Permutations preserving Cesàro mean, densities of natural numbers and uniform distribution of sequences

Blümlinger, M., and Obata, N.. "Permutations preserving Cesàro mean, densities of natural numbers and uniform distribution of sequences." Annales de l'institut Fourier 41.3 (1991): 665-678. <http://eudml.org/doc/74933>.

@article{Blümlinger1991,
abstract = {We are interested in permutations preserving certain distribution properties of sequences. In particular we consider $\mu $-uniformly distributed sequences on a compact metric space $X$, 0-1 sequences with densities, and Cesàro summable bounded sequences. It is shown that the maximal subgroups, respectively subsemigroups, of $Aut(\{\bf N\})$ leaving any of the above spaces invariant coincide. A subgroup of these permutation groups, which can be determined explicitly, is the Lévy group $\{\cal G\}$. We show that $\{\cal G\}$ is big in the sense that the Cesàro mean is characterized by its invariance under the Lévy group. As a result, any $\{\cal G\}$ -invariant positive normalized linear functional on $l^ \infty (\{\bf N\})$ is an extension of Cesàro means. Finally we prove that there exist $\{\cal G\}$ -invariant extensions of Cesàro mean to all of $l^ \infty (\{\bf N\})$.},
author = {Blümlinger, M., Obata, N.},
journal = {Annales de l'institut Fourier},
keywords = {densities; uniform distribution; permutation groups; Lévy group; linear functional; invariant extensions of Cesàro mean},
language = {eng},
number = {3},
pages = {665-678},
publisher = {Association des Annales de l'Institut Fourier},
title = {Permutations preserving Cesàro mean, densities of natural numbers and uniform distribution of sequences},
url = {http://eudml.org/doc/74933},
volume = {41},
year = {1991},
}

TY - JOUR
AU - Blümlinger, M.
AU - Obata, N.
TI - Permutations preserving Cesàro mean, densities of natural numbers and uniform distribution of sequences
JO - Annales de l'institut Fourier
PY - 1991
PB - Association des Annales de l'Institut Fourier
VL - 41
IS - 3
SP - 665
EP - 678
AB - We are interested in permutations preserving certain distribution properties of sequences. In particular we consider $\mu $-uniformly distributed sequences on a compact metric space $X$, 0-1 sequences with densities, and Cesàro summable bounded sequences. It is shown that the maximal subgroups, respectively subsemigroups, of $Aut({\bf N})$ leaving any of the above spaces invariant coincide. A subgroup of these permutation groups, which can be determined explicitly, is the Lévy group ${\cal G}$. We show that ${\cal G}$ is big in the sense that the Cesàro mean is characterized by its invariance under the Lévy group. As a result, any ${\cal G}$ -invariant positive normalized linear functional on $l^ \infty ({\bf N})$ is an extension of Cesàro means. Finally we prove that there exist ${\cal G}$ -invariant extensions of Cesàro mean to all of $l^ \infty ({\bf N})$.
LA - eng
KW - densities; uniform distribution; permutation groups; Lévy group; linear functional; invariant extensions of Cesàro mean
UR - http://eudml.org/doc/74933
ER -