平均函数集上的代数和拓扑结构及AGM平均的推广
Algebraic and topological structures on the set of mean functions and generalization of the AGM mean
摘要：
本文给出了$\mathbb{R}^2$的给定对称整环$\D$上的平均函数集$\M_\D$的新结构和结果。首先在$\M_\D$上构造了一个交换群的结构，其中的中性元仅为{ \it Arithmetic}平均，然后研究了该群中的一些对称性.其次，我们在$\M_\D$上构造了一个度量空间结构，在这个度量空间结构下，$\M_\D$就是以{ \it Arithmetic}平均为中心，半径为1/2的闭球。我们特别表明，{ \it几何}和{ \it谐波}的意思在于边界的$\M_\D$。最后，我们给出了推广$\AGM$均值构造的两个重要定理。粗略地说，这些定理表明，对于任意两个给定的满足某些正则条件的平均$M_1$和$M_2$，存在唯一的平均$M$满足函数方程：$M(M_1, M_2) = M$。
英文摘要：
In this paper, we present new structures and results on the set $\M_\D$ of mean functions on a given symmetric domain $\D$ of $\mathbb{R}^2$. First, we construct on $\M_\D$ a structure of abelian group in which the neutral element is simply the {\it Arithmetic} mean; then we study some symmetries in that group. Next, we construct on $\M_\D$ a structure of metric space under which $\M_\D$ is nothing else the closed ball with center the {\it Arithmetic} mean and radius 1/2. We show in particular that the {\it Geometric} and {\it Harmonic} means lie in the border of $\M_\D$. Finally, we give two important theorems generalizing the construction of the $\AGM$ mean. Roughly speaking, those theorems show that for any two given means $M_1$ and $M_2$, which satisfy some regular conditions, there exists a unique mean $M$ satisfying the functional equation: $M(M_1, M_2) = M$.
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