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Large-Scale Equivalence of Norms of the Radon Transform and Initial Function

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Abstract

This study aims to establish equivalences (in norm) of the problems of reconstructing computed tomography and computational (numerical) diameter (C(N)D), which was done in 2019 for functions of two variables. This was based on the equivalence of respective norms in the same two-dimensional Sobolev spaces proved by Frank Natterer. In this study, we prove the equivalence (in norm) of the Radon transform and the function that generated it for the case of functions of any dimension with large-scale prospects for application.

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REFERENCES

  1. N. Temirgaliev, Sh. Abikenova, Sh. Azhgaliev, and G. Taugynbaeva, “Theory of Radon transform in the concept of computational numerical diameter and methods of the quasi-Monte Carlo theory,” Bull. Gumilyov Eurasian Natl. Univ., Math. Comput. Sci. Mech. Ser., No. 4, 8–53 (2019). http://rep.enu.kz/handle/enu/2226.

  2. N. Temirgaliev, Sh. K. Abikenova, Sh. U. Azhgaliev, and G. E. Taugynbaeva, “The Radon transform in the scheme of C(N)D-investigations and the quasi-Monte Carlo theory,” Russ. Math. 64, 87–92 (2020). https://doi.org/10.3103/s1066369x2003010x

    Article  MATH  Google Scholar 

  3. F. Natterer, The Mathematics of Computerized Tomography, Classics of Applied Mathematics (SIAM, Philadelphia, 2001).

  4. F. Natterer, “A Sobolev space analysis of picture reconstruction,” SIAM J. Appl. Math. 39, 402–411 (1980). https://doi.org/10.1137/0139034

    Article  MathSciNet  MATH  Google Scholar 

  5. I. M. Gel’fand, M. I. Graev, and M. Ya. Vilenkin, Integral Geometry and Representation Theory, Generalized Functions, Vol. 5 (GIFML, Moscow, 1962; Academic, New York, 1966).

  6. V. A. Sharafutdinov, “Radon transform on Sobolev spaces,” Sib. Math. J. 62, 560–580 (2021). https://doi.org/10.1134/S0037446621030198

    Article  MathSciNet  MATH  Google Scholar 

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Funding

This work was supported by the Ministry of Science and Higher Education of the Republic of Kazakhstan, project no. AP09260484.

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Authors and Affiliations

  1. Institute of Theoretical Mathematics and Scientific Computations, Gumilev Eurasian National University, 010008, Astana, Republic of Kazakhstan

    N. Temirgaliyev, G. E. Taugynbayeva & A. Zh. Zhubanysheva

Authors
  1. N. Temirgaliyev
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  2. G. E. Taugynbayeva
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  3. A. Zh. Zhubanysheva
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Corresponding authors

Correspondence to N. Temirgaliyev, G. E. Taugynbayeva or A. Zh. Zhubanysheva.

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The authors declare that they have no conflicts of interest.

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Translated by V. Arutyunyan

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Temirgaliyev, N., Taugynbayeva, G.E. & Zhubanysheva, A.Z. Large-Scale Equivalence of Norms of the Radon Transform and Initial Function. Russ Math. 67, 62–66 (2023). https://doi.org/10.3103/S1066369X23080091

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  • DOI: https://doi.org/10.3103/S1066369X23080091

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