Geodesic
The Henstock-Kurzweil approach to Young integrals with integrators in ${\rm BV}\sb \phi$
Mathematica Bohemica, Tome 131 (2006) no. 3, pp. 233-260

Voir la notice de l'article provenant de la source Czech Digital Mathematics Library

MR Zbl
In 1938, L. C. Young proved that the Moore-Pollard-Stieltjes integral $\int _a^bf\mathrm{d}g$ exists if $f\in \mathop {{\mathrm BV}}_\phi [a,b]$, $g\in \mathop {{\mathrm BV}}_\psi [a,b]$ and $\sum _{n=1}^\infty \phi ^{-1}({1}/{n})\psi ^{-1} ({1}/{n})\infty $. In this note we use the Henstock-Kurzweil approach to handle the above integral defined by Young.
In 1938, L. C. Young proved that the Moore-Pollard-Stieltjes integral $\int _a^bf\mathrm{d}g$ exists if $f\in \mathop {{\mathrm BV}}_\phi [a,b]$, $g\in \mathop {{\mathrm BV}}_\psi [a,b]$ and $\sum _{n=1}^\infty \phi ^{-1}({1}/{n})\psi ^{-1} ({1}/{n})\infty $. In this note we use the Henstock-Kurzweil approach to handle the above integral defined by Young.
DOI : 10.21136/MB.2006.134138
Classification : 26A21, 26A39, 26A42, 28B15
Keywords: Henstock integral; Stieltjes integral; Young integral; $\phi $-variation 
Varayu, Boonpogkrong; Chew, Tuan Seng. The Henstock-Kurzweil approach to Young integrals with integrators in ${\rm BV}\sb \phi$. Mathematica Bohemica, Tome 131 (2006) no. 3, pp. 233-260. doi: 10.21136/MB.2006.134138
@article{10_21136_MB_2006_134138,
     author = {Varayu, Boonpogkrong and Chew, Tuan Seng},
     title = {The {Henstock-Kurzweil} approach to {Young} integrals with integrators in ${\rm BV}\sb \phi$},
     journal = {Mathematica Bohemica},
     pages = {233--260},
     year = {2006},
     volume = {131},
     number = {3},
     doi = {10.21136/MB.2006.134138},
     mrnumber = {2248593},
     zbl = {1112.26004},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.21136/MB.2006.134138/}
}
TY  - JOUR
AU  - Varayu, Boonpogkrong
AU  - Chew, Tuan Seng
TI  - The Henstock-Kurzweil approach to Young integrals with integrators in ${\rm BV}\sb \phi$
JO  - Mathematica Bohemica
PY  - 2006
SP  - 233
EP  - 260
VL  - 131
IS  - 3
UR  - http://geodesic.mathdoc.fr/articles/10.21136/MB.2006.134138/
DO  - 10.21136/MB.2006.134138
LA  - en
ID  - 10_21136_MB_2006_134138
ER  - 
%0 Journal Article
%A Varayu, Boonpogkrong
%A Chew, Tuan Seng
%T The Henstock-Kurzweil approach to Young integrals with integrators in ${\rm BV}\sb \phi$
%J Mathematica Bohemica
%D 2006
%P 233-260
%V 131
%N 3
%U http://geodesic.mathdoc.fr/articles/10.21136/MB.2006.134138/
%R 10.21136/MB.2006.134138
%G en
%F 10_21136_MB_2006_134138

[1] Boonpogkrong Varayu, Tuan Seng Chew: On integrals with integrators in $\mathop {{\mathrm BV}}_p$. Real Anal. Exch. 30 (2004/2005), 193–200. | DOI | MR

[2] R. M. Dudley, R. Norvaisa: Differentiability of Six Operators on Nonsmooth Functions and $p$-Variation. Springer, Berlin, 1999. | MR

[3] I. J. L. Garces, Peng-Yee Lee, Dongsheng Zhao: Moore-Smith limits and the Henstock integral. Real Anal. Exchange 24 (1998/1999), 447–455. | MR

[4] P. Y. Lee, R. Výborný: The Integral. An Easy Approach after Kurzweil and Henstock. Cambridge University Press, 2000. | MR

[5] R. Lesniewicz, W. Orlicz: On generalized variations (II). Stud. Math. 45 (1973), 71–109. | DOI | MR

[6] J. Musielak, W. Orlicz: On generalized variations (I). Stud. Math. 18 (1959), 11–41. | DOI | MR

[7] E. R. Love, L. C. Young: On fractional integration by parts. Proc. London Math. Soc., II. Ser., 44 (1938), 1–35. | MR

[8] E. R. Love: Integration by parts and other theorems for $R^3S$-integrals. Real Anal. Exch. 24 (1998/1999), 315–336. | MR

[9] R. Norvaisa: Quadratic Variation, p-Variation and Integration with Applications to Strock Price Modelling. Preprint, 2003. | MR

[10] Š. Schwabik: A note on integration by parts for abstract Perron-Stieltjes integrals. Math. Bohem. 126 (2001), 613–629. | MR | Zbl

[11] L. C. Young: An inequality of the Hölder type, connected with Stieltjes integration. Acta Math. 67 (1936), 251–282. | DOI | MR | Zbl

[12] L. C. Young: General inequalities for Stieltjes integrals and the convergence of Fourier series. Math. Ann. 115 (1938), 581–612. | DOI | MR

Cité par Sources :