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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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/

[1] Popov D. A., “Obobschennoe preobrazovanie Radona na ploskosti, ego obraschenie i usloviya Kavaleri”, Funkts. analiz i ego pril., 35:4 (2001), 38–53 | DOI | MR | Zbl

[2] Gelfand I. M., Gindikin S. G., Graev M. I., Izbrannye zadachi integralnoi geometrii, Dobrosvet, M., 2000 | MR

[3] Khelgason S., Preobrazovanie Radona, Mir, M., 1983 | MR | Zbl

[4] Lax P. D., Phillips R. S., “The Paley–Wiener Theorem for the Radon Transform”, Comm. Pure Appl. Math., 23:3 (1970), 409–424 | DOI | MR | Zbl

[5] Marr R. B., “On the reconstruction of a function on a circular domain from a sampling of its line integrals”, J. Math. Anal. Appl., 45 (1974), 357–374 | DOI | MR | Zbl

[6] Suetin P. K., Ortogonalnye polinomy, Nauka, M., 1979 | MR | Zbl