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The Journal of Geometric Analysis

, Volume 28, Issue 4, pp 3807–3828 | Cite as

An Example of Non-uniqueness for Radon Transforms with Continuous Positive Rotation Invariant Weights

  • F. O. Goncharov
  • R. G. NovikovEmail author
Article
  • 60 Downloads

Abstract

We consider weighted Radon transforms \(R_W\) along hyperplanes in \(\mathbb {R}^3\) with strictly positive weights W. We construct an example of such a transform with non-trivial kernel \(\mathrm {Ker}R_W\) in the space of infinitely smooth compactly supported functions and with continuous weight. Moreover, in this example the weight W is rotation invariant. In particular, by this result we continue studies of Quinto (J Math Anal Appl 91(2): 510–522, 1983), Markoe and Quinto (SIAM J Math Anal 16(5), 1114–1119, 1985), Boman (J Anal Math 61(1), 395–401, 1993) and Goncharov and Novikov (An example of non-uniqueness for the weighted Radon transforms along hyperplanes in multidimensions. arXiv:1709.04194v2, 2017). We also extend our example to the case of weighted Radon transforms along two-dimensional planes in \(\mathbb {R}^d, \, d\ge 3\).

Keywords

Radon transforms Integral geometry Injectivity Non-injectivity 

Mathematics Subject Classification

44A12 53C65 65R32 

Notes

Acknowledgements

This work is partially supported by the PRC No. 1545 CNRS/RFBR: Équations quasi-linéaires, problèmes inverses et leurs applications. The authors are also grateful to the referee for remarks that have helped to improve the presentation.

References

  1. 1.
    Beylkin, G.: The inversion problem and applications of the generalized Radon transform. Commun. Pure Appl. Math. 37(5), 579–599 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Beylkin, G.: Imaging of discontinuities in the inverse scattering problem by inversion of a causal generalized Radon transform. J. Math. Phys. 26(1), 99–108 (1985)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Boman, J., Quinto, E.T.: Support theorems for real-analytic Radon transforms. Duke Math. J. 55(4), 943–948 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Boman, J.: An example of non-uniqueness for a generalized Radon transform. J. Anal. Math. 61(1), 395–401 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Do Carmo, M.P.: Riemannian Geometry. Birkhäuser, Basel (1992)CrossRefzbMATHGoogle Scholar
  6. 6.
    Finch, D.: Uniqueness for the attenuated X-ray transform in the physical range. Inverse Probl. 2(2) (1986)Google Scholar
  7. 7.
    Goncharov, F.O., Novikov, R.G.: An analog of Chang inversion formula for weighted Radon transforms in multidimensions. Eurasian J. Math. Comput. Appl. 4(2), 23–32 (2016)Google Scholar
  8. 8.
    Goncharov, F.O.: An iterative inversion of weighted Radon transforms along hyperplanes. Inverse Probl. 33, 124005 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Goncharov, F.O., Novikov, R.G.: An example of non-uniqueness for the weighted Radon transforms along hyperplanes in multidimensions. arXiv:1709.04194v2 (2017)
  10. 10.
    Goncharov, F.O., Novikov, R.G.: A breakdown of injectivity for weighted ray transforms in multidimensions. hal-01635188, version 1 (2017)Google Scholar
  11. 11.
    Guillement, J.-P., Novikov, R.G.: Inversion of weighted Radon transforms via finite Fourier series weight approximations. Inverse Probl. Sci. Eng. 22(5), 787–802 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Kunyansky, L.: Generalized and attenuated Radon transforms: restorative approach to the numerical inversion. Inverse Probl. 8(5), 809 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lavrent’ev, M.M., Bukhgeim, A.L.: A class of operator equations of the first kind. Funct. Anal. Appl. 7(4), 290–298 (1973)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Markoe, A., Quinto, E.T.: An elementary proof of local invertibility for generalized and attenuated Radon transforms. SIAM J. Math. Anal. 16(5), 1114–1119 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Natterer, F.: The Mathematics of Computerized Tomography. SIAM (2001)Google Scholar
  16. 16.
    Novikov, R.G.: Weighted Radon transforms for which Chang’s approximate inversion formula is exact. Russ. Math. Surv. 66(2), 442–443 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Novikov, R.G.: Weighted Radon transforms and first order differential systems on the plane. Moscow Math. J. 14(4), 807–823 (2014)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Quinto, E.T.: The invertibility of rotation invariant Radon transforms. J. Math. Anal. Appl. 91(2), 510–522 (1983). Erratum: J. Math. Anal. Appl. 94(2), 602–603 (1983)Google Scholar

Copyright information

© Mathematica Josephina, Inc. 2018

Authors and Affiliations

  1. 1.CMAP, Ecole Polytechnique, CNRS, Université Paris-SaclayPalaiseauFrance
  2. 2.IEPT RASMoscowRussia

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