inverse fourier Definition and 53 Threads

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



f
:

R

→

C



{\displaystyle f:\mathbb {R} \to \mathbb {C} }
satisfying certain conditions, and we use the convention for the Fourier transform that




(


F


f
)
(
ξ
)
:=

∫


R




e

−
2
π
i
y
⋅
ξ



f
(
y
)

d
y
,


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




f
(
x
)
=

∫


R




e

2
π
i
x
⋅
ξ



(


F


f
)
(
ξ
)

d
ξ
.


{\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




f
(
x
)
=

∬



R


2





e

2
π
i
(
x
−
y
)
⋅
ξ



f
(
y
)

d
y

d
ξ
.


{\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



R


{\displaystyle R}
is the flip operator i.e.



(
R
f
)
(
x
)
:=
f
(
−
x
)


{\displaystyle (Rf)(x):=f(-x)}
, then







F



−
1


=


F


R
=
R


F


.


{\displaystyle {\mathcal {F}}^{-1}={\mathcal {F}}R=R{\mathcal {F}}.}
The theorem holds if both



f


{\displaystyle f}
and its Fourier transform are absolutely integrable (in the Lebesgue sense) and



f


{\displaystyle f}
is continuous at the point



x


{\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.

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