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Garrido, M. Isabel and Jaramillo Aguado, Jesús Ángel and Rangel, Yenny C.
(2013)
*Smooth approximation of Lipschitz functions on Finsler manifolds.*
Journal of function spaces and applications
.
ISSN 0972-6802

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Official URL: http://www.hindawi.com/journals/jfsa/2013/164571/abs/

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http://www.hindawi.com/ | Publisher |

## Abstract

We study the smooth approximation of Lipschitz functions on Finsler manifolds, keeping control on the corresponding Lipschitz constants. We prove that, given a Lipschitz function f : M -> R defined on a connected, second countable Finsler manifold M, for each positive continuous function epsilon : M -> (0, infinity) and each r > 0, there exists a C-1-smooth Lipschitz function g : M -> R such that vertical bar f(x) - g(x)vertical bar <= epsilon(x), for every x is an element of M, and Lip(g) <= Lip(f) + r. As a consequence, we derive a completeness criterium in the class of what we call quasi-reversible Finsler manifolds. Finally, considering the normed algebra C-b(1)(M) of all C-1 functions with bounded derivative on a complete quasi-reversible Finsler manifold M, we obtain a characterization of algebra isomorphisms T : C-b(1)(N) -> C-b(1)(M) as composition operators. From this we obtain a variant of Myers-Nakai Theorem in the context of complete reversible Finsler manifolds.

Item Type: | Article |
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Uncontrolled Keywords: | Riemannian-manifolds; isometries |

Subjects: | Sciences > Mathematics > Differential geometry |

ID Code: | 23180 |

Deposited On: | 15 Oct 2013 10:18 |

Last Modified: | 12 Dec 2018 15:12 |

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