| | |
Summary: arXiv:1012.4339v2[math.FA]31Dec2010
REAL ANALYTIC APPROXIMATIONS WHICH ALMOST
PRESERVE LIPSCHITZ CONSTANTS OF FUNCTIONS
DEFINED ON THE HILBERT SPACE
D. AZAGRA, R. FRY, AND L. KEENER
Abstract. Let X be a separable real Hilbert space. We show that for
every Lipschitz function f : X R, and for every > 0, there exists a
Lipschitz, real analytic function g : X R such that |f(x) - g(x)|
and Lip(g) Lip(f) + .
In a recent paper [AFK1] we proved that for every separable Banach
space X having a separating polynomial there exists a constant C 1 such
that, for every Lipschitz function f : X R and every > 0 there exists
a Lipschitz, real analytic function f : X R such that |f - g| and
Lip(g) CLip(f). It is natural to wonder whether the constant C can be
assumed to be 1 (as in the finite-dimensional case), or at least any number
greater that 1. The aim of this note is to prove that the latter is indeed true
in the case when X is a Hilbert space.
Theorem 1. Let X be a separable real Hilbert space. For every Lipschitz
function f : X R, and for every > 0, there exists a Lipschitz, real
analytic function g : X R such that |f(x) - g(x)| and Lip(g)
|
