Real analytic approximation of Lipschitz functions on Hilbert space and other Banach spaces

TL;DR

This paper proves that Lipschitz functions on certain Banach spaces, including Hilbert spaces, can be uniformly approximated by real analytic Lipschitz functions with controlled Lipschitz constants, extending previous results.

Contribution

It establishes the existence of real analytic approximations for Lipschitz functions on Banach spaces with separating polynomials, including Hilbert spaces, with explicit Lipschitz bounds.

Findings

Existence of real analytic approximations with controlled Lipschitz constants

Approximation results hold for Banach spaces with separating polynomials

In Hilbert spaces, the Lipschitz constant can be arbitrarily close to the original

Abstract

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.