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One of the integral transforms (cf. [[Integral transform|Integral transform]]). It is a linear operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f0411501.png" /> acting on a space whose elements are functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f0411502.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f0411503.png" /> real variables. The smallest domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f0411504.png" /> is the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f0411505.png" /> of all infinitely-differentiable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f0411506.png" /> of compact support. For such functions
+
{{TEX|done}}
  
<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/f041150/f0411507.png" /></td> <td valign="top" style="width:5%;text-align:right;">(1)</td></tr></table>
+
One of the integral transforms (cf. [[Integral transform|Integral transform]]). It is a linear operator $F$ acting on a space whose elements are functions $f$ of $n$ real variables. The smallest domain of definition of $F$ is the set $D=C_0^\infty$ of all infinitely-differentiable functions $\phi$ of compact support. For such functions
  
In a certain sense the most natural domain of definition of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f0411508.png" /> is the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f0411509.png" /> of all infinitely-differentiable functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115010.png" /> that, together with their derivatives, vanish at infinity faster than any power of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115011.png" />. Formula (1) still holds for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115012.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115013.png" />. Moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115014.png" /> is an isomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115015.png" /> onto itself, the inverse mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115016.png" /> (the inverse Fourier transform) is the inverse of the Fourier transform and is given by the formula:
+
\begin{equation} (F\phi)(x) = \frac{1}{(2\pi)^{\frac{n}{2}}} \cdot \int_{\mathbf R^n} \phi(\xi) e^{-i x \xi} \, \mathrm d\xi. \end{equation}
  
<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/f041150/f04115017.png" /></td> <td valign="top" style="width:5%;text-align:right;">(2)</td></tr></table>
+
In a certain sense the most natural domain of definition of $F$ is the set $S$ of all infinitely-differentiable functions $\phi$ that, together with their derivatives, vanish at infinity faster than any power of $\frac{1}{|x|}$. Formula (1) still holds for $\phi\in S$, and $(F \phi)(x) \equiv \psi(x)\in S$. Moreover, $F$ is an isomorphism of $S$ <u>onto</u> itself, the inverse mapping $F^{-1}$ (the inverse Fourier transform) is the inverse of the Fourier transform and is given by the formula:
  
Formula (1) also acts on the space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115018.png" /> of integrable functions. In order to enlarge the domain of definition of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115019.png" /> generalization of (1) is necessary. In classical analysis such a generalization has been constructed for locally integrable functions with some restriction on their behaviour as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115020.png" /> (see [[Fourier integral|Fourier integral]]). In the theory of generalized functions the definition of the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115021.png" /> is free of many requirements of classical analysis.
+
\begin{equation} \phi(x) = (F^{-1} \psi)(x) = \frac{1}{(2\pi)^{\frac{n}{2}}} \cdot \int_{\mathbf R^n} \psi(\xi) e^{i x \xi} \, \mathrm d\xi. \end{equation}
  
The basic problems connected with the study of the Fourier transform <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115022.png" /> are: the investigation of the domain of definition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115023.png" /> and the range of values <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115024.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115025.png" />; as well as studying properties of the mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115026.png" /> (in particular, conditions for the existence of the inverse operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115027.png" /> and its expression). The inversion formula for the Fourier transform is very simple:
+
Formula (1) also acts on the space $L_{1}\left(\mathbf{R}^{n}\right)$ of integrable functions. In order to enlarge the domain of definition of the operator $F$ generalization of (1) is necessary. In classical analysis such a generalization has been constructed for locally integrable functions with some restriction on their behaviour as $|x|\to\infty$ (see [[Fourier integral|Fourier integral]]). In the theory of generalized functions the definition of the operator $F$ is free of many requirements of classical analysis.
  
