A breakdown of injectivity for weighted ray transforms in multidimensions

TL;DR

This paper constructs examples of weighted ray transforms in multiple dimensions that have non-trivial kernels, challenging existing beliefs about their injectivity under certain regularity conditions.

Contribution

It provides explicit counterexamples of non-injective weighted ray transforms with smooth, rotation-invariant weights, relaxing regularity assumptions.

Findings

Counterexamples with non-trivial kernels for weighted ray transforms.

Existence of weights with arbitrarily large kernel dimension.

Smoothness properties of constructed weights vary with dimension.

Abstract

We consider weighted ray-transforms $P\_W$ (weighted Radon transforms along 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 $\dim \ker P\_W \geq n$ in the space of infinitely smooth compactly supported functions on $\mathbb{R}^d$ for arbitrary $n\in \mathbb{N}\cup \{\infty\}$, where $W$ are infinitely smooth for $d=2$ and infinitely smooth almost everywhere for $d\geq 3$.