Fourier inversion theorem

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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 and phase information about a wave then we may reconstruct the original wave precisely.

The theorem says that if we have a function ${\displaystyle f:\mathbb {R} \to \mathbb {C} }$ satisfying certain conditions, and we use the convention for the Fourier transform that

${\displaystyle ({\mathcal {F}}f)(\xi ):=\int _{\mathbb {R} }e^{-2\pi iy\cdot \xi }\,f(y)\,dy,}$

then

${\displaystyle f(x)=\int _{\mathbb {R} }e^{2\pi ix\cdot \xi }\,({\mathcal {F}}f)(\xi )\,d\xi .}$

In other words, the theorem says that

${\displaystyle f(x)=\iint _{\mathbb {R} ^{2}}e^{2\pi i(x-y)\cdot \xi }\,f(y)\,dy\,d\xi .}$

This last equation is called the Fourier integral theorem.

Another way to state the theorem is that if ${\displaystyle R}$ is the flip operator i.e. ${\displaystyle (Rf)(x):=f(-x)}$, then

${\displaystyle {\mathcal {F}}^{-1}={\mathcal {F}}R=R{\mathcal {F}}.}$

The theorem holds if both ${\displaystyle f}$ and its Fourier transform are absolutely integrable (in the Lebesgue sense) and ${\displaystyle f}$ is continuous at the point ${\displaystyle x}$. 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.