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