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.
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 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
- 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.
- 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.