Spherical means with centers on a hyperplane in even dimensions

TL;DR

This paper develops a new inversion formula for recovering functions from spherical means centered on a hyperplane in even dimensions, avoiding Fourier and Hilbert transforms, and extends related isometry identities.

Contribution

It introduces an inversion formula for even dimensions that simplifies previous methods by not requiring Fourier or Hilbert transforms, extending existing identities to even dimensions.

Findings

Derived an inversion formula for even n without Fourier or Hilbert transforms.

Extended the isometry identity for solutions of the wave equation to even n.

Provided a unified approach for spherical mean problems in all dimensions.

Abstract

Given a real valued function on R^n we study the problem of recovering the function from its spherical means over spheres centered on a hyperplane. An old paper of Bukhgeim and Kardakov derived an inversion formula for the odd n case with great simplicity and economy. We apply their method to derive an inversion formula for the even n case. A feature of our inversion formula, for the even n case, is that it does not require the Fourier transform of the mean values or the use of the Hilbert transform, unlike the previously known inversion formulas for the even n case. Along the way, we extend the isometry identity of Bukhgeim and Kardakov for odd n, for solutions of the wave equation, to the even n case.