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An extension of the Fourier transform from test functions to generalized functions (cf. [[Generalized function|Generalized function]]). Let  $  K $
be a space of test functions on which the Fourier transformation  $  F $,

$$ 
\phi  \rightarrow \ 
F [ \phi ]  = \ 
\int\limits \phi ( x)
e ^ {i ( \xi , x) } \ 
dx,\  \phi \in K,
$$

is defined and on which  $  F $
is an isomorphism of  $  K $
onto a space of test functions  $  \widetilde{K}  $. 
Then the Fourier transformation  $  f \rightarrow F [ f] $
is defined on the space of generalized functions  $  \widetilde{K}  {}  ^  \prime  $
by

$$ 
( F [ f], \phi )  = \ 
( f, F [ \phi ]),\ \ 
\phi \in K,
$$

and this is an isomorphism of  $  \widetilde{K}  {}  ^  \prime  $
onto the space of generalized functions  $  K  ^  \prime  $.

===Examples.===

1)  $  K = S = \widetilde{K}  $, 
$  K  ^  \prime  = S  ^  \prime  = \widetilde{K}  {}  ^  \prime  $. 
Here the inverse of  $  F $
is the operation

$$ 
F  ^ {-} 1 [ f]  = \ 
{
\frac{1}{( 2 \pi )  ^ {n} }
 }
F [ f (- \xi )],\ \ 
f \in S  ^  \prime  ,
$$

and the basic formulas for  $  f \in S  ^  \prime  $
are

$$ 
D  ^  \alpha  F [ f]  = \ 
F [( ix)  ^  \alpha  f],\ \ 
F [ D  ^  \alpha  f]  = \ 
(- i \xi )  ^  \alpha  F [ f].
$$

2) Let  $  K = \cap _ {s \geq  0 }  L _ {2}  ^ {s} $, 
$  \widetilde{K}  = D _ {L _ {2}  } = \cap _ {s \geq  0 }  H _ {s} $, 
$  \widetilde{K}  {}  ^  \prime  = D _ {L _ {2}  }  ^  \prime  = \cup _ {s \geq  0 }  H _ {-} s $, 
where  $  L _ {2}  ^ {s} $
is the set of all functions  $  \phi $
for which  $  ( 1 + ( \xi )  ^ {2} )  ^ {s/2} \phi \in L _ {2} $, 
and where  $  H _ {s} = \widetilde{L}  {} _ {2}  ^ {s} $, 
$  - \infty < s < \infty $.

3)  $  K = D $, 
$  \widetilde{K}  = Z $, 
where  $  Z $
is the set of all entire functions  $  \phi ( z) $
satisfying the growth condition: There is a number  $  a = a _  \phi  \geq  0 $
such that for any  $  N \geq  0 $
one can find a  $  C _ {N} > 0 $
such that

$$ 
| \phi ( z) |  \leq  \ 
C _ {N} e ^ {a |  \mathop{\rm Im}  z | }
( 1 + | z |)  ^ {-} N ,\ \ 
z \in \mathbf C  ^ {n} .
$$

==Fourier series of generalized functions.==
If a generalized function  $  f $
is periodic with  $  n $-
period  $  T = ( T _ {1} \dots T _ {n} ) $, 
$  T _ {j} > 0 $, 
then  $  f \in S  ^  \prime  $
and it can be expanded in a trigonometric series,

$$ 
f ( x)  = \ 
\sum _ {| k | = 0 } ^  \infty  
c _ {k} ( f  )
e ^ {i ( k \omega , x) } ,\ \ 
| c _ {k} ( f  ) |  \leq  \ 
A ( 1 + | k | )  ^ {m} ,
$$

converging to  $  f $
in  $  S  ^  \prime  $; 
here

$$ 
\omega  =  \left (

\frac{2 \pi }{T _ {1} }
 \dots

\frac{2 \pi }{T _ {n} }

\right ) ,\ \ 
k \omega  =  \left (

\frac{2 \pi k _ {1} }{T _ {1} }
 \dots

\frac{2 \pi k _ {n} }{T _ {n} }

\right ) .
$$

===Examples.===

4)  $  F ( x  ^  \alpha  ) = ( 2 \pi )  ^ {n} (- i) ^ {| \alpha | } D  ^  \alpha  \delta ( \xi ) $, 
in particular  $  F [ 1] = ( 2 \pi )  ^ {n} \delta ( \xi ) $.

5)  $  F [ D  ^  \alpha  \delta ] = (- i \xi )  ^  \alpha  $, 
in particular  $  F [ \delta ] = 1 $.

6)  $  F [ \theta ] = i / ( \xi + i0) = \pi \delta ( \xi ) + iP ( 1/ \xi ) $, 
where  $  \theta $
is the Heaviside function.

