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 r0 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 n1(r) is the "surface area" of this (n1)-sphere.

Equivalently, the spherical mean is given by

where n1 is the area of the (n1)-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

• is continuous, and that its limit

• Spherical means can be used to solve the Cauchy problem for the wave equation, and then using d'Alembert's formula. The expression itself is presented in wave equation article.

• is a C2 is harmonic This result can be used to prove the maximum principle for harmonic functions.

References

External links