Smooth Approximation of Lipschitz Functions on Finsler Manifolds
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 ea...
| Main Authors: | , , |
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| Format: | Article |
| Language: | English |
| Published: |
Hindawi Limited
2013-01-01
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| Series: | Journal of Function Spaces and Applications |
| Online Access: | http://dx.doi.org/10.1155/2013/164571 |
| Summary: | 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. |
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| ISSN: | 0972-6802 1758-4965 |