najoNanosystems: Physics, Chemistry, MathematicsНаносистемы: физика, химия, математика2220-80542305-7971Университет ИТМО10.17586/2220-8054-2023-14-1-22-27najo-172Research ArticleMATHEMATICSМАТЕМАТИКАAn inversion formula for the weighted Radon transform along family of conesФормула обращения для весового преобразования Радона вдоль семейства конусовhttps://orcid.org/0000-0003-0335-6558МуминовМ. И.MuminovM. I.

Мухиддин И. Муминов,

Самарканд;

Ташкент.

Mukhiddin I. Muminov,

140100, Samarkand;

100174, Tashkent.

mmuminov@mail.ruhttps://orcid.org/0000-0002-4803-538XОчиловЗ. Х.OchilovZ. Kh.

Зарифжон Х. Очилов,

Самарканд.

Zarifjon Kh. Ochilov,

140100, Samarkand.

zarifjonochilov@mail.ruSamarkand State University; V. I. Romanovskiy Institute of Mathematics, Uzbekistan Academy of SciencesUzbekistanSamarkand State UniversityUzbekistan2023050620251412227Copyright © Muminov M.I., Ochilov Z.K., 20252025Муминов М.И., Очилов З.Х.Muminov M.I., Ochilov Z.K.This work is licensed under a Creative Commons Attribution 4.0 License.https://nanojournal.ifmo.ru/jour/article/view/172

In this paper, an inversion problem for the weighted Radon transform along family of cones in threedimensional space is considered. An inversion formula for the weighted Radon transform is obtained for the case when the range is a space of infinitely smooth functions.

В этой статье решается задача обращения весового преобразования Радона вдоль семейства конусов в трехмерном пространстве. Получена формула обращения для взвешенного преобразования Радона для случая, когда областью значений является пространство бесконечно гладких функций.

