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

Abstract

We consider the weighted Radon transforms $R_W$ along hyperplanes in $R^d,d\ge 3$, 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).

References in corpus (1)

• An iterative inversion of weighted Radon transforms along hyperplanes

Cited by in corpus (1)

• Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems