RT Journal Article
T1 On some problems on smooth approximation and smooth extension of Lipschitz functions on Banach–Finsler manifolds
A1 Sánchez González, Luis Francisco
A1 Jiménez Sevilla, María Del Mar
AB Let us consider a Riemannian manifold M (either separable or non-separable). We prove that, for every ε>0, every Lipschitz function f:M→R can be uniformly approximated by a Lipschitz, C1-smooth function g with . As a consequence, every Riemannian manifold is uniformly bumpable. These results extend to the non-separable setting those given in [1] for separable Riemannian manifolds. The results are presented in the context of Cℓ Finsler manifolds modeled on Banach spaces. Sufficient conditions are given on the Finsler manifold M (and the Banach space X where M is modeled), so that every Lipschitz function f:M→R can be uniformly approximated by a Lipschitz, Ck-smooth function g with  (for some C depending only on X). Some applications of these results are also given as well as a characterization, on the separable case, of the class of Cℓ Finsler manifolds satisfying the above property of approximation. Finally, we give sufficient conditions on the C1 Finsler manifold M and X, to ensure the existence of Lipschitz and C1-smooth extensions of every real-valued function f defined on a submanifold N of M provided f is C1-smooth on N and Lipschitz with the metric induced by M.
PB Elsevier
SN 0362-546X
YR 2011
FD 2011-07
LK https://hdl.handle.net/20.500.14352/42295
UL https://hdl.handle.net/20.500.14352/42295
LA eng
NO The authors wish to thank Jesús Jaramillo for many helpful discussions. Supported in part by DGES (Spain) ProjectMTM2009-07848. L. Sánchez-González has also been supported by grant MEC AP2007-00868.
NO DGES
NO MEC
DS Docta Complutense
RD 9 abr 2025