Geodesic
Countable Additivity of Spreading the Differentiation Operator
Modelirovanie i analiz informacionnyh sistem, Tome 21 (2014) no. 3, pp. 81-90.

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

In this article, we continue the study of the properties acquired by the differentiation operator $ \Lambda $ with spreading beyond the space $ W_1^1 $. The study is conducted by introducing the family of spaces $ Y_p^1$, $0 $, having analogy with the family $ W_p^1$, $1 \le p \infty.$ Spaces $ Y_p^1 $ are equiped with quasinorms constructed on quasinorms spaces $ L_p $ as the basis; $ \Lambda: Y_p^1 \mapsto L_p $. We have given a sufficient condition for a function, piecewise belonging to the space $ Y_p^1 $ to be in this space (if $ f \in Y_p^1 [x_{i-1}; x_i]$, $i \in N$, $0 = x_0 $, then $ f\in Y_p^1[0;1] $). In other words, it is the sign when the equality: $ \Lambda (\bigcup f_i) = \bigcup \Lambda (f_i)$ is true. The bounded variation in the Jordan sense is closest to the sufficient condition among the classic characteristics of functions. As a corollary, it comes out that, if a function $ f $ piecewise belongs to the space of $ W_1^1 $ and has a bounded variation, $ f $ belongs to each space $ Y_p^1$, $0 $.
Keywords: differentiation operator
Mots-clés : quasinorm. 
@article{MAIS_2014_21_3_a4,
     author = {A. N. Morozov},
     title = {Countable {Additivity} of {Spreading} the {Differentiation} {Operator}},
     journal = {Modelirovanie i analiz informacionnyh sistem},
     pages = {81--90},
     publisher = {mathdoc},
     volume = {21},
     number = {3},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MAIS_2014_21_3_a4/}
}
TY  - JOUR
AU  - A. N. Morozov
TI  - Countable Additivity of Spreading the Differentiation Operator
JO  - Modelirovanie i analiz informacionnyh sistem
PY  - 2014
SP  - 81
EP  - 90
VL  - 21
IS  - 3
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MAIS_2014_21_3_a4/
LA  - ru
ID  - MAIS_2014_21_3_a4
ER  - 
%0 Journal Article
%A A. N. Morozov
%T Countable Additivity of Spreading the Differentiation Operator
%J Modelirovanie i analiz informacionnyh sistem
%D 2014
%P 81-90
%V 21
%N 3
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MAIS_2014_21_3_a4/
%G ru
%F MAIS_2014_21_3_a4
A. N. Morozov. Countable Additivity of Spreading the Differentiation Operator. Modelirovanie i analiz informacionnyh sistem, Tome 21 (2014) no. 3, pp. 81-90. http://geodesic.mathdoc.fr/item/MAIS_2014_21_3_a4/

[1] Morozov A. N., “Kusochno-polinomialnye priblizheniay i differentsiruemost v prostranstvakh $L_p \; (0 p 1)$”, Model. i analiz inform. system, 12:1 (2005), 18—21 (in Russian)

[2] Morozov A. N., “On Smoothness in $L_p, \; 0 p 1$”, Modeling and analysis of inform. systems, 19:3 (2012), 97—104 (in Russian)

[3] L. V. Kantorovich, G. P. Akilov, Functional analysis, translated by Howard L. Silcock, Pergamon Press, Oxford, New York, 1982 | MR | MR | Zbl | Zbl

[4] J. Berg, J. Lofsrom, Interpolation Spaces. An Introduction, Springer-Verlag, 1976 ; I. Berg, I. Lefstrem, Interpolyatsionnye prostranstva. Vvedenie, per. s angl. Kryuchkova V. S. i Lizorkina P. I., Mir, M., 1980 | MR | MR

[5] A. F. Timan, Theory of Approximation of Functions of a Real Variable, Courier Dover Publications, 1994 | MR