Real analytic approximation of Lipschitz functions on Hilbert space and other Banach spaces
| Autor: | Azagra, D., Fry, R., Keener, L. |
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| Rok vydání: | 2010 |
| Předmět: | |
| Druh dokumentu: | Working Paper |
| Popis: | Let $X$ be a separable Banach space with a separating polynomial. We show that there exists $C\geq 1$ (depending only on $X$) such that for every Lipschitz function $f:X\rightarrow\mathbb{R}$, and every $\epsilon>0$, there exists a Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that $|f(x)-g(x)|\leq \epsilon$ and $\textrm{Lip}(g)\leq C\textrm{Lip}(f)$. This result is new even in the case when $X$ is a Hilbert space. Furthermore, in the Hilbertian case we also show that $C$ can be assumed to be any number greater than 1. Comment: Updated version with a sharper result in the Hilbertian case. One thin tube is enough. Some misprints corrected |
| Databáze: | arXiv |
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