Smooth Approximation of Lipschitz Functions on Finsler Manifolds - M. I. Garrido; J. A. Jaramillo; Y. C. Rangel (Journal of Function Spaces and Applications)
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 ε:M→(0,∞) and each r>0, there exists a C1-smooth Lipschitz function g:M→ℝ such that |f(x)-g(x)|≤ε(x), for every x∈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 Cb1(M) of all C1 functions with bounded derivative on a complete quasi-reversible Finsler manifold M, we obtain a characterization of algebra isomorphisms T:Cb1(N)→Cb1(M) as composition operators. From this we obtain a variant of Myers-Nakai Theorem in the context of complete reversible Finsler manifolds.