Spherical mean

The spherical mean of a function

u

(shown in red) is the average of the values

u(y)

(top, in blue) with

y

on a "sphere" of given radius around a given point (bottom, in blue).

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 > 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 ∂B(x, r) is the (n−1)-sphere forming the boundary of B(x, r), dS 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 {1}{\omega _{{n-1}}}}\int \limits _{{\|y\|=1}}\!u(x+ry)\,{\mathrm {d}}S(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 _{\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 its limit as

r\to 0

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)\,\mathrm {d} S(y).

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

References

Evans, Lawrence C. (1998). Partial differential equations. American Mathematical Society. ISBN 0-8218-0772-2.

Sabelfeld, K. K.; Shalimova, I. A. (1997). Spherical means for PDEs. VSP. ISBN 90-6764-211-8.
Sunada, Toshikazu (1981). "Spherical means and geodesic chains in a Riemannian manifold". Trans. A.M.S. 267: 483–501.

External links

"Spherical mean". PlanetMath.

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