Spherical mean

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

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 ω_n−1(r) is the "surface area" of this (n−1)-sphere.

Equivalently, the spherical mean is given by

where ω_n−1 is the area of the (n−1)-sphere of radius 1.

The spherical mean is often denoted as

Properties and uses

• From the continuity of u it follows that the function

is continuous, and its limit as is u(x).

• Spherical means are used in finding the solution of the wave equation u_tt = c²Δu for t > 0 with prescribed boundary conditions at t = 0.

• If U is an open set in and u is a C² function defined on U, then u is harmonic if and only if for all x in U and all r > 0 such that the closed ball B(x,r) is contained in U one has

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

References

• Evans, Lawrence C. (1998). Partial differential equations. American Mathematical Society. ISBN 0821807722. 

• Sabelfeld, K. K.; Shalimova, I. A. (1997). Spherical means for PDEs. VSP. ISBN 9067642118. 

External links

• Spherical mean on PlanetMath