Real analytic approximations which almost preserve Lipschitz constants of functions defined on the Hilbert space

TL;DR

This paper demonstrates that any Lipschitz function on a separable real Hilbert space can be approximated arbitrarily closely by a real analytic Lipschitz function with nearly preserved Lipschitz constant.

Contribution

It introduces a method to approximate Lipschitz functions with real analytic functions while nearly maintaining the original Lipschitz constant.

Findings

Approximation of Lipschitz functions by real analytic functions within any epsilon

Preservation of Lipschitz constants within epsilon during approximation

Applicable to functions defined on separable real Hilbert spaces

Abstract

Let $X$ be a separable real Hilbert space. We show that for every Lipschitz function $f:X\rightarrow\mathbb{R}$, and for every $\epsilon>0$, there exists a Lipschitz, real analytic function $g:X\rightarrow\mathbb{R}$ such that $|f(x)-g(x)|\leq \epsilon$ and $\textrm{Lip}(g)\leq \textrm{Lip}(f)+\epsilon$.