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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 be a continuous function of the real variables that is decreasing sufficiently rapidly at infinity, , .

For any hyperplane in ,

and

the following integral is defined:

where is the Euclidean -dimensional volume in the hyperplane . The function

is called the Radon transform of the function . It is a homogeneous function of its variables of degree :

and is related to the Fourier transform , , of by

The Radon transform is immediately associated with the problem, going back to Radon, of the recovery of a function from the values of its integrals calculated over all hyperplanes of the space (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)