==The Fourier transform of the convolution of generalized functions.==
Let the direct product  $  f ( x) \times g ( y) $
of two generalized functions  $  f $
and  $  g $
in  $  D  ^  \prime  ( \mathbf R  ^ {n} ) $
admit an extension to functions of the form  $  \phi ( x + y) $, 
for all  $  \phi \in D ( \mathbf R  ^ {n} ) $. 
Namely, suppose that for any sequence  $  \eta _ {k} ( x;  y) $, 
$  k \rightarrow \infty $, 
in  $  D ( \mathbf R  ^ {2n} ) $
with the properties:  $  | D  ^  \alpha  \eta _ {k} ( x;  y) | \leq  c _  \alpha  $, 
$  \eta _ {k} ( x;  y) \rightarrow 1 $, 
$  D  ^  \alpha  \eta _ {k} ( x;  y) \rightarrow 0 $, 
$  | \alpha | \geq  1 $, 
$  k \rightarrow \infty $(
uniformly on any compact set), the sequence

$$ 
( f ( x) \times g ( y), \eta _ {k} ( x;  y) \phi ( x + y)),\ \ 
k \rightarrow \infty ,
$$

has a limit, denoted by  $  ( f ( x) \times g ( y) , \phi ( x + y)) $, 
which does not depend on the sequence  $  \{ \eta _ {k} \} $
from the class indicated. In this case the functional  $  f \star g $
that acts according to the formula  $  ( f \star g, \phi ) = ( f ( x) \times g ( y), \phi ( x + y)) $, 
$  \phi \in D ( \mathbf R  ^ {n} ) $, 
is called the convolution of the generalized functions  $  f $
and  $  g $, 
$  f \star g \in D  ^  \prime  ( \mathbf R  ^ {n} ) $. 
The convolution does not exist for all pairs of generalized functions  $  f $
and  $  g $. 
It automatically exists if for any  $  R > 0 $
the set

$$ 
T _ {R}  =  \{ {
( x, y) } : {
x \in \supp  f,\ 
y \in \supp  g,\ 
| x + y | \leq  R } \}
$$

is bounded in  $  \mathbf R  ^ {2n} $(
in particular if  $  f $
or  $  g $
has compact support). If the convolution  $  f \star g $
exists, then it is commutative:  $  f \star g = g \star f $; 
and it commutes with shifts and with derivatives:  $  f \star D  ^  \alpha  g = D  ^  \alpha  ( f \star g) = D  ^  \alpha  f \star g $; 
the Dirac  $  \delta $-
function plays the role of  "identity" :  $  f = \delta \star f = f \star \delta $. 
Convolution is a non-associative operation. However, there are associative (and commutative) convolution algebras. The Dirac delta-function  $  \delta $
serves as the identity in them. For example, the set  $  D _  \Gamma   ^  \prime  $
consisting of generalized functions from  $  D  ^  \prime  ( \mathbf R  ^ {n} ) $
with support in a convex, acute, closed cone  $  \Gamma $
with vertex at  $  0 $
is a convolution algebra. The set  $  S _  \Gamma   ^  \prime  = S  ^  \prime  \cap D _  \Gamma   ^  \prime  $
forms a convolution subalgebra of  $  D _  \Gamma   ^  \prime  $. 
Notation:  $  D _ {+}  ^  \prime  = D _ {[ 0, \infty ) }   ^  \prime  $, 
$  S _ {+}  ^  \prime  = S _ {[ 0, \infty ) }   ^  \prime  $(
when  $  n = 1 $). 
The formula for the Fourier transform of the convolution

$$ 
F [ f \star g]  =  F [ f] F [ g]
$$

is valid in the following cases:

a)  $  f \in S  ^  \prime  $, 
$  g $
has compact support;

b)  $  f , g \in D _ {L _ {2}  }  ^  \prime  $;

c)  $  f \in D  ^  \prime  $, 
$  g $
has compact support;

d)  $  f , g \in S _  \Gamma   ^  \prime  $. 
In this case the product  $  F [ f] F [ g] $
of the generalized functions  $  F [ f] $
and  $  F [ g] $
is understood to be the limit in  $  S  ^  \prime  $
of the product  $  \widetilde{f}  ( \zeta ) \widetilde{g}  ( \zeta ) $, 
$  \zeta = \xi + i \eta $, 
as  $  \eta \rightarrow 0 $, 
$  \eta \in  \mathop{\rm Int}  \Gamma  ^ {*} $, 
where  $  \widetilde{f}  $
and  $  \widetilde{g}  $
denote the Laplace transforms of  $  f $
and  $  g $(
see [[Generalized functions, product of|Generalized functions, product of]]).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  V.S. Vladimirov,   "Generalized functions in mathematical physics" , MIR  (1979)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Gel'fand,   G.E. Shilov,   "Generalized functions" , '''1''' , Acad. Press  (1964)  (Translated from Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  L. Schwartz,   "Théorie des distributions" , '''2''' , Hermann  (1951)</TD></TR><TR><TD valign="top">[4]</TD> <TD valign="top">  P. Antosik,   J. Mikusiński,   R. Sikorski,   "Theory of distributions. The sequential approach" , Elsevier  (1973)</TD></TR><TR><TD valign="top">[5]</TD> <TD valign="top">  L.V. Hörmander,   "The analysis of linear partial differential operators" , '''1''' , Springer  (1983)</TD></TR></table>

====Comments====
For other normalizations used in defining Fourier transforms, cf. [[Fourier transform|Fourier transform]].

The Heaviside function  $  \theta $
on  $  \mathbf R $
is defined by  $  \theta ( x) = 0 $
if  $  x < 0 $
and  $  \theta ( x) = 1 $
if  $  x > 0 $.

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  K. Yosida,   "Functional analysis" , Springer  (1980)  pp. Chapt. 8, Sect. 4; 5</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  D.S. Jones,   "The theory of generalized functions" , Cambridge Univ. Press  (1982)</TD></TR></table>