Spherical mean

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The spherical mean of a function (shown in red) is the average of the values (top, in blue) with 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

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

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

The spherical mean is often denoted as

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

Properties and uses[edit]

  • From the continuity of it follows that the function
is continuous, and its limit as is
  • Spherical means are used in finding the solution of the wave equation for with prescribed boundary conditions at
  • If is an open set in and is a C2 function defined on , then is harmonic if and only if for all in and all such that the closed ball is contained in one has
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]