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 Rn and a continuous function u defined on U with real or complex values. Let x be a point in U and r &gt; 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


 * $$\frac{1}{\omega_{n-1}(r)}\int\limits_{\partial B(x, r)} \! u(y) \, \mathrm{d} S(y) $$

where &part;B(x, r) is the (n &minus; 1)-sphere forming the boundary of B(x, r), dS denotes integration with respect to spherical measure and &omega;n&minus;1(r) is the "surface area" of this (n &minus; 1)-sphere.

Equivalently, the spherical mean is given by


 * $$\frac{1}{\omega_{n-1}}\int\limits_{\|y\|=1} \! u(x+ry) \, \mathrm{d}S(y) $$

where &omega;n&minus;1 is the area of the (n &minus; 1)-sphere of radius 1.

The spherical mean is often denoted as


 * $$\int\limits_{\partial B(x, r)}\!\!\!\!\!\!\!\!\!-\, u(y) \, \mathrm{d} S(y). $$

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 $$r\to \int\limits_{\partial B(x, r)}\!\!\!\!\!\!\!\!\!-\, u(y) \,\mathrm{d}S(y)$$ 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 C2 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 $$u(x)=\int\limits_{\partial B(x, r)}\!\!\!\!\!\!\!\!\!-\, u(y) \, \mathrm{d}S(y).$$ This result can be used to prove the maximum principle for harmonic functions.