Spherical mean

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File:Spherical mean.png
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(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

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \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

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \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

Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): \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
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 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).
  • 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
Failed to parse (Missing <code>texvc</code> executable. Please see math/README to configure.): 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

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External links