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Title: Curves in Banach spaces which allow a $C^{1,\rm BV}$ parametrization or a parametrization with finite convexity (English)
Author: Duda, Jakub
Author: Zajíček, Luděk
Language: English
Journal: Czechoslovak Mathematical Journal
ISSN: 0011-4642 (print)
ISSN: 1572-9141 (online)
Volume: 63
Issue: 4
Year: 2013
Pages: 1057-1085
Summary lang: English
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Category: math
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Summary: We give a complete characterization of those $f\colon [0,1] \to X$ (where $X$ is a Banach space) which allow an equivalent $C^{1,\rm BV}$ parametrization (i.e., a $C^1$ parametrization whose derivative has bounded variation) or a parametrization with bounded convexity. Our results are new also for $X= \mathbb R^n$. We present examples which show applicability of our characterizations. For example, we show that the $C^{1,\rm BV}$ and $C^2$ parametrization problems are equivalent for $X=\mathbb R$ but are not equivalent for $X = \mathbb R^2$. (English)
Keyword: curve in Banach spaces
Keyword: $C^{1,\rm BV}$ parametrization
Keyword: parametrization with bounded convexity
MSC: 26A51
MSC: 26E20
MSC: 53A04
idZBL: Zbl 06373962
idMR: MR3165515
DOI: 10.1007/s10587-013-0072-7
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Date available: 2014-01-28T14:18:13Z
Last updated: 2020-07-03
Stable URL: http://hdl.handle.net/10338.dmlcz/143617
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Reference: [1] Alexandrov, A. D., Reshetnyak, Yu. G.: General Theory of Irregular Curves. Transl. from the Russian by L. Ya. Yuzina.Mathematics and Its Applications: Soviet Series 29 Kluwer Academic Publishers, Dordrecht (1989). Zbl 0691.53002, MR 1117220
Reference: [2] Bourbaki, N.: Éléments de Mathématique. I: Les structures fondamentales de l'analyse. Livre IV: Fonctions d'une variable réelle (théorie élémentaire). Chapitres 1, 2 et 3: Dérievées. Primitives et intégrales. Fonctions élémentaires. Second ed.French Actualés Sci. Indust. 1074 Hermann, Paris (1958).
Reference: [3] Chistyakov, V. V.: On mappings of bounded variation.J. Dyn. Control Sys. 3 (1997), 261-289. Zbl 0940.26009, MR 1449984, 10.1007/BF02465896
Reference: [4] Duda, J.: Curves with finite turn.Czech. Math. J. 58 (2008), 23-49. Zbl 1167.46321, MR 2402524, 10.1007/s10587-008-0003-1
Reference: [5] Duda, J.: Second order differentiability of paths via a generalized $\frac{1}{2}$-variation.J. Math. Anal. Appl. 338 (2008), 628-638. Zbl 1135.46021, MR 2386444, 10.1016/j.jmaa.2007.05.046
Reference: [6] Duda, J.: Generalized $\alpha$-variation and Lebesgue equivalence to differentiable functions.Fundam. Math. 205 (2009), 191-217. Zbl 1191.26003, MR 2557935, 10.4064/fm205-3-1
Reference: [7] Duda, J., Zajíček, L.: Curves in Banach spaces---differentiability via homeomorphisms.Rocky Mt. J. Math. 37 (2007), 1493-1525. MR 2382898, 10.1216/rmjm/1194275931
Reference: [8] Duda, J., Zajíek, L.: Curves in Banach spaces which allow a $C^2$-parametrization or a parametrization with finite convexity.J. London Math. Soc., II. Ser. 83 (2011), 733-754. MR 2802508, 10.1112/jlms/jdq100
Reference: [9] Duda, J., Zajíek, L.: On vector-valued curves that allow a $C^{1,\alpha}$-parametrization.Acta Math. Hung. 127 (2010), 85-111. MR 2629670, 10.1007/s10474-010-9094-x
Reference: [10] Duda, J., Zajíček, L.: Curves in Banach spaces which allow a $C^2$-parametrization.J. Lond. Math. Soc., II. Ser. 83 (2011), 733-754. MR 2802508, 10.1112/jlms/jdq100
Reference: [11] Federer, H.: Geometric Measure Theory.Die Grundlehren der mathematischen Wissenschaften in Einzeldarstellungen, 153. Springer, New York (1969). Zbl 0176.00801, MR 0257325
Reference: [12] Kirchheim, B.: Rectifiable metric spaces: Local structure and regularity of the Hausdorff measure.Proc. Am. Math. Soc. 121 (1994), 113-123. Zbl 0806.28004, MR 1189747, 10.1090/S0002-9939-1994-1189747-7
Reference: [13] Kühnel, W.: Differential Geometry. Curves-Surfaces-Manifolds. Transl. from the German by Bruce Hunt.Student Mathematical Library 16. AMS Providence, RI (2002). Zbl 1009.53002, MR 1882174
Reference: [14] Laczkovich, M., Preiss, D.: $\alpha$-variation and transformation into $C\sp n$ functions.Indiana Univ. Math. J. 34 (1985), 405-424. Zbl 0634.26006, MR 0783923, 10.1512/iumj.1985.34.34024
Reference: [15] Lebedev, V. V.: Homeomorphisms of an interval and smoothness of a function.Math. Notes 40 (1986), 713-719 translation from Mat. Zametki 40 (1986), 364-373 Russian. Zbl 0637.26006, MR 0869927, 10.1007/BF01142475
Reference: [16] Massera, J. L., Schäffer, J. J.: Linear differential equations and functional analysis I.Ann. Math. (2) 67 (1958), 517-573. Zbl 0178.17701, MR 0096985, 10.2307/1969871
Reference: [17] Pogorelov, A. V.: Extrinsic Geometry of Convex Surfaces. Translated from the Russian by Israel Program for Scientific Translations.Translations of Mathematical Monographs 35 AMS, Providence, RI (1973). Zbl 0311.53067, MR 0346714, 10.1090/mmono/035
Reference: [18] Roberts, A. W., Varberg, D. E.: Convex Functions.Pure and Applied Mathematics 57 Academic Press, a subsidiary of Harcourt Brace Jovanovich, New York (1973). Zbl 0271.26009, MR 0442824
Reference: [19] Veselý, L.: On the multiplicity points of monotone operators on separable Banach spaces.Commentat. Math. Univ. Carol. 27 (1986), 551-570. MR 0873628
Reference: [20] Veselý, L., Zajíek, L.: Delta-convex mappings between Banach spaces and applications.Dissertationes Math. (Rozprawy Mat.) 289 1-48 (1989). MR 1016045
Reference: [21] Veselý, L., Zajíek, L.: On vector functions of bounded convexity.Math. Bohem. 133 (2008), 321-335. MR 2494785
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