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


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 \(\mathbb {R}^d, \, d\ge 3\).


Radon transforms Integral geometry Injectivity Non-injectivity 

Mathematics Subject Classification

44A12 53C65 65R32 



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.


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© Mathematica Josephina, Inc. 2018

Authors and Affiliations

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

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