Reparametrization invariant norms

This paper explores the concept of reparametrization invariantnorm (RPI-norm) for C^1-functions that vanish at −∞ and whose derivativehas compact support, such as C^1_c -functions. An RPI-norm is any norm invariantunder composition with orientation-preserving diffeomorphisms. TheL_∞-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 RPI-norms,for which we exhibit both an integral and a discrete characterization.Our main result states that for every piecewise monotone function ϕ in C^1_c (R)the standard RPI-norms of ϕ allow us to compute the value of any other RPI-normof ϕ. This is proved using the standard RPI-norms to reconstruct thefunction ϕ up to reparametrization, sign and an arbitrarily small error withrespect to the total variation norm.