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

F G Goncharov (1) , Roman Novikov (1)

1 CMAP - Centre de Mathématiques Appliquées - Ecole Polytechnique

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

Mots clés: non-injectivity; injectivity; weighted Radon transforms

Domaines: Analyse fonctionnelle [math.FA]; Physique mathématique [math-ph]

F G Goncharov, Roman Novikov. An example of non-uniqueness for the weighted Radon transforms along hyperplanes in multidimensions. Inverse Problems, 2018, 34 (5), pp.054001. ⟨10.1088/1361-6420/aab24d⟩. ⟨hal-01583755v2⟩