Abstract of a paper by D. Azagra, R. Fry, and L. Keener
This is an announcement for the paper "Real analytic approximation of Lipschitz functions on Hilbert space and other Banach spaces" by D. Azagra, R. Fry, and L. Keener. Abstract: Let $X$ be a separable Banach space with a separating polynomial. We show that there exists $C\geq 1$ such that for every Lipschitz function $f:X\rightarrow\mathbb{R}$, and every $\varepsilon>0$, there exists a Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that $|f(x)-g(x)|\leq \varepsilon$ and $\textrm{Lip}(g)\leq C\textrm{Lip}(f)$. This result is new even in the case when $X$ is a Hilbert space. Furthermore we characterize the class of Banach spaces having this approximation property as those Banach spaces $X$ having a Lipschitz, real-analytic separating function (meaning a Lipschitz, real analytic function $Q:X\to [0, +\infty)$ such that $Q(0)=0$ and $Q(x)\geq \|x\|$ for $\|x\|\geq 1$). Archive classification: math.FA Mathematics Subject Classification: 46B20 Remarks: 40 pages Submitted from: dazagra@gmail.com The paper may be downloaded from the archive by web browser from URL http://front.math.ucdavis.edu/1005.1050 or http://arXiv.org/abs/1005.1050
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alspach@fourier.math.okstate.edu