An example of non-uniqueness for the weighted Radon transforms along hyperplanes in multidimensions

TL;DR

This paper demonstrates that weighted Radon transforms along hyperplanes in multidimensional space can have non-unique solutions, extending known non-uniqueness examples from two dimensions to higher dimensions.

Contribution

It constructs an explicit example of a weighted Radon transform in higher dimensions with a non-trivial kernel, based on previous two-dimensional non-uniqueness results.

Findings

Existence of non-trivial kernel for weighted Radon transforms in R^d, d ≥ 3

The constructed weight is smooth, bounded, and almost everywhere smooth

Extends non-uniqueness results from 2D to higher dimensions

Abstract

We consider the weighted Radon transforms $R_W$ along hyperplanes in $R^d, \, d \geq 3$, with strictly positive weights $W = W (x, \theta), \, x \in R^d, \, \theta \in S^{d-1}$. We construct an example of such a transform with non-trivial kernel in the space of infinitely smooth compactly supported functions. In addition, the related weight $W$ is infinitely smooth almost everywhere and is bounded. Our construction is based on the famous example of non-uniqueness of J. Boman (1993) for the weighted Radon transforms in $R^2$ and on a recent result of F. Goncharov and R. Novikov (2016).