×

zbMATH — the first resource for mathematics

Second order differentiability of paths via a generalized \(\frac 1 2\)-variation. (English) Zbl 1135.46021
Summary: We find an equivalent condition for a continuous vector-valued path to be Lebesgue equivalent to a twice differentiable function. For that purpose, we introduce the notion of a \(VBG_{1/2}\) function, which plays an analogous role for the second order differentiability as the classical notion of a \(VBG_{*}\) function for the first order differentiability. In fact, for a function \(f:[a,b]\rightarrow X\), being Lebesgue equivalent to a twice differentiable function is the same as being Lebesgue equivalent to a differentiable function \(g\) with a pointwise Lipschitz derivative such that \(g^{\prime\prime }(x)\) exists whenever \(g^{\prime }(x)\neq 0\). We also consider the case when the first derivative can be taken non-zero almost everywhere.

MSC:
46G05 Derivatives of functions in infinite-dimensional spaces
26E20 Calculus of functions taking values in infinite-dimensional spaces
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Choquet, G., Application des propriétés descriptives de la fonction contingent à la théorie des fonctions de variable réelle et à la géométrie différentielle des variétés cartésiennes, J. math. pures appl., 26, 9, 115-226, (1947), (1948) · Zbl 0035.24201
[2] Deville, R.; Godefroy, G.; Zizler, V., Smoothness and renormings in Banach spaces, () · Zbl 0782.46019
[3] J. Duda, Generalized α-variation and Lebesgue equivalence to differentiable functions, preprint, submitted for publication, available from http://www.arxiv.org
[4] J. Duda, L. Zajíček, Curves in Banach spaces—differentiability via homeomorphisms, Rocky Mountain J. Math., in press · Zbl 1148.26008
[5] J. Duda, L. Zajíček, Curves in Banach spaces which allow a \(C^2\) parametrization or a parametrization with finite convexity, preprint, submitted for publication, available from http://www.arxiv.org
[6] Federer, H., Geometric measure theory, Grundlehren math. wiss., vol. 153, (1969), Springer New York · Zbl 0176.00801
[7] Fleissner, R.J.; Foran, J.A., A note on differentiable functions, Proc. amer. math. soc., 69, 56, (1978) · Zbl 0384.26003
[8] Goffman, C.; Nishiura, T.; Waterman, D., Homeomorphisms in analysis, Math. surveys monogr., vol. 54, (1997), Amer. Math. Soc. Providence, RI · Zbl 0890.26001
[9] Kirchheim, B., Rectifiable metric spaces: local structure and regularity of the Hausdorff measure, Proc. amer. math. soc., 121, 113-123, (1994) · Zbl 0806.28004
[10] Laczkovich, M.; Preiss, D., α-variation and transformation into cn functions, Indiana univ. math. J., 34, 405-424, (1985) · Zbl 0557.26004
[11] Lebedev, V.V., Homeomorphisms of a segment and smoothness of a function, Mat. zametki, 40, 364-373, (1986), 429 (in Russian)
[12] Saks, S., Theory of the integral, Monogr. mat., vol. 7, (1937), Hafner New York
[13] Tolstov, G.P., Curves allowing a differentiable parametric representation, Uspekhi mat. nauk (N.S.), 6, 135-152, (1951), (in Russian)
[14] Varberg, D.E., On absolutely continuous functions, Amer. math. monthly, 72, 831-841, (1965) · Zbl 0133.00502
[15] Zahorski, Z., On Jordan curves possessing a tangent at every point, Mat. sb. (N.S.), 22, 64, 3-26, (1948), (in Russian)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.