Spherical mean
In mathematics, the spherical mean of a function around a point is the average of all values of that function on a sphere of given radius centered at that point. Definition. Consider an open set "U" in the Euclidean space R and a continuous function "u" defined on "U" with real or complex values. Let "x" be a point in "U" and "r" > 0 be such that the closed ball "B"("x", "r") of center "x" and radius "r" is contained in "U". The spherical mean over the sphere of radius "r" centered at "x" is defined as  where ∂"B"("x", "r") is the ("n" − 1)-sphere forming the boundary of "B"("x", "r"), d"S" denotes integration with respect to spherical measure and "ω"("r") is the "surface area" of this ("n" − 1)-sphere. Equivalently, the spherical mean is given by where "ω" is the area of the ("n" − 1)-sphere of radius 1. The spherical mean is often denoted as The spherical mean is also defined for Riemannian manifolds in a natural manner.