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Homespherical mean

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# 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 $r$. 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.

## Mathematics Subject Classification

35L05*no label found*26E60

*no label found*

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## Comments

## classification

I'm not sure what classification this entry shold be given.

Suggestions are welcome.

## Re: classification

Primary 26E60 and secondary 35L05 perhaps?