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An integral transform of a function in several variables, related to the [[Fourier transform|Fourier transform]]. It was introduced by J. Radon (see [[#References|[1]]]).

Let  $  f ( x _ {1} \dots x _ {n} ) $
be a continuous function of the real variables  $  x _ {i} \in \mathbf R  ^ {1} $
that is decreasing sufficiently rapidly at infinity,  $  i = 1 \dots n $, 
$  n = 1 , 2 ,\dots $.

For any hyperplane in  $  \mathbf R  ^ {n} $,

$$ 
\Gamma  =  \{ {( x _ {1} \dots x _ {n} ) } : {\xi _ {1} x _ {1} + \dots + \xi _ {n} x _ {n} = C } \}
,
$$

$$ 
\xi _ {i}  \in  \mathbf R  ^ {1} ,\  i  =  1 \dots n ,
$$

and

$$ 
\sum _ { i= } 1 ^ { n }  \xi _ {i}  ^ {2}  >  0 ,\ \ 
C  \in  \mathbf R  ^ {1} ,
$$

the following integral is defined:

$$ 
F ( \xi _ {1} \dots \xi _ {n} ;  C )  = \ 

\frac{1}{\left ( \sum _ { i= } 1 ^ { n }  \xi _ {j} \right )  ^ {1/2} }

\int\limits _  \Gamma  f ( x _ {1} \dots x _ {n} )  d V _  \Gamma  ,
$$

where  $  V _  \Gamma  $
is the Euclidean  $  ( n - 1 ) $-
dimensional volume in the hyperplane  $  \Gamma $. 
The function

$$ 
F ( \xi _ {1} \dots \xi _ {n} ;  C ) ,\ \ 
( \xi _ {1} \dots x _ {n} , C ) \in \mathbf R  ^ {n+} 1 ,
$$

is called the Radon transform of the function  $  f $. 
It is a homogeneous function of its variables of degree  $  - 1 $:

$$ 
F ( \alpha \xi _ {1} \dots \alpha \xi _ {n} ;  \alpha C )  = \ 

\frac{1}{| \alpha | }
 F ( \xi _ {1} \dots \xi _ {n} ;  C ) ,
$$

and is related to the Fourier transform  $  \widetilde{f}  ( \xi _ {1} \dots \xi _ {n} ) $, 
$  \xi _ {i} \in \mathbf R  ^ {1} $, 
of  $  f $
by

$$ 
F ( \xi _ {1} \dots \xi _ {n} ;  C )  =  
\frac{1}{2 \pi }
 \int\limits _ {- \infty } ^  \infty   \widetilde{f} 
( \alpha \xi _ {1} \dots \alpha \xi _ {n} ) e ^ {- i \alpha C }  d \alpha .
$$

The Radon transform is immediately associated with the problem, going back to Radon, of the recovery of a function  $  f $
from the values of its integrals calculated over all hyperplanes of the space  $  \mathbf R  ^ {n} $(
that is, the problem of the inversion of the Radon transform).

====References====
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  J. Radon,   "Ueber die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten"  ''Ber. Verh. Sächs. Akad.'' , '''69'''  (1917)  pp. 262–277</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  I.M. Gel'fand,   M.I. Graev,   N.Ya. Vilenkin,   "Generalized functions" , '''5. Integral geometry and representation theory''' , Acad. Press  (1966)  (Translated from Russian)</TD></TR></table>

====Comments====
For the far-reaching generalizations of the Radon transform to homogeneous spaces see [[#References|[a3]]].

The Radon transform and, in particular, the corresponding inversion formula (i.e. the formula recovering  $  f $
from its Radon transform) is of central importance in [[Tomography|tomography]].

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  S.R. Deans,   "The Radon transform and some of its applications" , Wiley  (1983)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Helgason,   "The Radon transform" , Birkhäuser  (1980)</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  S. Helgason,   "Groups and geometric analysis" , Acad. Press  (1984)  pp. Chapt. II, Sect. 4</TD></TR></table>