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.

The spherical mean of a function ${\displaystyle u}$ (shown in red) is the average of the values ${\displaystyle u(y)}$ (top, in blue) with ${\displaystyle y}$ on a "sphere" of given radius around a given point (bottom, in blue).

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

${\displaystyle {\frac {1}{\omega _{n-1}(r)}}\int \limits _{\partial B(x,r)}\!u(y)\,\mathrm {d} S(y)}$

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

${\displaystyle {\frac {1}{\omega _{n-1}}}\int \limits _{\|y\|=1}\!u(x+ry)\,\mathrm {d} S(y)}$

where ω_n−1 is the area of the (n − 1)-sphere of radius 1.

The spherical mean is often denoted as

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

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

Properties and uses

• From the continuity of ${\displaystyle u}$ it follows that the function

${\displaystyle r\to \int \limits _{\partial B(x,r)}\!\!\!\!\!\!\!\!\!-\,u(y)\,\mathrm {d} S(y)}$

is continuous, and that its limit as ${\displaystyle r\to 0}$ is ${\displaystyle u(x).}$

• Spherical means can be used to solve the Cauchy problem for the wave equation ${\displaystyle \partial _{t}^{2}u=c^{2}\,\Delta u}$ in odd space dimension. The result, known as Kirchoff's formula, is derived by using spherical means to reduce the wave equation in ${\displaystyle \mathbb {R} ^{n}}$ (for odd ${\displaystyle n}$) to the wave equation in ${\displaystyle \mathbb {R} }$, and then using d'Alembert's formula. The expression itself is presented in wave equation article.

• If ${\displaystyle U}$ is an open set in ${\displaystyle \mathbb {R} ^{n}}$ and ${\displaystyle u}$ is a C² function defined on ${\displaystyle U}$, then ${\displaystyle u}$ is harmonic if and only if for all ${\displaystyle x}$ in ${\displaystyle U}$ and all ${\displaystyle r>0}$ such that the closed ball ${\displaystyle B(x,r)}$ is contained in ${\displaystyle U}$ one has

${\displaystyle u(x)=\int \limits _{\partial B(x,r)}\!\!\!\!\!\!\!\!\!-\,u(y)\,\mathrm {d} S(y).}$

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 978-0-8218-0772-9.

• Sabelfeld, K. K.; Shalimova, I. A. (1997). Spherical means for PDEs. VSP. ISBN 978-90-6764-211-8.
• Sunada, Toshikazu (1981). "Spherical means and geodesic chains in a Riemannian manifold". Trans. Am. Math. Soc. 267: 483–501.

External links

• Spherical mean at PlanetMath.