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 (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).

Definition

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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

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  • From the continuity of   it follows that the function   is continuous, and that its limit as   is  
  • Spherical means can be used to solve the Cauchy problem for the wave equation   in odd space dimension. The result, known as Kirchhoff's formula, is derived by using spherical means to reduce the wave equation in   (for odd  ) to the wave equation in  , and then using d'Alembert's formula. The expression itself is presented in wave equation article.
  • 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

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  • Evans, Lawrence C. (1998). Partial differential equations. American Mathematical Society. ISBN 978-0-8218-0772-9.
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