Geodesic
On generalization of mean value
Čebyševskij sbornik, Tome 21 (2020) no. 1, pp. 374-380.

Voir la notice de l'article provenant de la source Math-Net.Ru

In paper we discuss the solution of mean value general form problem in case of all variables symmetry absence. In 1930 A. N. Kolmogorov proved the formula for general form of mean value. He formulated four axioms: continuity and monotony on each variable, symmetry on each variable, mean value of equal variables is equal to these variables, any substitution of any group of variables with their mean value does not change the mean value. In Kolmogorov's theorem all arguments are equitable, this means that the mean value is symmetric on each variable. V. N. Chubarikov set the task of generalization to this result in case of all variables symmetry absence. We divide all the variables on groups and the mean value is a symmetric function for variables in each group separately. For example, if we have only one group the mean value will be Kolmogorov's mean value, so we have a generalization of Kolmogorov's theorem. In paper we show the general form of mean value in our case and we note the connection with uniform distribution modulo 1.
Keywords: Kolmogorov's mean value theorem, uniform distribution modulo 1. 
@article{CHEB_2020_21_1_a25,
     author = {R. V. Pocherevin},
     title = {On generalization of mean value},
     journal = {\v{C}eby\v{s}evskij sbornik},
     pages = {374--380},
     publisher = {mathdoc},
     volume = {21},
     number = {1},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/CHEB_2020_21_1_a25/}
}
TY  - JOUR
AU  - R. V. Pocherevin
TI  - On generalization of mean value
JO  - Čebyševskij sbornik
PY  - 2020
SP  - 374
EP  - 380
VL  - 21
IS  - 1
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/CHEB_2020_21_1_a25/
LA  - ru
ID  - CHEB_2020_21_1_a25
ER  - 
%0 Journal Article
%A R. V. Pocherevin
%T On generalization of mean value
%J Čebyševskij sbornik
%D 2020
%P 374-380
%V 21
%N 1
%I mathdoc
%U http://geodesic.mathdoc.fr/item/CHEB_2020_21_1_a25/
%G ru
%F CHEB_2020_21_1_a25
R. V. Pocherevin. On generalization of mean value. Čebyševskij sbornik, Tome 21 (2020) no. 1, pp. 374-380. http://geodesic.mathdoc.fr/item/CHEB_2020_21_1_a25/

[1] Kolmogorov A. N., Selected Works. Mathematics and Mechanics, Nauka, M., 1985, 136–138

[2] Chubarikov V. N., “A remark on definition of mean value”, Vestnik Moscovskogo Univ. Seriya 1: Matematika. Mehanika, 2003, no. 1, 30–33 | Zbl