<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/f041150/f04115028.png" /></td> </tr></table>
+
The basic problems connected with the study of the Fourier transform $F$ are: the investigation of the domain of definition  $  \Phi $
 +
and the range of values  $  F \Phi = \Psi $
 +
of  $  F $;  
 +
as well as studying properties of the mapping  $  F: \  \Phi \rightarrow \Psi $(
 +
in particular, conditions for the existence of the inverse operator  $  F ^ {\  -1} $
 +
and its expression). The inversion formula for the Fourier transform is very simple:
  
Under the action of the Fourier transform linear operators on the original space, which are invariant with respect to a shift, become (under certain conditions) multiplication operators in the image space. In particular, the convolution of two functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115030.png" /> goes over into the product of the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115032.png" />:
+
$$
 +
F ^ {\  -1} [g (x)] \  = \  F [g (-x)].
 +
$$
  
<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/f041150/f04115033.png" /></td> </tr></table>
+
Under the action of the Fourier transform linear operators on the original space, which are invariant with respect to a shift, become (under certain conditions) multiplication operators in the image space. In particular, the convolution of two functions  $  f $
 +
and  $  g $
 +
goes over into the product of the functions  $  Ff $
 +
and  $  Fg $:
 +
 
 +
$$
 +
F (f * g) \  = \  Ff \cdot Fg;
 +
$$
  
 
and differentiation induces multiplication by the independent variable:
 
and differentiation induces multiplication by the independent variable:
  
<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/f041150/f04115034.png" /></td> </tr></table>
+
$$
 +
F (D^ \alpha  f \  ) \  = \  (ix)^ \alpha  Ff.
 +
$$
  
In the spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115035.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115036.png" />, the operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115037.png" /> is defined by the formula (1) on the set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115038.png" /> and is a bounded operator from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115039.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115040.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115041.png" />:
+
In the spaces $  L _{p} ( \mathbf R^{n} ) $,  
 +
$  1 \leq  p \leq  2 $,  
 +
the operator $  F $
 +
is defined by the formula (1) on the set $  D _{F} = (L _{1} \cap L _{p} ) ( \mathbf R^{n} ) $
 +
and is a bounded operator from $  L _{p} ( \mathbf R^{n} ) $
 +
into $  L _{q} ( \mathbf R^{n} ) $,  
 +
$  p^{-1} + q^{-1} = 1 $:
  
<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/f041150/f04115042.png" /></td> </tr></table>
+
\begin{equation} \left\{\frac{1}{(2 \pi)^{n / 2}} \int_{\mathbf{R}^{n}}|(F f)(x)|^{q} d x\right\}^{1 / q} \leq\left\{\frac{1}{(2 \pi)^{n / 2}} \int_{\mathbf{R}^{n}}|f(x)|^{p} d x\right\}^{1 / p} \end{equation}
  
(the Hausdorff–Young inequality). <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115043.png" /> admits a continuous extension onto the whole space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115044.png" /> which (for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115045.png" />) is given by
+
(the Hausdorff–Young inequality). $  F $
 +
admits a continuous extension onto the whole space $  L _{p} ( \mathbf R^{n} ) $
 +
which (for $  1 < p \leq  2 $)  
 +
is given by
  
<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/f041150/f04115046.png" /></td> <td valign="top" style="width:5%;text-align:right;">(3)</td></tr></table>
+
$$ \tag{3}
 +
(Ff \  ) (x) \  = \
 +
\lim\limits _ {R \rightarrow \infty} {}^{q} \
 +
{
 +
\frac{1}{(2 \pi ) ^ n/2}
 +
}
 +
\int\limits _ {| \xi | < R}
 +
f ( \xi ) e ^ {-i \xi x} \
 +
d \xi \  = \  \widetilde{f(x).
 +
$$
  
Convergence is understood to be in the norm of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115047.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115048.png" />, the image of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115049.png" /> under the action of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115050.png" /> does not coincide with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115051.png" />, that is, the imbedding <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115052.png" /> is strict when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115053.png" /> (for the case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115054.png" /> see [[Plancherel theorem|Plancherel theorem]]). The inverse operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115055.png" /> is defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115056.png" /> by
+
Convergence is understood to be in the norm of $  L _{q} ( \mathbf R^{n} ) $.  
 +
If $  p \neq 2 $,  
 +
the image of $  L _{p} $
 +
under the action of $  F $
 +
does not coincide with $  L _{q} $,  
 +
that is, the imbedding $  FL _{p} \subset  L _{q} $
 +
is strict when $  1 \leq  p < 2 $(
 +
for the case $  p = 2 $
 +
see [[Plancherel theorem|Plancherel theorem]]). The inverse operator $  F ^ {\  -1} $
 +
is defined on $  FL _{p} $
 +
by
  
