Algebraic and topological structures on the set of mean functions and generalization of the AGM mean

Bakir Farhi. "Algebraic and topological structures on the set of mean functions and generalization of the AGM mean." Colloquium Mathematicae 132.1 (2013): 139-149. <http://eudml.org/doc/283598>.

@article{BakirFarhi2013,
abstract = {We present new structures and results on the set $ _ $ of mean functions on a given symmetric domain in ℝ². First, we construct on $ _ $ a structure of abelian group in which the neutral element is the arithmetic mean; then we study some symmetries in that group. Next, we construct on $ _ $ a structure of metric space under which $ _ $ is the closed ball with center the arithmetic mean and radius 1/2. We show in particular that the geometric and harmonic means lie on the boundary of $ _ $. Finally, we give two theorems generalizing the construction of the AGM mean. Roughly speaking, those theorems show that for any two given means M₁ and M₂, which satisfy some regularity conditions, there exists a unique mean M satisfying the functional equation M(M₁,M₂) = M.},
author = {Bakir Farhi},
journal = {Colloquium Mathematicae},
keywords = {means; abelian groups; metric spaces; symmetries},
language = {eng},
number = {1},
pages = {139-149},
title = {Algebraic and topological structures on the set of mean functions and generalization of the AGM mean},
url = {http://eudml.org/doc/283598},
volume = {132},
year = {2013},
}

TY - JOUR
AU - Bakir Farhi
TI - Algebraic and topological structures on the set of mean functions and generalization of the AGM mean
JO - Colloquium Mathematicae
PY - 2013
VL - 132
IS - 1
SP - 139
EP - 149
AB - We present new structures and results on the set $ _ $ of mean functions on a given symmetric domain in ℝ². First, we construct on $ _ $ a structure of abelian group in which the neutral element is the arithmetic mean; then we study some symmetries in that group. Next, we construct on $ _ $ a structure of metric space under which $ _ $ is the closed ball with center the arithmetic mean and radius 1/2. We show in particular that the geometric and harmonic means lie on the boundary of $ _ $. Finally, we give two theorems generalizing the construction of the AGM mean. Roughly speaking, those theorems show that for any two given means M₁ and M₂, which satisfy some regularity conditions, there exists a unique mean M satisfying the functional equation M(M₁,M₂) = M.
LA - eng
KW - means; abelian groups; metric spaces; symmetries
UR - http://eudml.org/doc/283598
ER -