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arXiv · 2502.07249

d-plane transform: unique and non-unique continuation

Divyansh Agrawal·Nisha Singhal
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Abstract

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.

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Divyansh Agrawal, Nisha Singhal. 2025-02-11. d-plane transform: unique and non-unique continuation. https://arxiv.org/abs/2502.07249

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