Curves in Banach spaces which allow a $C^2$ parametrization

Details Curves in Banach spaces which allow a $C^2$ parametrization Curves in Banach spaces which allow a $C^2$ parametrization

2006-03-31

arxiv.org/abs/math/0603735v2

Mathematics

Classical Analysis and ODEs

22 pages; We split the original paper into two parts. This is the first part ($C^2$ paths), the second part (paths with finite

Scientific paper

We give a complete characterization of those $f: [0,1] \to X$ (where $X$ is a Banach space which admits an equivalent Fr\'echet smooth norm) which allow an equivalent $C^2$ parametrization. For $X=\R$, a characterization is well-known. However, even in the case $X=\R^2$, several quite new ideas are needed. Moreover, the very close case of parametrizations with a bounded second derivative is solved.

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