Reparametrization invariant norms

This paper explores the concept of reparametrization invariant norm (RPI-norm) for C1-functions that vanish at -infinity and whose derivative has compact support, such as C-c(1)-functions. An RPI-norm is any norm invariant under composition with orientation-preserving diffeomorphisms. The L-infinity-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 phi in C-c(1) (R) the standard RPI-norms of. allow us to compute the value of any other RPI-norm of phi. This is proved using the standard RPI-norms to reconstruct the function phi up to reparametrization, sign and an arbitrarily small error with respect to the total variation norm.