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For any square-summable function  $  f \in L _ {2} (- \infty , + \infty ) $
the integral

$$ 
\widehat{f}  _  \omega  ( x)  =  
\frac{1}{\sqrt {2 \pi } }
 \int\limits _ {- \omega } ^ { {+ }  \omega }
f( y) e  ^ {-} ixy  dy
$$

converges in  $  L _ {2} $
to some function  $  \widehat{f}  \in L _ {2} $
as  $  \omega \rightarrow \infty $, 
i.e.

$$ \tag{1 }
\lim\limits _ {\omega \rightarrow \infty }  \int\limits _ {- \infty } ^ { {+ }  \infty } | \widehat{f}  ( x) - \widehat{f}  _  \omega  ( x) |  ^ {2}  dx  =  0.
$$

Here the function  $  f $
itself is representable as the limit in  $  L _ {2} $
of the integrals

$$ 
f _  \eta  ( x)  =  
\frac{1}{\sqrt {2 \pi } }
 \int\limits _ {- \eta } ^ { {+ }  \eta } \widehat{f}  ( y)
e  ^ {ixy}  dy,\  \eta > 0,
$$

as  $  \eta \rightarrow \infty $, 
i.e.

$$ 
\lim\limits _ {\eta \rightarrow \infty }  \int\limits _ {- \infty } ^  \infty  
| f( x) - f _  \eta  ( x) |  ^ {2}  dx  =  0.
$$

Also, the following relation holds:

$$ 
\int\limits _ {- \infty } ^ { {+ }  \infty } | f( x) |  ^ {2}  dx  =  \int\limits _ {- \infty } ^ { {+ }  \infty } |
\widehat{f}  ( \lambda ) |  ^ {2}  d \lambda
$$

(the Parseval–Plancherel formula).

The function

$$ 
\widehat{f}  ( x)  =  \lim\limits _ {\omega \rightarrow \infty }  
\frac{1}{\sqrt {2 \pi } }
 \int\limits _ {-
\omega } ^ { {+ }  \omega } f( y) e  ^ {-} iyx  dy,
$$

where the limit is understood in the sense of convergence in  $  L _ {2} $(
as in (1)), is called the [[Fourier transform|Fourier transform]] of  $  f $; 
it is sometimes denoted by the symbolic formula:

$$ \tag{2 }
\widehat{f}  ( x)  =  
\frac{1}{\sqrt {2 \pi } }
 \int\limits _ {- \infty } ^ { {+ }  \infty } f( y) e  ^ {-} ixy  dy ,
$$

where the integral in (2) must be understood in the sense of the principal value at  $  \infty $
in the metric of  $  L _ {2} $. 
One similarly interprets the equation

$$ \tag{3 }
f( x)  =  
\frac{1}{\sqrt {2 \pi } }
 \int\limits _ {- \infty } ^ { {+ }  \infty } \widetilde{f}  ( y) e  ^ {ixy}
 dy.
$$

For functions  $  f \in L _ {2} $, 
the integrals (2) and (3) exist in the sense of the principal value for almost all  $  x $.

The functions  $  f $
and  $  \widehat{f}  $
also satisfy the following equations for almost-all  $  x $:

$$ 
\widehat{f}  ( x)  =  
\frac{d}{dx}
 \left \{ 
\frac{1}{\sqrt {2 \pi } }
 \int\limits _ {- \infty } ^ { {+ } 
\infty } f( y) 
\frac{e  ^ {-} ixy - 1 }{-}
 iy  dy \right \} ,
$$

$$ 
f( x)  =  
\frac{d}{dx}
 \left \{ 
\frac{1}{\sqrt {2 \pi
 } }
 \int\limits _ {- \infty } ^ { {+ }  \infty } \widehat{f}  ( y) 
\frac{e  ^ {ixy} - 1 }{iy}
  dy \right \} .
$$

If Fourier transformation is denoted by  $  {\mathcal F} $
and if  $  {\mathcal F}  ^ {-} 1 $
denotes the inverse, then Plancherel's theorem can be rephrased as follows:  $  {\mathcal F} $
and  $  {\mathcal F}  ^ {-} 1 $
are mutually-inverse unitary operators on  $  L _ {2} $(
cf. [[Unitary operator|Unitary operator]]).

The theorem was established by M. Plancherel (1910).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  A. Zygmund,   "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  E.C. Titchmarsh,   "Introduction to the theory of Fourier integrals" , Oxford Univ. Press  (1948)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S. Bochner,   "Lectures on Fourier integrals" , Princeton Univ. Press  (1959)  (Translated from German)</TD></TR></table>

====Comments====
The heart of Plancherel's theorem is the assertion that if  $  f \in L _ {1} ( \mathbf R ) \cap L _ {2} ( \mathbf R ) $, 
then: a)  $  \widehat{f}  \in L _ {2} ( \mathbf R ) $, 
where  $  \widehat{f}  ( y) $
is defined by (2) for  $  y \in \mathbf R $; 
b)  $  \| \widehat{f}  \| _ {2} = \| f \| _ {2} $; 
and c) the set of all such  $  \widehat{f}  $
is dense in  $  L _ {2} ( \mathbf R ) $. 
Then one extends this mapping  $  f \rightarrow \widehat{f}  $
to a unitary mapping  $  {\mathcal F} $
of  $  L _ {2} ( \mathbf R ) $
onto itself which satisfies  $  ( {\mathcal F}  ^ {-} 1 f  ) ( y) = ( {\mathcal F} f  )(- y) $
for almost every  $  y \in \mathbf R $. 
There are generalizations of Plancherel's theorem in which  $  \mathbf R $
is replaced by  $  \mathbf R  ^ {n} $
or by any locally compact Abelian group. Cf. also [[Harmonic analysis, abstract|Harmonic analysis, abstract]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,   "Fourier analysis on groups" , Wiley  (1962)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  A. Weil,   "l'Intégration dans les groupes topologiques et ses applications" , Hermann  (1940)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  L.S. Pontryagin,   "Topological groups" , Princeton Univ. Press  (1958)  (Translated from Russian)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  E. Hewitt,   K.A. Ross,   "Abstract harmonic analysis" , '''1–2''' , Springer  (1979)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  H. Reiter,   "Classical harmonic analysis and locally compact groups" , Oxford Univ. Press  (1968)</TD></TR></table>