Second order differentiability of paths via a generalized 1/2-variation

2005/11/21 by Jakub Duda, Duda, Jakub
Computer Science · Engineering · Mathematics · #14H50 #53A04 #Advanced Numerical Analysis Techniques #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #Mathematical and Theoretical Analysis #Optimization and Variational Analysis #math.CA #msc:14H50 #msc:53A04

paper · pdf · doi:10.48550/arxiv.math/0511518

generalized version; new title; 11 pages

openalex publication_date 2005/11/21 · arxiv created 2006/07/16 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

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]→ X, being Lebesgue equivalent to a twice differentiable function is the same as being Lebesgue equivalent to a differentiable function with a pointwise Lipschitz derivative. We also consider the case when the first derivative can be taken non-zero almost everywhere.

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