<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/f041150/f04115057.png" /></td> </tr></table>
+
$$
 +
(F ^ {\  -1} \widetilde{f}  \  ) \  = \
 +
\lim\limits _ {R \rightarrow \infty} {}^{p} \
 +
{
 +
\frac{1}{(2 \pi ) ^ n/2}
 +
}
 +
\int\limits _ {| \xi | < R}
 +
\widetilde{f}  ( \xi )
 +
e ^ {i \xi x} \  d \xi ,\ \
 +
1 < p \leq  2.
 +
$$
  
 
The problem of extending the Fourier transform to a larger class of functions arises constantly in analysis and its applications. See, for example, [[Fourier transform of a generalized function|Fourier transform of a generalized function]].
 
The problem of extending the Fourier transform to a larger class of functions arises constantly in analysis and its applications. See, for example, [[Fourier transform of a generalized function|Fourier transform of a generalized function]].
Line 37: Line 91:
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.C. Titchmarsh,  "Introduction to the theory of Fourier integrals" , Oxford Univ. Press  (1948)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.M. Stein,  G. Weiss,  "Fourier analysis on Euclidean spaces" , Princeton Univ. Press  (1971)</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  E.C. Titchmarsh,  "Introduction to the theory of Fourier integrals" , Oxford Univ. Press  (1948)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  A. Zygmund,  "Trigonometric series" , '''2''' , Cambridge Univ. Press  (1988)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  E.M. Stein,  G. Weiss,  "Fourier analysis on Euclidean spaces" , Princeton Univ. Press  (1971)</TD></TR></table>
 
 
  
 
====Comments====
 
====Comments====
 
Instead of  "generalized function"  the term  "distributiondistribution"  is often used.
 
Instead of  "generalized function"  the term  "distributiondistribution"  is often used.
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115058.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115059.png" /> then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115060.png" /> denotes the scalar product <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115061.png" />.
+
If $  x = (x _{1} \dots x _{n} ) $
 +
and $  \xi = ( \xi _{1} \dots \xi _{n} ) $
 +
then $  x \cdot \xi $
 +
denotes the scalar product $  \sum _{ {i = 1}^{n}} x _{i} \xi _{i} $.
  
If in (1) the  "normalizing factor"  <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115062.png" /> is replaced by some constant <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115063.png" />, then in (2) it must be replaced by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115064.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115065.png" />.
+
If in (1) the  "normalizing factor"   $ (1/ {2 \pi} )^{n/2} $
 +
is replaced by some constant $  \alpha $,  
 +
then in (2) it must be replaced by $  \beta $
 +
with $  \alpha \beta = (1/ {2 \pi} )^{n} $.
  
 
At least two other conventions for the  "normalization factor"  are in common use:
 
At least two other conventions for the  "normalization factor"  are in common use:
  
<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/f041150/f04115066.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
+
$$ \tag{a1}
 +
(F \phi ) (x) \  = \
 +
\int\limits _ {\mathbf R ^ n}
 +
\phi ( \xi )
 +
e ^ {- ix \cdot \xi} \
 +
d \xi ,
 +
$$
  
<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/f041150/f04115067.png" /></td> </tr></table>
+
$$
 +
(F ^ {\  -1} \phi ) (x) \  =
 +
\frac{1}{(2 \pi ) ^ n}
 +
\int\limits _
 +
{\mathbf R ^ n} \phi ( \xi ) e ^ {ix \cdot \xi} \  d \xi ,
 +
$$
  
<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/f041150/f04115068.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a2)</td></tr></table>
+
$$ \tag{a2}
 +
(F \phi ) (x) \  = \  \int\limits _ {\mathbf R ^ n} \phi ( \xi ) e ^ {- 2 \pi ix \cdot \xi} \  d \xi ,
 +
$$
  
<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/f041150/f04115069.png" /></td> </tr></table>
+
$$
 +
(F ^ {\  -1} \phi ) (x) \  = \  \int\limits _ {\mathbf R^{n} } \phi ( \xi ) e ^ {2 \pi ix \cdot \xi} \  d \xi .
 +
$$
  
