Reparametrization Invariant Norms

This paper explores the concept of reparametrization invariant norm (RPI-norm) for C1-functions that vanish at −∞ and whose derivative has compact support, such as C1 c -functions. An RPI-norm is any norm invariant under composition with orientation-preserving diffeomorphisms. The L∞-norm and the total variation norm are well-known instances of RPI-norms. We prove the existence of an infinite family of RPI-norms, called standard RPInorms, for which we exhibit both an integral and a discrete characterization. Our main result states that for every piecewise monotone function φ in C1 c (R) the standard RPI-norms of φ allow us to compute the value of any other RPInorm of φ. This is proved using the standard RPI-norms to reconstruct the function φ up to reparametrization, sign and an arbitrarily small error with respect to the total variation norm.