d-plane transform: unique and non-unique continuation

TL;DR

This paper investigates the unique continuation properties of the $d$-plane transform, revealing that such properties depend on whether $d$ is odd or even, with counterexamples for even $d$ and positive results for odd $d$.

Contribution

It provides explicit counterexamples for even $d$ and establishes strong unique continuation results for odd $d$, clarifying the transform's behavior in these cases.

Findings

Counterexamples for even $d$ show non-uniqueness.

Unique continuation holds for odd $d$ under certain conditions.

Normal operator's properties differ based on the parity of $d$.

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.