Curves in Banach spaces which allow a C 1 , BV parametrization or a parametrization with finite convexity

Jakub Duda; Luděk Zajíček

Czechoslovak Mathematical Journal (2013)

  • Volume: 63, Issue: 4, page 1057-1085
  • ISSN: 0011-4642

Abstract

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We give a complete characterization of those f : [ 0 , 1 ] → X (where X is a Banach space) which allow an equivalent C 1 , 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 = ℝ n . We present examples which show applicability of our characterizations. For example, we show that the C 1 , BV and C 2 parametrization problems are equivalent for X = ℝ but are not equivalent for X = ℝ 2 .

How to cite

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Duda, Jakub, and Zajíček, Luděk. "Curves in Banach spaces which allow a $C^{1,\rm BV}$ parametrization or a parametrization with finite convexity." Czechoslovak Mathematical Journal 63.4 (2013): 1057-1085. <http://eudml.org/doc/260775>.

@article{Duda2013,
abstract = {We give a complete characterization of those $f\colon [0,1] \rightarrow X$ (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 $X = \mathbb \{R\}^2$.},
author = {Duda, Jakub, Zajíček, Luděk},
journal = {Czechoslovak Mathematical Journal},
keywords = {curve in Banach spaces; $C^\{1,\rm BV\}$ parametrization; parametrization with bounded convexity; curves in Banach spaces; parametrization; parametrization with bounded convexity},
language = {eng},
number = {4},
pages = {1057-1085},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Curves in Banach spaces which allow a $C^\{1,\rm BV\}$ parametrization or a parametrization with finite convexity},
url = {http://eudml.org/doc/260775},
volume = {63},
year = {2013},
}

TY - JOUR
AU - Duda, Jakub
AU - Zajíček, Luděk
TI - Curves in Banach spaces which allow a $C^{1,\rm BV}$ parametrization or a parametrization with finite convexity
JO - Czechoslovak Mathematical Journal
PY - 2013
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 63
IS - 4
SP - 1057
EP - 1085
AB - We give a complete characterization of those $f\colon [0,1] \rightarrow X$ (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 $X = \mathbb {R}^2$.
LA - eng
KW - curve in Banach spaces; $C^{1,\rm BV}$ parametrization; parametrization with bounded convexity; curves in Banach spaces; parametrization; parametrization with bounded convexity
UR - http://eudml.org/doc/260775
ER -

References

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