Spherical mean

Not to be confused with Spherical mean (statistics).

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 $u$ (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).

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 that its limit as $r\to 0$ is $u(x).$
• Spherical means can be used to solve the Cauchy problem for the wave equation $\partial _{t}^{2}u=c^{2}\,\Delta u$ in odd space dimension. The result, known as Kirchhoff's formula, is derived by using spherical means to reduce the wave equation in $\mathbb {R} ^{n}$ (for odd $n$) to the wave equation in $\mathbb {R}$, 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.

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 (2): 483–501. doi:10.1090/S0002-9947-1981-0626485-6.

External links

• Spherical mean at PlanetMath.