# spherical mean

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

 $M_{h}(x,r)=\frac{1}{A(n-1)}\int_{\|\xi\|=1}h(x+r\xi)\,dS=\frac{1}{A(n-1,r)}% \int_{\|\xi\|=|r|}h(x+\xi)\,dS,$

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 is independent 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 $C^{2}(\mathbb{R}^{n})$) then the following identity holds:

 $\nabla^{2}_{x}M_{h}(x,r)=\left(\frac{\partial^{2}}{\partial r^{2}}+\frac{n-1}{% r}\frac{\partial}{\partial r}\right)M_{h}(x,r),$

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

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

Title spherical mean SphericalMean 2013-03-22 14:09:04 2013-03-22 14:09:04 rspuzio (6075) rspuzio (6075) 7 rspuzio (6075) Definition msc 35L05 msc 26E60 WaveEquation