions versions of the Fourier inversion theorem hold. In these cases the integrals above may not converge in an ordinary sense.