Geodesic
Representation of the Stieltjes Integral in Terms of the Riemann Integral Depending on a Parameter
Matematičeskie zametki, Tome 78 (2005) no. 6, pp. 919-933.

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

In this paper, we obtain a new formula for the representation of the Riemann–Stieltjes integral of a continuous function in terms of the passage to the limit with respect to the parameter in a Riemann integral depending on this parameter. The derivation of this formula is based on the study of the functional properties of the solution of the auxiliary difference equation of first order representing the weighted first difference of a given function in the form of a simple first difference of an unknown function. The result obtained can be used for the analytic and approximate calculation of Stieltjes integrals. 
@article{MZM_2005_78_6_a10,
     author = {V. A. Chernyatin},
     title = {Representation of the {Stieltjes} {Integral} in {Terms} of the {Riemann} {Integral} {Depending} on a {Parameter}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {919--933},
     publisher = {mathdoc},
     volume = {78},
     number = {6},
     year = {2005},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2005_78_6_a10/}
}
TY  - JOUR
AU  - V. A. Chernyatin
TI  - Representation of the Stieltjes Integral in Terms of the Riemann Integral Depending on a Parameter
JO  - Matematičeskie zametki
PY  - 2005
SP  - 919
EP  - 933
VL  - 78
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2005_78_6_a10/
LA  - ru
ID  - MZM_2005_78_6_a10
ER  - 
%0 Journal Article
%A V. A. Chernyatin
%T Representation of the Stieltjes Integral in Terms of the Riemann Integral Depending on a Parameter
%J Matematičeskie zametki
%D 2005
%P 919-933
%V 78
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2005_78_6_a10/
%G ru
%F MZM_2005_78_6_a10
V. A. Chernyatin. Representation of the Stieltjes Integral in Terms of the Riemann Integral Depending on a Parameter. Matematičeskie zametki, Tome 78 (2005) no. 6, pp. 919-933. http://geodesic.mathdoc.fr/item/MZM_2005_78_6_a10/

[1] Natanson I. P., Teoriya funktsii veschestvennoi peremennoi, Gostekhizdat, M., 1957 | MR

[2] Fikhtengolts G. M., Kurs differentsialnogo i integralnogo ischisleniya, T. 1–3, Fizmatgiz, M., 1962

[3] Gelfond A. O., Ischislenie konechnykh raznostei, Fizmatgiz, M., 1967 | MR

[4] Kondurar V., “Sur l'integrale de Stieltjes”, Recueil Math., 2 (1937), 361–366 | Zbl