Spherical mean explained

Spherical mean should not 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.

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

(1)/(\omega _{n-1} (r)) \int\limits _{\partial} u(y) 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

(1)/(\omega _{n-1}) \int\limits _{\|y\|=1} u(x + ry) 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} - u(y) 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 $$r\backslash to\; \backslash int\backslash limits\_\backslash !\backslash !\backslash !\backslash !\backslash !\backslash !\backslash !\backslash !\backslash !-\backslash ,\; u(y)\; \backslash ,\backslash mathrmS(y)$$ is continuous, and that its limit as r\to 0 is u(x). \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 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 $$u(x)=\backslash int\backslash limits\_\backslash !\backslash !\backslash !\backslash !\backslash !\backslash !\backslash !\backslash !\backslash !-\backslash ,\; u(y)\; \backslash ,\; \backslash mathrmS(y).$$ This result can be used to prove the maximum principle for harmonic functions.

References

• Book: Evans , Lawrence C. . Partial differential equations . American Mathematical Society . 1998 . 978-0-8218-0772-9 .
• Book: Sabelfeld , K. K. . Shalimova, I. A. . Spherical means for PDEs . VSP . 1997 . 978-90-6764-211-8.
• Sunada . Toshikazu . Toshikazu Sunada . Trans. Am. Math. Soc. . 483–501 . Spherical means and geodesic chains in a Riemannian manifold . 267 . 1981. 2 . 10.1090/S0002-9947-1981-0626485-6 . free.