# spherical mean

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

$${M}_{h}(x,r)=\frac{1}{A(n-1)}{\int}_{\parallel \xi \parallel =1}h(x+r\xi )\mathit{d}S=\frac{1}{A(n-1,r)}{\int}_{\parallel \xi \parallel =|r|}h(x+\xi )\mathit{d}S,$$ |

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:

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

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

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

Title | spherical mean |
---|---|

Canonical name | SphericalMean |

Date of creation | 2013-03-22 14:09:04 |

Last modified on | 2013-03-22 14:09:04 |

Owner | rspuzio (6075) |

Last modified by | rspuzio (6075) |

Numerical id | 7 |

Author | rspuzio (6075) |

Entry type | Definition |

Classification | msc 35L05 |

Classification | msc 26E60 |

Related topic | WaveEquation |