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 -
