A breakdown of injectivity for weighted ray transforms in multidimensions

Fedor O Goncharov (1) , Roman Novikov (1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

We consider weighted ray-transforms $P_W$ (weighted Radon transforms along straight lines) in $R^d , d \geq 2$, 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 $R^d$. In addition, the constructed weight W is rotation-invariant continuous and is infinitely smooth almost everywhere on $R^d \times S^{d−1}$. 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.

Mots clés: Radon transforms; ray transforms; integral geometry; injectivity

Domaines: Analyse fonctionnelle [math.FA]; Analyse classique [math.CA]

Fedor O Goncharov, Roman Novikov. A breakdown of injectivity for weighted ray transforms in multidimensions. 2017. ⟨hal-01635188v1⟩