d-plane transform: unique and non-unique continuation

TL;DR

This work analyzes unique continuation for the $d$-plane transform $\mathcal{R}_d$ and its normal operator $\mathcal{N}_d$ in $\mathbb{R}^n$. It demonstrates a parity-driven dichotomy: for even $d$, there exist nontrivial compactly supported functions $f$ with $f|_U=0$ yet $\mathcal{R}_d f$ (and $\mathcal{N}_d f$) vanishing on all $d$-planes intersecting $U$, showing lack of UCP; for odd $d$, a stronger form holds: if $f|_U=0$ and $\mathcal{N}_d f$ vanishes to infinite order at a point in $U$, then $f\equiv 0$, yielding strong UCP for $\mathcal{N}_d$ and hence UCP for $\mathcal{R}_d$. The even-$d$ construction leverages radial counterexamples related to 1D Radon-type results, while the odd-$d$ result relies on a density argument for derivatives of the $|x|^{-(n-d)}$ kernel and a Kelvin transform. These findings clarify the role of the Laplacian’s powers in inversion and local uniqueness, with implications for inverse problems and imaging with partial data.

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.

Theorems & Definitions (17)

• Theorem 2.1
• Corollary 2.2
• Theorem 2.3
• Remark 2.4
• Theorem 2.5
• Remark 2.6
• Theorem 2.7
• proof : Proof of Theorem \ref{['lucp-rad']}
• proof : Proof of Theorem \ref{['lucp-rd']}
• proof : Proof of Corollary \ref{['lucp-nd']}