An example of non-uniqueness for Radon transforms with continuous positive rotation invariant weights

TL;DR

This paper constructs a continuous, rotation-invariant weight for the Radon transform in three or more dimensions that has a non-trivial kernel, challenging the uniqueness typically expected in such transforms.

Contribution

It provides the first explicit example of a non-unique weighted Radon transform with a continuous, rotation-invariant weight in higher dimensions.

Findings

Existence of non-trivial kernel for weighted Radon transforms with positive weights.

Construction of a continuous, rotation-invariant weight with non-uniqueness.

Extension of the example to higher dimensions (d ≥ 3).

Abstract

We consider weighted Radon transforms $R_W$ along hyperplanes in $R^3$ with strictly positive weights $W$. We construct an example of such a transform with non-trivial kernel $\mathrm{Ker}R_W$ in the space of infinitely smooth compactly supported functions and with continuous weight. Moreover, in this example the weight $W$ is rotation invariant. In particular, by this result we continue studies of Quinto (1983), Markoe, Quinto (1985), Boman (1993) and Goncharov, Novikov (2017). We also extend our example to the case of weighted Radon transforms along two-dimensional planes in $R^d$ , $d \geq 3$.