Geodesic
On a family of analogs of the Perron–Stieltjes integral
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 3 (2011), pp. 95-106
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

We define the concept of a quasi-integral for two regulated functions defined on a segment and for a special parameter called a defect. In case there exists the Riemann–Stieltjes integral, there is a quasi-integral for any defect, and all quasi-integrals are equal. The Perron–Stieltjes integral, if it exists, coincides with one of quasi-integrals where the defect is defined in a special way. We give proofs of necessary and sufficient conditions for the existence of quasi-integrals and of their basic properties, in particular, of the analogue of the formula of integration by parts.
Keywords: regulated function, Riemann–Stieltjes integral, Perron–Stieltjes integral. 
@article{VUU_2011_3_a8,
     author = {V. I. Rodionov},
     title = {On a~family of analogs of the {Perron{\textendash}Stieltjes} integral},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {95--106},
     year = {2011},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/VUU_2011_3_a8/}
}
TY  - JOUR
AU  - V. I. Rodionov
TI  - On a family of analogs of the Perron–Stieltjes integral
JO  - Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
PY  - 2011
SP  - 95
EP  - 106
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/VUU_2011_3_a8/
LA  - ru
ID  - VUU_2011_3_a8
ER  - 
%0 Journal Article
%A V. I. Rodionov
%T On a family of analogs of the Perron–Stieltjes integral
%J Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
%D 2011
%P 95-106
%N 3
%U http://geodesic.mathdoc.fr/item/VUU_2011_3_a8/
%G ru
%F VUU_2011_3_a8
V. I. Rodionov. On a family of analogs of the Perron–Stieltjes integral. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, no. 3 (2011), pp. 95-106. http://geodesic.mathdoc.fr/item/VUU_2011_3_a8/

[1] Hönig Ch. S., Volterra-Stieltjes integral equations, Mathematics Studies, 16, North-Holland, Amsterdam, 1975, 152 pp. | MR | Zbl

[2] Rodionov V. I., “Ob odnom semeistve podprostranstv prostranstva preryvistykh funktsii”, Vestnik Udmurtskogo universiteta. Matematika. Mekhanika. Kompyuternye nauki, 2009, no. 4, 7–24

[3] Kolmogorov A. N., Fomin S. V., Elementy teorii funktsii i funktsionalnogo analiza, uchebnik dlya vuzov, 5-e izd., Nauka, M., 1981, 544 pp. | MR

[4] Fikhtengolts G. M., Kurs differentsialnogo i integralnogo ischisleniya, uchebnik dlya vuzov, v. 3, 5-e izd., Nauka, M., 1969, 656 pp.

[5] Tvrdý M., “Regulated functions and the Perron-Stieltjes integral”, Časopis pĕst. mat., 114 (1989), 187–209 | MR | Zbl