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.

Consider an open set U in the Euclidean space Rn 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), dS denotes integration with respect to spherical measure and ωn−1(r) is the "surface area" of this (n − 1)-sphere.

Equivalently, the spherical mean is given by where ωn−1 is the area of the (n − 1)-sphere of radius 1.

The spherical mean of a function (shown in red) is the average of the values (top, in blue) with on a "sphere" of given radius around a given point (bottom, in blue).