Spherical mean

From Wikipedia, the free encyclopedia

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.

1 Definition
2 Properties and uses
3 References
4 External links

[edit] Definition

Consider an open set $U$ in the Euclidean space $\mathbb R^n$ 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

$\frac{1}{\omega_{n-1}(r)}\int\limits_{\partial B(x, r)} \! u(y) \,dS(y)$

where $\partial B(x, r)$ is the (n−1)-sphere forming the boundary of $B (x, r)$ and $ω n - 1 (r)$ is the "surface area" of this $(n - 1)$ -sphere.

Equivalently, the spherical mean is given by

$\frac{1}{\omega_{n-1}}\int\limits_{\|y\|=1} \! u(x+ry) \,dS(y)$

where $ω n - 1$ is the area of the $(n - 1)$ -sphere of radius 1.

The spherical mean is often denoted as

$\int\limits_{\partial B(x, r)}\!\!\!\!\!\!\!\!\!\!\!-\, u(y) \,dS(y).$

[edit] Properties and uses

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

$r\to \int\limits_{\partial B(x, r)}\!\!\!\!\!\!\!\!\!\!\!-\, u(y) \,dS(y)$

is continuous, and its limit as $r\to 0$ is

u (x).

Spherical means are used in finding the solution of the wave equation $u t t = c 2 Δ u$ 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