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PreprintarXiv licencearXiv

Counterexamples to uniqueness for a discrete X-ray transform under quadratic decay

  • Shubham R. Jathar
  • Jesse Railo
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Abstract

We show nonuniqueness for a discrete X-ray transform on ℤ2 under quadratic decay. We construct an explicit nonzero function in ℓ1(ℤ2) with pointwise decay O(|z|−2) whose sums over integer lines vanish. We also obtain an infinite-dimensional null space with the same decay assumption. Furthermore, we show a result which indicates that the decay rate |z|−2 may be sharp and faster decay may imply uniqueness.

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Cite

BibTeXRIS
BibTeX
@misc{jathar2026counterexamples,
  title = {{Counterexamples to uniqueness for a discrete X-ray transform under quadratic decay}},
  author = {Shubham R. Jathar and Jesse Railo},
  howpublished = {arXiv},
  year = {2026},
  month = nov,
  doi = {10.48550/arxiv.2611.01294},
  eprint = {2611.01294},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2611.01294},
}
RIS
TY  - UNPB
TI  - Counterexamples to uniqueness for a discrete X-ray transform under quadratic decay
AU  - Shubham R. Jathar
AU  - Jesse Railo
PY  - 2026
DA  - 2026/11/01
PB  - arXiv
DO  - 10.48550/arxiv.2611.01294
UR  - https://arxiv.org/abs/2611.01294
AB  - We show nonuniqueness for a discrete X-ray transform on ℤ^2 under quadratic decay. We construct an explicit nonzero function in ℓ^1(ℤ^2) with pointwise decay O(|z|^−2) whose sums over integer lines vanish. We also obtain an infinite-dimensional null space with the same decay assumption. Furthermore, we show a result which indicates that the decay rate |z|^−2 may be sharp and faster decay may imply uniqueness.
ER  -