View source for Fourier transform

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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

\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 $S$ <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:

\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|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|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|Fourier transform of a generalized function]].

====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>

====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|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====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  W. Rudin,   "Functional analysis" , McGraw-Hill  (1973)</TD></TR></table>