## Reparametrization invariant norms

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- by P. Frosini and C. Landi PDF
- Trans. Amer. Math. Soc.
**361**(2009), 407-452 Request permission

## 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.## References

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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
- Email: frosini@dm.unibo.it
**C. Landi**- Affiliation: Dismi, Università di Modena e Reggio Emilia, via Amendola 2, Pad. Morselli, I-42100 Reggio Emilia, Italia
- Received by editor(s): March 21, 2007
- Published electronically: July 24, 2008
- © Copyright 2008
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication. - Journal: Trans. Amer. Math. Soc.
**361**(2009), 407-452 - MSC (2000): Primary 46E10, 46B20
- DOI: https://doi.org/10.1090/S0002-9947-08-04581-9
- MathSciNet review: 2439412