A breakdown of injectivity for weighted ray transforms in multidimensions

Fedor Goncharov CMAP, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris Roman Novikov CMAP, CNRS, Ecole Polytechnique, Institut Polytechnique de Paris

Functional Analysis mathscidoc:1912.43041

Arkiv for Matematik, 57, (2), 333 – 371, 2019
We consider weighted ray-transforms PW (weighted Radon transforms along oriented straight lines) in Rd,d≥2, with strictly positive weights W. We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions on Rd. In addition, the constructed weight W is rotation-invariant continuous and is infinitely smooth almost everywhere on Rd×Sd−1. In particular, by this construction we give counterexamples to some well-known injectivity results for weighted ray transforms for the case when the regularity of W is slightly relaxed. We also give examples of continous strictly positive W such that dimkerPW≥n in the space of infinitely smooth compactly supported functions on Rd for arbitrary n∈N∪{∞}, where W are infinitely smooth for d=2 and infinitely smooth almost everywhere for d≥3.
radon transforms, ray transforms, integral geometry, injectivity, non-injectivity
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@inproceedings{fedor2019a,
  title={A breakdown of injectivity for weighted ray transforms in multidimensions},
  author={Fedor Goncharov, and Roman Novikov},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191204140119446967597},
  booktitle={Arkiv for Matematik},
  volume={57},
  number={2},
  pages={333 – 371},
  year={2019},
}
Fedor Goncharov, and Roman Novikov. A breakdown of injectivity for weighted ray transforms in multidimensions. 2019. Vol. 57. In Arkiv for Matematik. pp.333 – 371. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20191204140119446967597.
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