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| [[File:Spherical mean.png|right|thumb|The spherical mean of a function <math>u</math> (shown in red) is the average of the values <math>u(y)</math> (top, in blue) with <math>y</math> on a "sphere" of given radius around a given point (bottom, in blue).]]
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| In [[mathematics]], the '''spherical mean''' of a [[function (mathematics)|function]] around a point is the average of all values of that function on a sphere of given radius centered at that point.
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| ==Definition==
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| Consider an [[open set]] ''U'' in the [[Euclidean space]] '''R'''<sup>''n''</sup> and a [[continuous function]] ''u'' defined on ''U'' with [[real number|real]] or [[complex number|complex]] values. Let ''x'' be a point in ''U'' and ''r'' > 0 be such that the [[closed set|closed]] [[ball (mathematics)|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
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| : <math>\frac{1}{\omega_{n-1}(r)}\int\limits_{\partial B(x, r)} \! u(y) \, \mathrm{d} S(y) </math>
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| where ∂''B''(''x'', ''r'') is the [[n-sphere|(''n''−1)-sphere]] forming the [[boundary (topology)|boundary]] of ''B''(''x'', ''r''), d''S'' denotes integration with respect to [[spherical measure]] and ''ω''<sub>''n''−1</sub>(''r'') is the "surface area" of this (''n''−1)-sphere. | |
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| Equivalently, the spherical mean is given by
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| : <math>\frac{1}{\omega_{n-1}}\int\limits_{\|y\|=1} \! u(x+ry) \, \mathrm{d}S(y) </math>
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| where ''ω''<sub>''n''−1</sub> is the area of the (''n''−1)-sphere of radius 1.
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| The spherical mean is often denoted as
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| : <math>\int\limits_{\partial B(x, r)}\!\!\!\!\!\!\!\!\!\!\!-\, u(y) \, \mathrm{d} S(y). </math>
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| The spherical mean is also defined for Riemannian manifolds in a natural manner.
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| ==Properties and uses==
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| * From the continuity of <math>u</math> it follows that the function
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| ::<math>r\to \int\limits_{\partial B(x, r)}\!\!\!\!\!\!\!\!\!\!\!-\, u(y) \,\mathrm{d}S(y)</math> | |
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| :is continuous, and its [[Limit of a function|limit]] as <math>r\to 0</math> is <math>u(x).</math>
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| * Spherical means are used in finding the solution of the [[wave equation]] <math>u_{tt}=c^2\Delta u</math> for <math>t>0</math> with prescribed [[boundary conditions]] at <math>t=0.</math>
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| * If <math>U</math> is an open set in <math>\mathbb R^n</math> and <math>u</math> is a [[smooth function|''C''<sup>2</sup>]] function defined on <math>U</math>, then <math>u</math> is [[harmonic function|harmonic]] if and only if for all <math>x</math> in <math>U</math> and all <math>r>0</math> such that the closed ball <math>B(x, r)</math> is contained in <math>U</math> one has
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| ::<math>u(x)=\int\limits_{\partial B(x, r)}\!\!\!\!\!\!\!\!\!\!\!-\, u(y) \,\mathrm{d}S(y).</math>
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| : This result can be used to prove the [[maximum principle]] for harmonic functions.
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| ==References==
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| *{{cite book
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| | last = Evans
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| | first = Lawrence C.
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| | title = Partial differential equations
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| | publisher = American Mathematical Society
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| | year = 1998
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| | pages =
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| | isbn = 0-8218-0772-2
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| }}
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| *{{cite book
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| | last = Sabelfeld
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| | first = K. K.
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| | coauthors = Shalimova, I. A.
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| | title = Spherical means for PDEs
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| | publisher = VSP
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| | year = 1997
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| | pages =
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| | isbn = 90-6764-211-8
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| }}
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| *{{cite journal
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| | last = Sunada | first = Toshikazu | journal = Trans. A.M.S.
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| | pages = 483–501
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| | title = Spherical means and geodesic chains in a Riemannian manifold
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| | volume = 267
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| | year = 1981}}
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| ==External links==
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| * {{planetmath reference|id=5568|title=Spherical mean}}
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| [[Category:Partial differential equations]]
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| [[Category:Means]]
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