Applications of a Result on Spherical Integration to the Theory of Convex Sets
Applications of a Result on Spherical Integration to the Theory of Convex Sets
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DOI：
10.1080/00029890.1983.11971314
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发表时间：
1983-12
影响因子：
0.5
通讯作者：
K. Falconer
中科院分区：
文献类型：
--
作者：
K. Falconer
本文讨论了凸体几何的五个结果，证明了它们几乎是上函数积分定理的直接推论！J是单位球面在n维欧氏空间中的曲面。这个基本定理是关于n维球调和函数的经典Funk-Hecke定理的推论。Seeley的文章[4]提供了球谐函数的可读性描述，Erdelyi[2]给出了这类函数的更详细的理论。从本质上讲，球谐函数Sm(O)可以看作是cos mt和sin mt在傅里叶级数中的高维类比。ICF(O)是在！上的有理行为函数！J，则我们可以写出f(O)=I：(fcmSm(O)，其中Sm(O)是m次的球谐函数。
This note relates five results on the geometry of convex bodies by demonstrating that they are almost immediate consequences of a theorem on the integration of functions over! J, the surface of the unit sphere in n-dimensional Euclidean space. This basic theorem is a corollary to the classical Funk-Hecke theorem concerning n-dimensional spherical harmonic functions. The article by Seeley [4] provides a readable account of spherical harmonics, and more detailed theory of such functions is given in Erdelyi [2]. Essentially the spherical harmonic functions Sm (O) may be thought of as the higher dimensional analogues of cos mt and sin mt in Fourier series. ICf (O) is a reasonably behaved function on! J, then we may writef (O)= I:(fcmSm (O) where Sm (O) is some spherical harmonic of degree m.