%0 Journal Article
%A Sánchez González, Luis Francisco
%A Jiménez Sevilla, María Del Mar
%T On some problems on smooth approximation and smooth extension of Lipschitz functions on Banach–Finsler manifolds
%D 2011
%@ 0362-546X
%U https://hdl.handle.net/20.500.14352/42295
%X 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.
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