The convention of the article leads to the Fourier transform as a [[Unitary operator|unitary operator]] from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/f/f041/f041150/f04115070.png" /> into itself, and so does the convention (a2). Convention (a1) is more in line with [[Harmonic analysis|harmonic analysis]].
+
The convention of the article leads to the Fourier transform as a [[Unitary operator|unitary operator]] from $  L _{2} ( \mathbf R^{n} ) $
 +
into itself, and so does the convention (a2). Convention (a1) is more in line with [[Harmonic analysis|harmonic analysis]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Functional analysis" , McGraw-Hill  (1973)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,  "Functional analysis" , McGraw-Hill  (1973)</TD></TR></table>

Latest revision as of 12:28, 1 February 2020


One of the integral transforms (cf. Integral transform). It is a linear operator $F$ acting on a space whose elements are functions $f$ of $n$ real variables. The smallest domain of definition of $F$ is the set $D=C_0^\infty$ of all infinitely-differentiable functions $\phi$ of compact support. For such functions

\begin{equation} (F\phi)(x) = \frac{1}{(2\pi)^{\frac{n}{2}}} \cdot \int_{\mathbf R^n} \phi(\xi) e^{-i x \xi} \, \mathrm d\xi. \end{equation}

In a certain sense the most natural domain of definition of $F$ is the set $S$ of all infinitely-differentiable functions $\phi$ that, together with their derivatives, vanish at infinity faster than any power of $\frac{1}{|x|}$. Formula (1) still holds for $\phi\in S$, and $(F \phi)(x) \equiv \psi(x)\in S$. Moreover, $F$ is an isomorphism of $S$ onto itself, the inverse mapping $F^{-1}$ (the inverse Fourier transform) is the inverse of the Fourier transform and is given by the formula:

\begin{equation} \phi(x) = (F^{-1} \psi)(x) = \frac{1}{(2\pi)^{\frac{n}{2}}} \cdot \int_{\mathbf R^n} \psi(\xi) e^{i x \xi} \, \mathrm d\xi. \end{equation}

Formula (1) also acts on the space $L_{1}\left(\mathbf{R}^{n}\right)$ of integrable functions. In order to enlarge the domain of definition of the operator $F$ generalization of (1) is necessary. In classical analysis such a generalization has been constructed for locally integrable functions with some restriction on their behaviour as $|x|\to\infty$ (see Fourier integral). In the theory of generalized functions the definition of the operator $F$ is free of many requirements of classical analysis.

The basic problems connected with the study of the Fourier transform $F$ are: the investigation of the domain of definition $ \Phi $ and the range of values $ F \Phi = \Psi $ of $ F $; as well as studying properties of the mapping $ F: \ \Phi \rightarrow \Psi $( in particular, conditions for the existence of the inverse operator $ F ^ {\ -1} $ and its expression). The inversion formula for the Fourier transform is very simple:

$$ F ^ {\ -1} [g (x)] \ = \ F [g (-x)]. $$

Under the action of the Fourier transform linear operators on the original space, which are invariant with respect to a shift, become (under certain conditions) multiplication operators in the image space. In particular, the convolution of two functions $ f $ and $ g $ goes over into the product of the functions $ Ff $ and $ Fg $:

$$ F (f * g) \ = \ Ff \cdot Fg; $$

and differentiation induces multiplication by the independent variable:

$$ F (D^ \alpha f \ ) \ = \ (ix)^ \alpha Ff. $$

In the spaces $ L _{p} ( \mathbf R^{n} ) $, $ 1 \leq p \leq 2 $, the operator $ F $ is defined by the formula (1) on the set $ D _{F} = (L _{1} \cap L _{p} ) ( \mathbf R^{n} ) $ and is a bounded operator from $ L _{p} ( \mathbf R^{n} ) $ into $ L _{q} ( \mathbf R^{n} ) $, $ p^{-1} + q^{-1} = 1 $:

\begin{equation} \left\{\frac{1}{(2 \pi)^{n / 2}} \int_{\mathbf{R}^{n}}|(F f)(x)|^{q} d x\right\}^{1 / q} \leq\left\{\frac{1}{(2 \pi)^{n / 2}} \int_{\mathbf{R}^{n}}|f(x)|^{p} d x\right\}^{1 / p} \end{equation}

