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

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

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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.

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Goncharov, F.O., Novikov, R.G. An Example of Non-uniqueness for Radon Transforms with Continuous Positive Rotation Invariant Weights. J Geom Anal 28, 3807–3828 (2018). https://doi.org/10.1007/s12220-018-0001-y

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Keywords

  • Radon transforms
  • Integral geometry
  • Injectivity
  • Non-injectivity

Mathematics Subject Classification

  • 44A12
  • 53C65
  • 65R32