Geodesic
The Paley--Wiener Theorem for the Generalized Radon Transform on the Plane
Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 3, pp. 65-72

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We consider the problem of reconstructing a function on the disk $\mathbb{D}\subset\mathbb{R}^2$ from its integrals over curves close to straight lines, i.e., the inversion problem for the generalized Radon transform. Necessary and sufficient conditions on the range of the generalized Radon transform are obtained for functions supported in a smaller disk $\mathbb{D}'\subset\mathbb{D}$ under the additional condition that the curves that do not meet $\mathbb{D}'$ coincide with the corresponding straight lines.
Keywords: Paley–Winer theorem, Fourier integral operator, Zernike polynomial.
Mots-clés : Radon transform 
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     title = {The {Paley--Wiener} {Theorem} for the {Generalized} {Radon} {Transform} on the {Plane}},
     journal = {Funkcionalʹnyj analiz i ego prilo\v{z}eni\^a},
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D. A. Popov. The Paley--Wiener Theorem for the Generalized Radon Transform on the Plane. Funkcionalʹnyj analiz i ego priloženiâ, Tome 37 (2003) no. 3, pp. 65-72. http://geodesic.mathdoc.fr/item/FAA_2003_37_3_a4/