(the Hausdorff–Young inequality). $ F $ admits a continuous extension onto the whole space $ L _{p} ( \mathbf R^{n} ) $ which (for $ 1 < p \leq 2 $) is given by

$$ \tag{3} (Ff \ ) (x) \ = \ \lim\limits _ {R \rightarrow \infty} {}^{q} \ { \frac{1}{(2 \pi ) ^ n/2} } \int\limits _ {| \xi | < R} f ( \xi ) e ^ {-i \xi x} \ d \xi \ = \ \widetilde{f} (x). $$

Convergence is understood to be in the norm of $ L _{q} ( \mathbf R^{n} ) $. If $ p \neq 2 $, the image of $ L _{p} $ under the action of $ F $ does not coincide with $ L _{q} $, that is, the imbedding $ FL _{p} \subset L _{q} $ is strict when $ 1 \leq p < 2 $( for the case $ p = 2 $ see Plancherel theorem). The inverse operator $ F ^ {\ -1} $ is defined on $ FL _{p} $ by

$$ (F ^ {\ -1} \widetilde{f} \ ) \ = \ \lim\limits _ {R \rightarrow \infty} {}^{p} \ { \frac{1}{(2 \pi ) ^ n/2} } \int\limits _ {| \xi | < R} \widetilde{f} ( \xi ) e ^ {i \xi x} \ d \xi ,\ \ 1 < p \leq 2. $$

The problem of extending the Fourier transform to a larger class of functions arises constantly in analysis and its applications. See, for example, Fourier transform of a generalized function.

References

[1] E.C. Titchmarsh, "Introduction to the theory of Fourier integrals" , Oxford Univ. Press (1948)
[2] A. Zygmund, "Trigonometric series" , 2 , Cambridge Univ. Press (1988)
[3] E.M. Stein, G. Weiss, "Fourier analysis on Euclidean spaces" , Princeton Univ. Press (1971)

Comments

Instead of "generalized function" the term "distributiondistribution" is often used.

If $ x = (x _{1} \dots x _{n} ) $ and $ \xi = ( \xi _{1} \dots \xi _{n} ) $ then $ x \cdot \xi $ denotes the scalar product $ \sum _{ {i = 1}^{n}} x _{i} \xi _{i} $.

If in (1) the "normalizing factor" $ (1/ {2 \pi} )^{n/2} $ is replaced by some constant $ \alpha $, then in (2) it must be replaced by $ \beta $ with $ \alpha \beta = (1/ {2 \pi} )^{n} $.

At least two other conventions for the "normalization factor" are in common use:

$$ \tag{a1} (F \phi ) (x) \ = \ \int\limits _ {\mathbf R ^ n} \phi ( \xi ) e ^ {- ix \cdot \xi} \ d \xi , $$

$$ (F ^ {\ -1} \phi ) (x) \ = \ \frac{1}{(2 \pi ) ^ n} \int\limits _ {\mathbf R ^ n} \phi ( \xi ) e ^ {ix \cdot \xi} \ d \xi , $$

$$ \tag{a2} (F \phi ) (x) \ = \ \int\limits _ {\mathbf R ^ n} \phi ( \xi ) e ^ {- 2 \pi ix \cdot \xi} \ d \xi , $$

$$ (F ^ {\ -1} \phi ) (x) \ = \ \int\limits _ {\mathbf R^{n} } \phi ( \xi ) e ^ {2 \pi ix \cdot \xi} \ d \xi . $$

The convention of the article leads to the Fourier transform as a unitary operator from $ L _{2} ( \mathbf R^{n} ) $ into itself, and so does the convention (a2). Convention (a1) is more in line with harmonic analysis.

References

[a1] W. Rudin, "Functional analysis" , McGraw-Hill (1973)
How to Cite This Entry:
Fourier transform. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fourier_transform&oldid=12659
This article was adapted from an original article by P.I. Lizorkin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article