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Article Dans Une Revue Arkiv för Matematik Année : 2019

A breakdown of injectivity for weighted ray transforms in multidimensions

Résumé

We consider weighted ray-transforms $P_W$ (weighted Radon transforms along straight lines) in $\mathbb{R}^d, \, d\geq 2,$ 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 $\mathbb{R}^d$. In addition, the constructed weight $W$ is rotation-invariant continuous and is infinitely smooth almost everywhere on $\mathbb{R}^d \times \mathbb{S}^{d-1}$. 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. We also give examples of continous strictly positive $W$ such that $\dim \ker P_W \geq n$ in the space of infinitely smooth compactly supported functions on $\mathbb{R}^d$ for arbitrary $n\in \mathbb{N}\cup \{\infty\}$, where $W$ are infinitely smooth for $d=2$ and infinitely smooth almost everywhere for $d\geq 3$.
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Dates et versions

hal-01635188 , version 1 (14-11-2017)
hal-01635188 , version 2 (13-03-2018)
hal-01635188 , version 3 (25-03-2018)

Identifiants

Citer

Fedor O Goncharov, Roman G Novikov. A breakdown of injectivity for weighted ray transforms in multidimensions. Arkiv för Matematik, 2019, 57, pp.333-371. ⟨10.4310/ARKIV.2019.v57.n2.a5⟩. ⟨hal-01635188v3⟩
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