spherical mean

Let $h$ be a function (usually real or complex valued) on ${ℝ}^n$ ($n\ge 1$).Its spherical mean at point $x$ over a sphere of radius $r$ is defined as

$$M_h(x,r)=\frac{1}{A(n-1)}{\int }_{\parallel \xi \parallel =1}h(x+r\xi )𝑑S=\frac{1}{A(n-1,r)}{\int }_{\parallel \xi \parallel =|r|}h(x+\xi )𝑑S,$$

where the integral is over the surface of the unit $n-1$-sphere. Here $A(n-1)$ is is the area of the unit sphere, while $A(n-1,r)=r^{n-1}A(n-1)$ is the area of a sphere of radius $r$ (http://planetmath.org/AreaOfTheNSphere). In essense, the spherical mean $M_h(x,r)$ is just the average of $h$ over the surface of a sphere of radius $r$ centered at $x$, as the name suggests.

The spherical mean is defined for both positive and negative $r$ and isindependent of its sign. As $r\to 0$, if $h$ is continuous, $M_h(x,r)\to h(x)$. If $h$ has two continuous derivatives (is in $C^2({ℝ}^n)$) then thefollowing identity holds:

$${\nabla }_x^2{M}_h(x,r)=\left(\frac{{\partial }^2}{\partial r^2}+\frac{n-1}{r}\frac{\partial }{\partial r}\right)M_h(x,r),$$

where ${\nabla }^2$ is the Laplacian.

Spherical means are used to obtain an explicit general solution for the waveequation in $n$ space and one time dimensions.