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 $$\mathbb R^n$$ 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


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

where $$\partial B(x, r)$$ is the (n&minus;1)-sphere forming the boundary of $$B(x, r)$$ and $$\omega_{n-1}(r)$$ is the "surface area" of this $$(n-1)$$-sphere.

Equivalently, the spherical mean is given by


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

where $$\omega_{n-1}$$ is the area of the $$(n-1)$$-sphere of radius 1.

The spherical mean is often denoted as


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

Properties and uses

 * From the continuity of $$u$$ it follows that the function


 * $$r\to \int\limits_{\partial B(x, r)}\!\!\!\!\!\!\!\!\!\!\!-\, u(y) \,dS(y)$$


 * is continuous, and its limit as $$r\to 0$$ is $$u(x).$$


 * Spherical means are used in finding the solution of the wave equation $$u_{tt}=c^2\Delta u$$ for $$t>0$$ with prescribed boundary conditions at $$t=0.$$


 * 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


 * $$u(x)=\int\limits_{\partial B(x, r)}\!\!\!\!\!\!\!\!\!\!\!-\, u(y) \,dS(y).$$


 * This result can be used to prove the maximum principle for harmonic functions.