| Title:
|
The Henstock-Kurzweil approach to Young integrals with integrators in ${\rm BV}\sb \phi$ (English) |
| Author:
|
Varayu, Boonpogkrong |
| Author:
|
Chew, Tuan Seng |
| Language:
|
English |
| Journal:
|
Mathematica Bohemica |
| ISSN:
|
0862-7959 (print) |
| ISSN:
|
2464-7136 (online) |
| Volume:
|
131 |
| Issue:
|
3 |
| Year:
|
2006 |
| Pages:
|
233-260 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
Henstock integral |
| Keyword:
|
Stieltjes integral |
| Keyword:
|
Young integral |
| Keyword:
|
$\phi $-variation |
| MSC:
|
26A21 |
| MSC:
|
26A39 |
| MSC:
|
26A42 |
| MSC:
|
28B15 |
| idZBL:
|
Zbl 1112.26004 |
| idMR:
|
MR2248593 |
| DOI:
|
10.21136/MB.2006.134138 |
| . |
| Date available:
|
2009-09-24T22:26:09Z |
| Last updated:
|
2020-07-29 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/134138 |
| . |
| Reference:
|
[1] Boonpogkrong Varayu, Tuan Seng Chew: On integrals with integrators in $\mathop {{\mathrm BV}}_p$.Real Anal. Exch. 30 (2004/2005), 193–200. MR 2127525, 10.14321/realanalexch.30.1.0193 |
| Reference:
|
[2] R. M. Dudley, R. Norvaisa: Differentiability of Six Operators on Nonsmooth Functions and $p$-Variation.Springer, Berlin, 1999. MR 1705318 |
| Reference:
|
[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 1691764 |
| Reference:
|
[4] P. Y. Lee, R. Výborný: The Integral. An Easy Approach after Kurzweil and Henstock.Cambridge University Press, 2000. MR 1756319 |
| Reference:
|
[5] R. Lesniewicz, W. Orlicz: On generalized variations (II).Stud. Math. 45 (1973), 71–109. MR 0346509, 10.4064/sm-45-1-71-109 |
| Reference:
|
[6] J. Musielak, W. Orlicz: On generalized variations (I).Stud. Math. 18 (1959), 11–41. MR 0104771, 10.4064/sm-18-1-11-41 |
| Reference:
|
[7] E. R. Love, L. C. Young: On fractional integration by parts.Proc. London Math. Soc., II. Ser., 44 (1938), 1–35. MR 1575481 |
| Reference:
|
[8] E. R. Love: Integration by parts and other theorems for $R^3S$-integrals.Real Anal. Exch. 24 (1998/1999), 315–336. MR 1691754 |
| Reference:
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[9] R. Norvaisa: Quadratic Variation, p-Variation and Integration with Applications to Strock Price Modelling.Preprint, 2003. MR 2251444 |
| Reference:
|
[10] Š. Schwabik: A note on integration by parts for abstract Perron-Stieltjes integrals.Math. Bohem. 126 (2001), 613–629. Zbl 0980.26005, MR 1970264 |
| Reference:
|
[11] L. C. Young: An inequality of the Hölder type, connected with Stieltjes integration.Acta Math. 67 (1936), 251–282. Zbl 0016.10404, MR 1555421, 10.1007/BF02401743 |
| Reference:
|
[12] L. C. Young: General inequalities for Stieltjes integrals and the convergence of Fourier series.Math. Ann. 115 (1938), 581–612. MR 1513204, 10.1007/BF01448958 |
| . |
