spherical mean

Let h be a function (usually real or complex valued) on n (n1). Its spherical mean at point x over a sphere of radius r is defined as


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)=rn-1A(n-1) is the area of a sphere of radius r (http://planetmath.org/AreaOfTheNSphere). In essense, the spherical mean Mh(x,r) is just the averageMathworldPlanetmath 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 r0, if h is continuousMathworldPlanetmath, Mh(x,r)h(x). If h has two continuous derivatives (is in C2(n)) then the following identity holds:


where 2 is the Laplacian.

Spherical means are used to obtain an explicit general solution for the wave equationMathworldPlanetmath 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