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Summary: arXiv:1005.1050v2[math.FA]8May2010
REAL ANALYTIC APPROXIMATION OF LIPSCHITZ
FUNCTIONS ON HILBERT SPACE AND OTHER BANACH
SPACES
D. AZAGRA, R. FRY, AND L. KEENER
Abstract. Let X be a separable Banach space with a separating poly-
nomial. We show that there exists C 1 such that for every Lips-
chitz function f : X R, and every > 0, there exists a Lipschitz,
real analytic function g : X R such that |f(x) - g(x)| and
Lip(g) CLip(f). This result is new even in the case when X is
a Hilbert space. Furthermore, the proof of this results works for all
Banach spaces X having a separating function Q (meaning a function
Q : X [0, +) such that Q(0) = 0 and Q(x) x for x 1)
which has a Lipschitz, holomorphic extension Q to a uniformly wide
neighborhood V = {x + iy : x X, y < } of X in the complexifica-
tion X.
1. Introduction and main results
Not much is known about the natural question of approximating func-
tions by real analytic functions on a real Banach space X. When X is
finite dimensional, a famous paper of Whitney's [W] provides a completely
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