On the null space of the backprojection operator and Rubin's conjecture for the spherical mean transform

TL;DR

This paper characterizes the null space of the backprojection operator for the spherical mean transform in odd dimensions and provides a counterexample that disproves Rubin's conjecture, advancing understanding of the transform's properties.

Contribution

It offers a detailed null space characterization and refutes Rubin's conjecture by constructing an explicit counterexample, building on recent range characterizations.

Findings

Null space of backprojection operator fully characterized.

Rubin's conjecture is disproved with an explicit counterexample.

Range characterization for the spherical mean transform in odd dimensions is utilized.

Abstract

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 Erd\'elyi-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.