On the null space of the backprojection operator and Rubin's conjecture for the spherical mean transform
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| Format: | Preprint |
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2024
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| _version_ | 1866909229512654848 |
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| author | Agrawal, Divyansh Ambartsoumian, Gaik Krishnan, Venkateswaran P. Singhal, Nisha |
| author_facet | Agrawal, Divyansh Ambartsoumian, Gaik Krishnan, Venkateswaran P. Singhal, Nisha |
| contents | The spherical mean transform associates to a function $f$ its integral averages over all spheres. We consider the spherical mean transform for functions supported in the unit ball $\mathbb{B}$ in $\mathbb{R}^n$ for odd $n$, with the centers of integration spheres restricted to the unit sphere $\mathbb{S}^{n-1}$. In this setup, Rubin employed properties of Erdélyi-Kober fractional integrals and analytic continuation to re-derive the explicit inversion formulas proved earlier by Finch, Patch, and Rakesh using wave equation techniques. As part of his work, Rubin stated a conjecture relating spherical mean transform, its associated backprojection operator and the Riesz potential. Furthermore, he pointed to the necessity of a detailed analysis of injectivity of the backprojection operator as a crucial step toward the resolution of his conjecture. This article addresses both questions posed by Rubin by providing a characterization of the null space of the backprojection operator, and disproving the conjecture through the construction of an explicit counterexample. Crucial to the proofs is the range characterization for the spherical mean transform in odd dimensions derived recently by the authors. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2406_15815 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On the null space of the backprojection operator and Rubin's conjecture for the spherical mean transform Agrawal, Divyansh Ambartsoumian, Gaik Krishnan, Venkateswaran P. Singhal, Nisha Classical Analysis and ODEs Mathematical Physics Functional Analysis 44A12, 44A15, 44A20, 45Q05, 33C10 The spherical mean transform associates to a function $f$ its integral averages over all spheres. We consider the spherical mean transform for functions supported in the unit ball $\mathbb{B}$ in $\mathbb{R}^n$ for odd $n$, with the centers of integration spheres restricted to the unit sphere $\mathbb{S}^{n-1}$. In this setup, Rubin employed properties of Erdélyi-Kober fractional integrals and analytic continuation to re-derive the explicit inversion formulas proved earlier by Finch, Patch, and Rakesh using wave equation techniques. As part of his work, Rubin stated a conjecture relating spherical mean transform, its associated backprojection operator and the Riesz potential. Furthermore, he pointed to the necessity of a detailed analysis of injectivity of the backprojection operator as a crucial step toward the resolution of his conjecture. This article addresses both questions posed by Rubin by providing a characterization of the null space of the backprojection operator, and disproving the conjecture through the construction of an explicit counterexample. Crucial to the proofs is the range characterization for the spherical mean transform in odd dimensions derived recently by the authors. |
| title | On the null space of the backprojection operator and Rubin's conjecture for the spherical mean transform |
| topic | Classical Analysis and ODEs Mathematical Physics Functional Analysis 44A12, 44A15, 44A20, 45Q05, 33C10 |
| url | https://arxiv.org/abs/2406.15815 |