Reparametrization invariant norms

Frosini, P.; Landi, C.

doi:10.1090/S0002-9947-08-04581-9

Reparametrization invariant norms
HTML articles powered by AMS MathViewer

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

Luigi Ambrosio, Nicola Fusco, and Diego Pallara, Functions of bounded variation and free discontinuity problems, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 2000. MR 1857292
F. L. Bauer, J. Stoer, and C. Witzgall, Absolute and monotonic norms, Numer. Math. 3 (1961), 257–264. MR 130104, DOI 10.1007/BF01386026
J. Bergh, J. Löfström, Interpolation Spaces, Grundlehren der mathematischen Wissenschaften 223, 1976.
Pietro Donatini and Patrizio Frosini, Natural pseudodistances between closed manifolds, Forum Math. 16 (2004), no. 5, 695–715. MR 2096683, DOI 10.1515/form.2004.032
Pietro Donatini and Patrizio Frosini, Lower bounds for natural pseudodistances via size functions, Arch. Inequal. Appl. 2 (2004), no. 1, 1–12. MR 2043046
Pietro Donatini and Patrizio Frosini, Natural pseudodistances between closed surfaces, J. Eur. Math. Soc. (JEMS) 9 (2007), no. 2, 331–353. MR 2293959
A. Dumitrescu and G. Rote, On the Fréchet distance of a set of curves In: Proceedings of the 16th Canadian Conference on Computational Geometry (CCCG’04), Montreal, August 9-11, 2004, 162–165.
Tord Holmstedt and Jaak Peetre, On certain functionals arising in the theory of interpolation spaces. , J. Functional Analysis 4 (1969), 88–94. MR 0241966, DOI 10.1016/0022-1236(69)90023-8
Krzysztof Jarosz, Uniqueness of translation invariant norms, J. Funct. Anal. 174 (2000), no. 2, 417–429. MR 1768981, DOI 10.1006/jfan.2000.3593
Peter W. Michor and David Mumford, Riemannian geometries on spaces of plane curves, J. Eur. Math. Soc. (JEMS) 8 (2006), no. 1, 1–48. MR 2201275, DOI 10.4171/JEMS/37
E. Moreno and A. R. Villena, Uniqueness of dilation invariant norms, Proc. Amer. Math. Soc. 132 (2004), no. 7, 2067–2073. MR 2053979, DOI 10.1090/S0002-9939-04-07327-7
Bruno Sévennec, Normes invariantes par difféomorphismes sur $C^k(X)$, Ann. Fac. Sci. Toulouse Math. (6) 7 (1998), no. 2, 335–355 (French, with English and French summaries). MR 1656173