Spherical Mean

In mathematics, the spherical mean of a function around a point is the average of all values of that function on a sphere of given radius centered at that point.

Definition

Consider an open set ''U'' in the Euclidean space R^''n'' and a continuous function ''u'' defined on ''U'' with real or complex values. Let ''x'' be a point in ''U'' and ''r'' > 0 be such that the closed ball ''B''(''x'', ''r'') of center ''x'' and radius ''r'' is contained in ''U''. The spherical mean over the sphere of radius ''r'' centered at ''x'' is defined as

: $\frac\int\limits_ \! u(y) \, \mathrm S(y)$

where ∂''B''(''x'', ''r'') is the (''n'' − 1)-sphere forming the boundary of ''B''(''x'', ''r''), d''S'' denotes integration with respect to spherical measure and ''ω''_''n''−1(''r'') is the "surface area" of this (''n'' − 1)-sphere.

Equivalently, the spherical mean is given by

: $\frac\int\limits_ \! u(x+ry) \, \mathrmS(y)$

where ''ω''_''n''−1 is the area of the (''n'' − 1)-sphere of radius 1.

The spherical mean is often denoted as

: $\int\limits_\!\!\!\!\!\!\!\!\!-\, u(y) \, \mathrm S(y).$

The spherical mean is also defined for Riemannian manifolds in a natural manner.

Properties and uses

* From the continuity of $u$ it follows that the function $r\to \int\limits_\!\!\!\!\!\!\!\!\!-\, u(y) \,\mathrmS(y)$ is continuous, and that its limit as $r\to 0$ is $u(x).$
* Spherical means can be used to solve the Cauchy problem for the wave equation $\partial^2_t u=c^2\,\Delta u$ in odd space dimension. The result, known as Kirchhoff's formula, is derived by using spherical means to reduce the wave equation in $\R^n$ (for odd $n$) to the wave equation in $\R$, and then using d'Alembert's formula. The expression itself is presented in wave equation article.
* If $U$ is an open set in $\mathbb R^n$ and $u$ is a ''C''² function defined on $U$, then $u$ is harmonic if and only if for all $x$ in $U$ and all $r>0$ such that the closed ball $B(x, r)$ is contained in $U$ one has $u(x)=\int\limits_\!\!\!\!\!\!\!\!\!-\, u(y) \, \mathrmS(y).$ This result can be used to prove the maximum principle for harmonic functions.

References

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External links

* {{PlanetMath , urlname=sphericalmean , title=Spherical mean

Partial differential equations
Means