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An extension of the Fourier transform from test functions to generalized functions (cf. [[Generalized function|Generalized function]]). Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f0411701.png" /> be a space of test functions on which the Fourier transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f0411702.png" />,
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f0411703.png" /></td> </tr></table>
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is defined and on which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f0411704.png" /> is an isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f0411705.png" /> onto a space of test functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f0411706.png" />. Then the Fourier transformation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f0411707.png" /> is defined on the space of generalized functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f0411708.png" /> by
+
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 $,
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f0411709.png" /></td> </tr></table>
+
$$
 +
\phi  \rightarrow \
 +
F [ \phi ]  = \
 +
\int\limits \phi ( x)
 +
e ^ {i ( \xi , x) } \
 +
dx,\  \phi \in K,
 +
$$
  
and this is an isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117010.png" /> onto the space of generalized functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117011.png" />.
+
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.===
 
===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
  
1) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117013.png" />. Here the inverse of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117014.png" /> is the operation
+
$$
 +
| \phi ( z) |  \leq  \
 +
C _ {N} e ^ {a |  \mathop{\rm Im}  z | }
 +
( 1 + | z |) ^ {-} N ,\ \
 +
z \in \mathbf C  ^ {n} .
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117015.png" /></td> </tr></table>
+
==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,
  
and the basic formulas for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117016.png" /> are
+
$$
 +
f ( x)  = \
 +
\sum _ {| k | = 0 } ^  \infty 
 +
c _ {k} ( f  )
 +
e ^ {i ( k \omega , x) } ,\ \
 +
| c _ {k} ( f ) |  \leq  \
 +
A ( 1 + | k | )  ^ {m} ,
 +
$$
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117017.png" /></td> </tr></table>
+
converging to  $  f $
 +
in  $  S  ^  \prime  $;  
 +
here
  
2) Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117018.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117019.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117020.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117021.png" /> is the set of all functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117022.png" /> for which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117023.png" />, and where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117025.png" />.
+
$$
 +
\omega  = \left (
  
3) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117026.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117027.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117028.png" /> is the set of all entire functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117029.png" /> satisfying the growth condition: There is a number <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117030.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117031.png" /> one can find a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117032.png" /> such that
+
\frac{2 \pi }{T _ {1} }
 +
\dots
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117033.png" /></td> </tr></table>
+
\frac{2 \pi }{T _ {n} }
  
==Fourier series of generalized functions.==
+
\right ) ,\ \
If a generalized function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117034.png" /> is periodic with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117035.png" />-period <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117036.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117037.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117038.png" /> and it can be expanded in a trigonometric series,
+
k \omega  = \left (
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117039.png" /></td> </tr></table>
+
\frac{2 \pi k _ {1} }{T _ {1} }
 +
\dots
  
converging to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117040.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117041.png" />; here
+
\frac{2 \pi k _ {n} }{T _ {n} }
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117042.png" /></td> </tr></table>
+
\right ) .
 +
$$
  
 
===Examples.===
 
===Examples.===
  
 +
4)  $  F ( x  ^  \alpha  ) = ( 2 \pi )  ^ {n} (- i) ^ {| \alpha | } D  ^  \alpha  \delta ( \xi ) $,
 +
in particular  $  F [ 1] = ( 2 \pi )  ^ {n} \delta ( \xi ) $.
  
4) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117043.png" />, in particular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117044.png" />.
+
5) $  F [ D  ^  \alpha  \delta ] = (- i \xi )  ^  \alpha  $,  
 
+
in particular $  F [ \delta ] = 1 $.
5) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117045.png" />, in particular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117046.png" />.
 
