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October 2019 A breakdown of injectivity for weighted ray transforms in multidimensions
Fedor Goncharov, Roman Novikov
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Ark. Mat. 57(2): 333-371 (October 2019). DOI: 10.4310/ARKIV.2019.v57.n2.a5

Abstract

We consider weighted ray-transforms $P_W$ (weighted Radon transforms along oriented 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 $\mathrm{dim} \: \mathrm{ker} \: P_W \geq n$ in the space of infinitely smooth compactly supported functions on $\mathbb{R}^d$ for arbitrary $n \in \mathbb{N} \cup \lbrace \infty \rbrace$, where $W$ are infinitely smooth for $d=2$ and infinitely smooth almost everywhere for $d \geq 3$.

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Fedor Goncharov. Roman Novikov. "A breakdown of injectivity for weighted ray transforms in multidimensions." Ark. Mat. 57 (2) 333 - 371, October 2019. https://doi.org/10.4310/ARKIV.2019.v57.n2.a5

Information

Received: 22 March 2019; Published: October 2019
First available in Project Euclid: 16 April 2020

zbMATH: 07114509
MathSciNet: MR4018757
Digital Object Identifier: 10.4310/ARKIV.2019.v57.n2.a5

Subjects:
Primary: 44A12, 53C65, 65R32

Rights: Copyright © 2019 Institut Mittag-Leffler

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Vol.57 • No. 2 • October 2019
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