Abstract
We consider the problem of reconstructing a function on the disk \(\mathbb{D}\prime \subset \mathbb{R}^2 \)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 \(\mathbb{D}\prime \subset \mathbb{D}\) under the additional condition that the curves that do not meet \(\mathbb{D}\prime \) coincide with the corresponding straight lines.
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References
D. A. Popov, “The generalized Radon transform on the plane, the inverse transform, and the Cavalieri conditions,” Funkts. Anal. Prilozhen., 35, No. 4, 38–53 (2001).
I. M Gelfand, S. G. Gindikin, and M. I. Graev, Selected Problems of Integral Geometry [in Russian], Dobrosvet, Moscow, 2000.
S. Helgason, The Radon Transform, Birkhäuser, Boston-Basel-Stuttgart, 1980.
P. D. Lax and R. S. Phillips, “The Paley-Wiener Theorem for the Radon Transform,” Comm. Pure Appl. Math., 23, No. 3, 409–424 (1970).
R. B. Marr, “On the reconstruction of a function on a circular domain from a sampling of its line integrals,” J. Math. Anal. Appl., 45, 357–374 (1974).
P. K. Suetin, Orthogonal Polynomials [in Russian], Nauka, Moscow, 1979. Moscow State University, Belozerskii Institute of Physical-Chemical Biology
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Popov, D.A. The Paley--Wiener Theorem for the Generalized Radon Transform on the Plane. Functional Analysis and Its Applications 37, 215–220 (2003). https://doi.org/10.1023/A:1026036701110
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DOI: https://doi.org/10.1023/A:1026036701110