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

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

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.