Article
| Title: | Elementary construction of Hölder functions such that the Kurzweil-Stieltjes integral does not exist (English) |
| Author: | Rmoutil, Martin |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 75 |
| Issue: | 1 |
| Year: | 2025 |
| Pages: | 345-356 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | For any $\alpha , \beta >0$ with $\alpha +\beta <1$ we provide a simple construction of an $\alpha $-Hölde function $f\colon [0,1]\to {\mathbb R}$ and a $\beta $-Hölder function $g\colon [0,1]\to {\mathbb R}$ such that the integral $\int _0^1 f {\rm d} g$ fails to exist even in the Kurzweil-Stieltjes sense. (English) |
| Keyword: | Kurzweil-Stieltjes integral |
| Keyword: | Hölder function |
| Keyword: | counterexample |
| MSC: | 26A16 |
| MSC: | 26A39 |
| MSC: | 26A42 |
| DOI: | 10.21136/CMJ.2024.0296-23 |
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| Date available: | 2025-03-11T16:05:44Z |
| Last updated: | 2025-03-19 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/152912 |
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| Reference: | [2] Dudley, R. M., Norvaiša, R.: Concrete Functional Calculus.Springer Monographs in Mathematics. Springer, New York (2011). Zbl 1218.46003, MR 2732563, 10.1007/978-1-4419-6950-7 |
| Reference: | [3] Hardy, G. H.: Weierstrass's non-differentiable function.Trans. Am. Math. Soc. 17 (1916), 301-325 \99999JFM99999 46.0401.03. MR 1501044, 10.1090/S0002-9947-1916-1501044-1 |
| Reference: | [4] Lacina, F.: Stochastic Integrals: Master's Thesis.Charles University, Prague (2016), Available at http://hdl.handle.net/20.500.11956/75283\kern0pt. |
| Reference: | [5] Leśniewicz, R., Orlicz, W.: On generalized variations. II.Stud. Math. 45 (1973), 71-109. Zbl 0218.26007, MR 346509, 10.4064/sm-45-1-71-109 |
| Reference: | [6] Monteiro, G. A., Slavík, A., Tvrdý, M.: Kurzweil-Stieltjes Integral: Theory and Applications.Series in Real Analysis 15. World Scientific, Hackensack (2019). Zbl 1437.28001, MR 3839599, 10.1142/9432 |
| Reference: | [7] Young, L. C.: 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: | [8] Zygmund, A.: Trigonometric Series. Volumes I and II Combined.Cambridge Mathematical Library. Cambridge University Press, Cambridge (2002). Zbl 1084.42003, MR 1963498, 10.1017/CBO9781316036587 |
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