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.

Reparametrization invariant norms / P., Frosini; Landi, Claudia. - In: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9947. - STAMPA. - 361:(2009), pp. 407-452. [10.1090/S0002-9947-08-04581-9]

Reparametrization invariant norms

LANDI, Claudia
2009

Abstract

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.
361
407
452
Reparametrization invariant norms / P., Frosini; Landi, Claudia. - In: TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY. - ISSN 0002-9947. - STAMPA. - 361:(2009), pp. 407-452. [10.1090/S0002-9947-08-04581-9]
P., Frosini; Landi, Claudia
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11380/584761
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