Spherical mean transform from the pde point of view

TL;DR

This paper analyzes the spherical mean transform on Euclidean space using PDE methods, providing new proofs for known problems, characterizations of the kernel, and a reconstruction procedure, with extensions to hyperbolic and spherical spaces.

Contribution

It introduces a PDE-based approach via harmonic expansions to study the spherical mean transform, simplifying proofs and extending results to other geometries.

Findings

A simplified proof for local uniqueness under weaker conditions.

A characterization of the kernel on annular regions with necessary and sufficient conditions.

A reconstruction method using additional interior or exterior information.

Abstract

We study the spherical mean transform on $\rN^n$. The transform is characterized by the Euler-Poisson-Darboux equation. By looking at the spherical harmonic expansions, we obtain a system of 1+1-dimension hyperbolic equations, which provide a good machinery to attack problems of spherical mean transform. As showcases, we discuss two known problems. The first one is a local uniqueness problem investigated by M. Agranovsky and P. Kuchment, [{\em Memoirs on Differential Equations and Mathematical Physics}, 52:1--16, 2011]. We present a simple proof which works even under a weaker condition. The second problem is to characterize the kernel of spherical mean transform on annular regions, which was studied by C. Epstein and B. Kleiner [{\em Comm. Pure Appl. Math.}, 46(3):441--451, 1993]. We present a short proof that simultaneously obtains the necessity and sufficiency for the characterization. As a consequence, we derive a reconstruction procedure for the transform with additional interior (or exterior) information. We also discuss how the approach works for the hyperbolic and spherical spaces.