Spherical mean

From Wikipedia, the free encyclopedia
Jump to: navigation, search
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[edit]

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(xr) 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(xr) is the (n−1)-sphere forming the boundary of B(xr), dS denotes integration with respect to spherical measure and ωn−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 ωn−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[edit]

  • 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 is u(x).
  • 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[edit]

  • 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[edit]