Spherical mean

File:Spherical mean.png

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

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{1}{\omega_{n-1}(r)}\int\limits_{\partial B(x, r)} \! u(y) \, \mathrm{d} S(y)

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

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \frac{1}{\omega_{n-1}}\int\limits_{\|y\|=1} \! u(x+ry) \, \mathrm{d}S(y)

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

The spherical mean is often denoted as

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \int\limits_{\partial B(x, r)}\!\!\!\!\!\!\!\!\!\!\!-\, u(y) \, \mathrm{d} S(y).

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

Properties and uses

• From the continuity of $u$ it follows that the function

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): r\to \int\limits_{\partial B(x, r)}\!\!\!\!\!\!\!\!\!\!\!-\, u(y) \,\mathrm{d}S(y)

is continuous, and its limit as $r\to 0$ is $u(x).$

• Spherical means are used in finding the solution of the wave equation $u_{tt}=c^2\Delta u$ for $t>0$ with prescribed boundary conditions at $t=0.$
• If $U$ is an open set in $\mathbb R^n$ 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

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): u(x)=\int\limits_{\partial B(x, r)}\!\!\!\!\!\!\!\!\!\!\!-\, u(y) \,\mathrm{d}S(y).

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

External links

• Spherical mean at PlanetMath.org.