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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2511.08871 |
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| _version_ | 1866909899198300160 |
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| author | Kay, Jonathan Monard, François |
| author_facet | Kay, Jonathan Monard, François |
| contents | If $d$ is a boundary defining function for the Euclidean unit disk and $I$ denotes the geodesic X-ray transform, for $γ\in (-1,1)$, we study the singularly-weighted X-ray transforms $I_m d^γ$ acting on symmetric $m$-tensors. For any $m$, we provide a sharp range decomposition and characterization in terms of a distinguished Hilbert basis of the data space, that comes from earlier studies of the Singular Value Decomposition for the case $m=0$. Since for $m\ge 1$, the transform considered has an infinite-dimensional kernel, we fully characterize this kernel, and propose a representative for an $m$-tensor to be reconstructed modulo kernel, along with efficient procedures to do so. This representative is based on a new generalization of the potential/conformal/transverse-tracefree decomposition of tensor fields in the context of singularly weighted $L^2$-topologies. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_08871 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Singularly Weighted X-ray Tensor Tomography Kay, Jonathan Monard, François Analysis of PDEs If $d$ is a boundary defining function for the Euclidean unit disk and $I$ denotes the geodesic X-ray transform, for $γ\in (-1,1)$, we study the singularly-weighted X-ray transforms $I_m d^γ$ acting on symmetric $m$-tensors. For any $m$, we provide a sharp range decomposition and characterization in terms of a distinguished Hilbert basis of the data space, that comes from earlier studies of the Singular Value Decomposition for the case $m=0$. Since for $m\ge 1$, the transform considered has an infinite-dimensional kernel, we fully characterize this kernel, and propose a representative for an $m$-tensor to be reconstructed modulo kernel, along with efficient procedures to do so. This representative is based on a new generalization of the potential/conformal/transverse-tracefree decomposition of tensor fields in the context of singularly weighted $L^2$-topologies. |
| title | Singularly Weighted X-ray Tensor Tomography |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2511.08871 |