d-plane transform: unique and non-unique continuation
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866910822065766400 |
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| author | Agrawal, Divyansh Singhal, Nisha |
| author_facet | Agrawal, Divyansh Singhal, Nisha |
| contents | The $d$-plane transform maps functions to their integrals over $d$-planes in $\mathbb{R}^n$. We study the following question: if a function vanishes in a bounded open set, and its $d$-plane transform vanishes on all $d$-planes intersecting the same set, does the function vanish identically? For $d$ an even integer, we show by producing an explicit counterexample, that neither the $d$-plane transform, nor its normal operator has this property. On the other hand, an even stronger property holds when $d$ is odd, where the normal operator vanishing to infinite order at a point, along with the function vanishing on an open set containing that point, is sufficient to conclude that the function vanishes identically. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_07249 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | d-plane transform: unique and non-unique continuation Agrawal, Divyansh Singhal, Nisha Classical Analysis and ODEs 44A12, 45Q05, 44A35 The $d$-plane transform maps functions to their integrals over $d$-planes in $\mathbb{R}^n$. We study the following question: if a function vanishes in a bounded open set, and its $d$-plane transform vanishes on all $d$-planes intersecting the same set, does the function vanish identically? For $d$ an even integer, we show by producing an explicit counterexample, that neither the $d$-plane transform, nor its normal operator has this property. On the other hand, an even stronger property holds when $d$ is odd, where the normal operator vanishing to infinite order at a point, along with the function vanishing on an open set containing that point, is sufficient to conclude that the function vanishes identically. |
| title | d-plane transform: unique and non-unique continuation |
| topic | Classical Analysis and ODEs 44A12, 45Q05, 44A35 |
| url | https://arxiv.org/abs/2502.07249 |