A simple range characterization for spherical mean transform in odd dimensions and its applications

TL;DR

This paper introduces a new, simple range description for the spherical mean transform in odd dimensions, revealing symmetry relations and a key identity for Bessel functions, with applications to uniqueness questions.

Contribution

It provides a novel range characterization for the spherical mean transform in odd dimensions using symmetry relations and a new Bessel function identity.

Findings

Derived a simple range description involving symmetry relations.

Proved a cross product identity for spherical Bessel functions.

Constructed a counterexample showing failure of unique continuation.

Abstract

This article provides a novel and simple range description for the spherical mean transform of functions supported in the unit ball of an odd dimensional Euclidean space. The new description comprises a set of symmetry relations between the values of certain differential operators acting on the coefficients of the spherical harmonics expansion of the function in the range of the transform. As a central part of the proof of our main result, we derive a remarkable cross product identity for the spherical Bessel functions of the first and second kind, which may be of independent interest in the theory of special functions. Finally, as one application of the range characterization, we construct an explicit counterexample proving that unique continuation type results cannot hold for the spherical mean transform in odd dimensional spaces.