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Transactions of the American Mathematical Society

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Reparametrization invariant norms

Authors: P. Frosini and C. Landi
Journal: Trans. Amer. Math. Soc. 361 (2009), 407-452
MSC (2000): Primary 46E10, 46B20
Published electronically: July 24, 2008
MathSciNet review: 2439412
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Abstract: 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.

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Additional Information

P. Frosini
Affiliation: Arces, Università di Bologna, via Toffano 2/2, I-40135 Bologna, Italia – and – Dipartimento di Matematica, Università di Bologna, P.zza di Porta S. Donato 5, I-40126 Bologna, Italia

C. Landi
Affiliation: Dismi, Università di Modena e Reggio Emilia, via Amendola 2, Pad. Morselli, I-42100 Reggio Emilia, Italia

Keywords: Reparametrization invariant norm, standard reparametrization invariant norm
Received by editor(s): March 21, 2007
Published electronically: July 24, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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