Spherical mean transform

Summary

The spherical mean transform sends a function $f$ on $\R^n$ to a function $M f(x, r)$. Its value is the average of $f$ over the sphere of center $x$ and radius $r$. A continuous function is harmonic exactly when all its spherical means equal its value at the center. For a function of class $C^2$ the means satisfy the Darboux equation. This reduces the wave equation in $\R^n$ to an equation in one space variable. Recovering $f$ from its means over spheres with centers on a fixed surface is the mathematical core of thermoacoustic and photoacoustic tomography.