  
6) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117047.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117048.png" /> is the Heaviside function.
+
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.==
 
==The Fourier transform of the convolution of generalized functions.==
Let the direct product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117049.png" /> of two generalized functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117050.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117051.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117052.png" /> admit an extension to functions of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117053.png" />, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117054.png" />. Namely, suppose that for any sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117055.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117056.png" />, in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117057.png" /> with the properties: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117058.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117059.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117060.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117061.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117062.png" /> (uniformly on any compact set), the sequence
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117063.png" /></td> </tr></table>
+
$$
 +
( f ( x) \times g ( y), \eta _ {k} ( x; y) \phi ( x + y)),\ \
 +
k \rightarrow \infty ,
 +
$$
  
has a limit, denoted by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117064.png" />, which does not depend on the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117065.png" /> from the class indicated. In this case the functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117066.png" /> that acts according to the formula <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117067.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117068.png" />, is called the convolution of the generalized functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117069.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117070.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117071.png" />. The convolution does not exist for all pairs of generalized functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117072.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117073.png" />. It automatically exists if for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117074.png" /> the set
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117075.png" /></td> </tr></table>
+
$$
 +
T _ {R}  = \{ {
 +
( x, y) } : {
 +
x \in \supp  f,\
 +
y \in \supp  g,\
 +
| x + y | \leq  R } \}
 +
$$
  
is bounded in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117076.png" /> (in particular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117077.png" /> or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117078.png" /> has compact support). If the convolution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117079.png" /> exists, then it is commutative: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117080.png" />; and it commutes with shifts and with derivatives: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117081.png" />; the Dirac <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117082.png" />-function plays the role of  "identity" : <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117083.png" />. Convolution is a non-associative operation. However, there are associative (and commutative) convolution algebras. The Dirac delta-function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117084.png" /> serves as the identity in them. For example, the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117085.png" /> consisting of generalized functions from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117086.png" /> with support in a convex, acute, closed cone <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117087.png" /> with vertex at <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117088.png" /> is a convolution algebra. The set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117089.png" /> forms a convolution subalgebra of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117090.png" />. Notation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117091.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117092.png" /> (when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117093.png" />). The formula for the Fourier transform of the convolution
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117094.png" /></td> </tr></table>
+
$$
 +
F [ f \star g]  = F [ f] F [ g]
 +
$$
  
 
is valid in the following cases:
 
is valid in the following cases:
  
a) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117095.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117096.png" /> has compact support;
+
a) f \in S  ^  \prime  $,  
 +
$  g $
 +
has compact support;
  
b) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117097.png" />;
+
b) f , g \in D _ {L _ {2}  }  ^  \prime  $;
  
c) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117098.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f04117099.png" /> has compact support;
+
c) f \in D  ^  \prime  $,  
 +
$  g $
 +
has compact support;
  
d) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f041170100.png" />. In this case the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f041170101.png" /> of the generalized functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f041170102.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f041170103.png" /> is understood to be the limit in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f041170104.png" /> of the product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f041170105.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f041170106.png" />, as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f041170107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f041170108.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f041170109.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f041170110.png" /> denote the Laplace transforms of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f041170111.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f041170112.png" /> (see [[Generalized functions, product of|Generalized functions, product of]]).
+
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====
 
====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>
 
<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====
 
====Comments====
 
For other normalizations used in defining Fourier transforms, cf. [[Fourier transform|Fourier transform]].
 
For other normalizations used in defining Fourier transforms, cf. [[Fourier transform|Fourier transform]].
  
The Heaviside function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f041170113.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f041170114.png" /> is defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f041170115.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f041170116.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f041170117.png" /> if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041170/f041170118.png" />.
+
The Heaviside function $  \theta $
 +
on $  \mathbf R $
 +
is defined by $  \theta ( x) = 0 $
 +
if $  x < 0 $
 +
and $  \theta ( x) = 1 $
 +
if  $  x > 0 $.
  
 
====References====
 
====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>
 
<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>

Latest revision as of 19:39, 5 June 2020


An extension of the Fourier transform from test functions to generalized functions (cf. 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).

References

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

Comments

For other normalizations used in defining Fourier transforms, cf. 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

[a1] K. Yosida, "Functional analysis" , Springer (1980) pp. Chapt. 8, Sect. 4; 5
[a2] D.S. Jones, "The theory of generalized functions" , Cambridge Univ. Press (1982)
How to Cite This Entry:
Fourier transform of a generalized function. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fourier_transform_of_a_generalized_function&oldid=12995
This article was adapted from an original article by V.S. Vladimirov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article