Determining a Function from Its Mean Values Over a Family of Spheres

Source abstract

Suppose D is a bounded, connected, open set in R n and f is a smooth function on R n with support in $\overD$. We study the recovery of f from the mean values of f over spheres centered on a part or the whole boundary of D. For strictly convex $\overline{D}$, we prove uniqueness when the centers are restricted to an open subset of the boundary. We provide an inversion algorithm (with proof) when the mean values are known for all spheres centered on the boundary of D, with radii in the interval [0, diam(D)/2]. We also give an inversion formula when D is a ball in R n , $n \geq 3$ and odd, and the mean values are known for all spheres centered on the boundary.