Spherical mean

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

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

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 ωn1(r) is the "surface area" of this (n  1)-sphere.

Equivalently, the spherical mean is given by

where ωn1 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

  • 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 Kirchoff'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

  • Evans, Lawrence C. (1998). Partial differential equations. American Mathematical Society. ISBN 978-0-8218-0772-9.
  • Sabelfeld, K. K.; Shalimova, I. A. (1997). Spherical means for PDEs. VSP. ISBN 978-90-6764-211-8.
  • Sunada, Toshikazu (1981). "Spherical means and geodesic chains in a Riemannian manifold". Trans. Am. Math. Soc. 267: 483–501.
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