задача интегральной геометриивесовое преобразование Радонаформула обращенияintegral geometry problemweighted Radon transforminversion formulaAuthors partially supported by the grant FZ-20200929224 of Fundamental Science Foundation of Uzbekistan. Authors thank the referee for valuable comments and gratefully acknowledge.ReferencesRadon J. Uber die bestimmung von funktionen durch ihre integralwerte lngs gewisser mannigfaltigkeiten. Berichte¨ uber die Verhandlungen der¨ Koniglich-S¨ achsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse, 1917,¨ 69, P. 262–277.Radon J. Uber die bestimmung von funktionen durch ihre integralwerte lngs gewisser mannigfaltigkeiten. Berichte¨ uber die Verhandlungen der¨ Koniglich-S¨ achsischen Akademie der Wissenschaften zu Leipzig, Mathematisch-Physische Klasse, 1917,¨ 69, P. 262–277.Vassholz M., Koberstein-Schwarz B., Ruhlandt A., Krenkel M. and Salditt T. New X-ray tomography method based on the 3d Radon transform compatible with anisotropic sources. Physical Review Letters, 2016, 116(8), P. 088101.Vassholz M., Koberstein-Schwarz B., Ruhlandt A., Krenkel M. and Salditt T. New X-ray tomography method based on the 3d Radon transform compatible with anisotropic sources. Physical Review Letters, 2016, 116(8), P. 088101.Frikel J., Quinto E.T. Limited data problems for the generalized Radon transform in Rn, SIAM J. Math. Anal., 2016, 48(4), P. 2301–2318.Frikel J., Quinto E.T. Limited data problems for the generalized Radon transform in Rn, SIAM J. Math. Anal., 2016, 48(4), P. 2301–2318.Seeck O.H., Murphy B. (Eds.) X-Ray Diffraction: Modern Experimental Techniques (1st ed.). Jenny Stanford Publishing, 2014.Seeck O.H., Murphy B. (Eds.) X-Ray Diffraction: Modern Experimental Techniques (1st ed.). Jenny Stanford Publishing, 2014.Goncharov F.O., Novikov R.G. An example of non-uniqueness for the weighted Radon transforms along hyperplanes in multidimensions. Inverse Problem, 2018, 34, P. 054001.Goncharov F.O., Novikov R.G. An example of non-uniqueness for the weighted Radon transforms along hyperplanes in multidimensions. Inverse Problem, 2018, 34, P. 054001.Goncharov F.O., Novikov R.G. An analog of Chang inversion formula for weighted Radon transforms in multidimensions. Eurasian Journal of Mathematical and Computer Applications, 2016, 4(2), P. 23–32.Goncharov F.O., Novikov R.G. An analog of Chang inversion formula for weighted Radon transforms in multidimensions. Eurasian Journal of Mathematical and Computer Applications, 2016, 4(2), P. 23–32.Goncharov F.O. A geometric based preprocessing for weighted ray transforms with applications in SPECT, Journal of Inverse and Ill-posed Problems, 2021, 29(3), P. 435–457.Goncharov F.O. A geometric based preprocessing for weighted ray transforms with applications in SPECT, Journal of Inverse and Ill-posed Problems, 2021, 29(3), P. 435–457.Lavrentyev M.M., Savelyev L.Ya. Operator theory and ill-posed problems. Publishing house of the Institute of Mathematics, Moscow, 2010. [9] Kabanikhin S.I. Inverse and ill-posed problems. Siberian Scientific Publishing House, 2009.Lavrentyev M.M., Savelyev L.Ya. Operator theory and ill-posed problems. Publishing house of the Institute of Mathematics, Moscow, 2010. [9] Kabanikhin S.I. Inverse and ill-posed problems. Siberian Scientific Publishing House, 2009.Begmatov A.Kh., Muminov M.E., Ochilov Z.H. The problem of integral geometry of Volterra type with a weight function of a special type. Mathematics and Statistics, 2015, 3, P. 113–120.Begmatov A.Kh., Muminov M.E., Ochilov Z.H. The problem of integral geometry of Volterra type with a weight function of a special type. Mathematics and Statistics, 2015, 3, P. 113–120.Ochilov Z.X. The uniqueness of solution problems of integral geometry a family of parabolas with a weight function of a special type. Uzbek Mathematical Journal, 2020, 3, P. 107–116.Ochilov Z.X. The uniqueness of solution problems of integral geometry a family of parabolas with a weight function of a special type. Uzbek Mathematical Journal, 2020, 3, P. 107–116.Ochilov Z.Kh. Existence of solutions to problems of integral geometry by a family of parabolas with a weight function of a special form. Bull. Inst. Math., 2021, 4(4), P. 28–33.Ochilov Z.Kh. Existence of solutions to problems of integral geometry by a family of parabolas with a weight function of a special form. Bull. Inst. Math., 2021, 4(4), P. 28–33.Helgason S. Integral Geometry and Radon Transform. Springer, New York, 2011.Helgason S. Integral Geometry and Radon Transform. Springer, New York, 2011.Natterer F. The mathematics of computerized tomography, Classics in Mathematics. Society for Industrial and Applied Mathematics, New York, 2001.Natterer F. The mathematics of computerized tomography, Classics in Mathematics. Society for Industrial and Applied Mathematics, New York, 2001.Polyanin A.D., Manzhirov A.V. Handbook of Integral Equations. CRC Press LLC, N.W. Corporate Blvd., Boca Raton, Florida, 2000.Polyanin A.D., Manzhirov A.V. Handbook of Integral Equations. CRC Press LLC, N.W. Corporate Blvd., Boca Raton, Florida, 2000.Gradsteyn I.S., Ryzhik I.M. Table of Integrals. Series, and Products, Academic Press, New York, 2007.Gradsteyn I.S., Ryzhik I.M. Table of Integrals. Series, and Products, Academic Press, New York, 2007.

The authors declare that there are no conflicts of interest present.