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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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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.
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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