Inversion of Radon Transformation for Discontinuous Functions in Unbounded Sets

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Home Editorial board Publication ethics Peer review guidelines Latest issue All issues Rules for authors Online submission system's guidelines Submit manuscript Contacts Address: Vatutina st. 53, Vladikavkaz, 362025, RNO-A, Russia Phone: (8672)23-00-54 E-mail: rio@smath.ru	DOI: 10.46698/e9041-1168-6207-j Inversion of Radon Transformation for Discontinuous Functions in Unbounded Sets Anikonov, D. S. , Konovalova, D. S. Vladikavkaz Mathematical Journal 2024. Vol. 26. Issue 4. Abstract: This work relates to the theory of integral geometry in Euclidean space. The object to be searched is an information about the integrand for some given set of integrals. Similar settings are in demand in the theory of differential equations. Such investigations have been presented, for example, in the works of D. Radon, R. Courant, F. Jon, I. M. Gelfand. Later the use of integral geometry was associated with the study of inverse problems for differential equations. In particular, some formulations of inverse problems coincided with the problems of integral geometry. This circumstance has been widely used in works of the mathematical school of M. M. Lavrentiev and V. G. Romanov. From related areas of research, we note first of all probing environment with physical signals. Probably currently the most well-known area is $X$-ray tomography for needs of medicine and technology. More specifically, we mean theory of classical and generalized Radon transformations. In this area numerous results are received for Radon transformations. Moreover, some of the uniqueness theorems have been proven for fairly weak constraints. But the inversion formulas have been proven only for smooth functions. This circumstance somewhat reduces their applied value. This prompted the authors of this work to study specifically the cases of discontinuous integrands. An essential element of the proposed research is to introduce the concept of pseudoconvex sets on which unknown discontinuous functions are defined. Such sets turned out to be, on the one hand, not burdensome for the theory of probing, and on the other hand, convenient for research. So far it has been possible to study only the case of odd-dimensional Euclidean space. Keywords: Radon transform, pseudoconvex sets, integral geometry, probing, tomography, discontinuous functions, inversion formula Language: Russian Download the full text For citation: Anikonov, D. S. and Konovalova, D. S. Inversion of Radon Transformation for Discontinuous Functions in Unbounded Sets, Vladikavkaz Math. J., 2024, vol. 26, no. 4, pp. 21-27. (in Russian). DOI 10.46698/e9041-1168-6207-j + References 1. John, F. Plane Waves and Spherical Means Applied to Partial Differential Equations, Dover Publications, 1955. 2. Markoe, A. Analytic Tomography in Encyclopedia of Mathematics and its Applications, Cambridge, Cambridge University Press, 2006, 315 p. 3. Natterer, F. The Mathematics of Computerized Tomography, Wiley, 1986, 232 p. 4. Anikonov, D. S., Kovtanyuk, A. E. and Prokhorov, I. V. Ispol'zovanie uravneniya perenosa v tomografii [Using Transport Equations in Tomography], Moscow, Logos, 2000, 224 p. (in Russian). 5. Anikonov, D. S., Balakina, E. Yu. and Konovalova, D. S. An Inverse Problem for Generalized Radon Transformation, St. Petersburg Polytechnical State University Journal: Physics and Mathematics, 2022, vol. 15, no. 1, pp. 41-51. DOI: 10.18721/JPM15104. 6. Anikonov, D. S. and Konovalova, D. S. A Problem of Integral Geometry for a Family of Curves with Incomplete Data, Doklady Mathematics, 2015, vol. 92, no. 2, pp. 221-224. DOI: 10.1134/S1064562415050026. 7. Kalnin, T. G., Ivonin, D. A., Abrosimov, K. N., Grachev, E. A. and Sorokina, N. V. 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Series Mathematical Modeling, Programming and Computer Software, 2019, vol. 12, no. 3, pp. 102-114 (in Russian). DOI: 10.14529/mmp190309. 11. Derevtsov, E. Yu., Volkov, Yu. S. and Schuster, T. Differential Equations and Uniqueness Theorems for the Generalized Attenuated Ray Transforms of Tensor Fields, Numerical Computations: Theory and Algorithms. Part II, Lecture Notes in Computer Science, Eds. Ya. D. Sergeyev, D. E. Kvasov, 2020, vol. 11974, pp. 97-111. DOI: 10.1007/978-3-030-40616-5_8. The sole copyright holder of the published work is the Founder of the Vladikavkaz Mathematical Journal. The terms of use of this work are governed by an open license (Creative Commons Attribution-NonCommercial 4.0 International). 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