Reparametrization invariant norms

This paper explores the concept of reparametrization invariant norm (RPI-norm) for $C^1$-functions that vanish at $-\infty$ and whose derivative has compact support, such as $C^1_c$-functions. An RPI-norm is any norm invariant under composition with orientation-preserving diffeomorphisms. The $L_\infty$-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 $\varphi$ in $C^1_c(\mathbb{R})$ the standard RPI-norms of $\varphi$ allow us to compute the value of any other RPI-norm of $\varphi$. This is proved using the standard RPI-norms to reconstruct the function $\varphi$ up to reparametrization, sign and an arbitrarily small error with respect to the total variation norm.