2024/06/22 by Divyansh Agrawal, Gaik Ambartsoumian, Agrawal, Divyansh +5 · 1 citation
Mathematics · #33C10 #44A12 #44A15 #44A20 #45Q05 #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical Physics (math-ph) #Numerical methods in inverse problems #Point processes and geometric inequalities
paper · pdf · doi:10.48550/arxiv.2406.15815
openalex publication_date 2024/06/22 · openalex created_date 2024/06/26 · openalex updated_date 2026/07/28
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 \mathbbB in ℝn for odd n, with the centers of integration spheres restricted to the unit sphere \mathbbSn-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.