A breakdown of injectivity for weighted ray transforms in multidimensions

Abstract

We consider weighted ray-transforms $P_W$ (weighted Radon transforms along oriented straight lines) in $\mathbb{R}^d, d \geq 2$, with strictly positive weights $W$. We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions on $\mathbb{R}^d$. In addition, the constructed weight $W$ is rotation-invariant continuous and is infinitely smooth almost everywhere on $\mathbb{R}^d \times \mathbb{S}^{d-1}$. In particular, by this construction we give counterexamples to some well-known injectivity results for weighted ray transforms for the case when the regularity of $W$ is slightly relaxed. We also give examples of continous strictly positive $W$ such that $\mathrm{dim} \: \mathrm{ker} \: P_W \geq n$ in the space of infinitely smooth compactly supported functions on $\mathbb{R}^d$ for arbitrary $n \in \mathbb{N} \cup \lbrace \infty \rbrace$, where $W$ are infinitely smooth for $d=2$ and infinitely smooth almost everywhere for $d \geq 3$.