Permutations preserving Cesàro mean, densities of natural numbers and uniform distribution of sequences
Annales de l'Institut Fourier, Tome 41 (1991) no. 3, pp. 665-678.

Nous considérons les permutations de N qui conservent la μ-répartition des suites ou la densité des parties de N ou la somme de Cesàro des suites sommables, et montrons que le groupe (resp. semi-groupe) de ces permutations sont les mêmes. Il est prouvé qu’il y a des fonctionnelles de l (N) qui sont invariantes sous l’action du groupe de Lévy et que toutes ces fonctionnelles sont des extensions de la somme de Cesàro.

We are interested in permutations preserving certain distribution properties of sequences. In particular we consider μ-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(N) leaving any of the above spaces invariant coincide. A subgroup of these permutation groups, which can be determined explicitly, is the Lévy group 𝒢. We show that 𝒢 is big in the sense that the Cesàro mean is characterized by its invariance under the Lévy group. As a result, any 𝒢 -invariant positive normalized linear functional on l (N) is an extension of Cesàro means. Finally we prove that there exist 𝒢 -invariant extensions of Cesàro mean to all of l (N).

@article{AIF_1991__41_3_665_0,
     author = {Bl\"umlinger, M. and Obata, N.},
     title = {Permutations preserving {Ces\`aro} mean, densities of natural numbers and uniform distribution of sequences},
     journal = {Annales de l'Institut Fourier},
     pages = {665--678},
     publisher = {Institut Fourier},
     address = {Grenoble},
     volume = {41},
     number = {3},
     year = {1991},
     doi = {10.5802/aif.1269},
     mrnumber = {92j:43002},
     zbl = {0735.11004},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.1269/}
}
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
SP  - 665
EP  - 678
VL  - 41
IS  - 3
PB  - Institut Fourier
PP  - Grenoble
UR  - http://www.numdam.org/articles/10.5802/aif.1269/
DO  - 10.5802/aif.1269
LA  - en
ID  - AIF_1991__41_3_665_0
ER  - 
%0 Journal Article
%A Blümlinger, M.
%A Obata, N.
%T Permutations preserving Cesàro mean, densities of natural numbers and uniform distribution of sequences
%J Annales de l'Institut Fourier
%D 1991
%P 665-678
%V 41
%N 3
%I Institut Fourier
%C Grenoble
%U http://www.numdam.org/articles/10.5802/aif.1269/
%R 10.5802/aif.1269
%G en
%F AIF_1991__41_3_665_0
Blümlinger, M.; Obata, N. Permutations preserving Cesàro mean, densities of natural numbers and uniform distribution of sequences. Annales de l'Institut Fourier, Tome 41 (1991) no. 3, pp. 665-678. doi : 10.5802/aif.1269. http://www.numdam.org/articles/10.5802/aif.1269/

[C] J. Coquet, Permutations des entiers et répartition des suites, Publ. Math. Orsay (Univ. Paris XI, Orsay), 86-1 (1986), 25-39. | MR | Zbl

[K] V.L. Klee, Jr., Invariant extensions of linear functionals, Pacific J. Math., 14 (1954), 37-46. | MR | Zbl

[KN] L. Kuipers, H. Niederreiter, Uniform distribution of sequences Wiley, New York, 1974. | MR | Zbl

[L] P. Lévy, Problèms Concretes d'Analyse Fonctionelle, Gauthier-Villars, Paris, 1951. | Zbl

[O1] N. Obata, A note on certain permutation groups in the infinite dimensional rotation groups, Nagoya Math. J., 109 (1988), 91-107. | MR | Zbl

[O2] N. Obata, Density of natural numbers and the Lévy group J. Number Theory, 30 (1988), 288-297. | MR | Zbl

[P] A. Paterson, Amenability, A.M.S., Providence, 1988. | MR | Zbl

[R] H. Rindler, Eine Charakterisierung gleichverteilter Folgen, Arch. Math., 32 (1979), 185-188. | MR | Zbl

[S] Q. Stout, On Levi's duality between permutations and convergent series J. London Math. Soc., (2) 34 (1986), 67-80. | MR | Zbl

Cité par Sources :