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 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 is often denoted as

 

The spherical mean is also defined for Riemannian manifolds in a natural manner.

Properties and uses

 From the continuity of  it follows that the function

is continuous, and its limit as  is 

 Spherical means are used in finding the solution of the wave equation  for  with prescribed boundary conditions at 
 If  is an open set in  and  is  a C2 function defined on , then  is harmonic if and only if for all  in  and all  such that the closed ball  is contained in  one has

 This result can be used to prove the maximum principle for harmonic functions.