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Summary: Approximation of functions and their derivatives
by analytic maps on certain Banach spaces
D. Azagra, R. Fry, and L. Keener
Abstract. Let X be a separable Banach space which admits a sepa-
rating polynomial. Let f : X R be bounded, Lipschitz, and C1
with
uniformly continuous derivative. Then for each > 0, there exists an
analytic function g : X R with |g - f| < and g - f < .
1. Introduction
The problem of approximating a smooth function and its derivatives
by a function of higher order smoothness on a Banach space X has been
investigated by several authors, although the number of such results is lim-
ited. When X is finite dimensional excellent results are known, and in
fact Whitney in his classical paper [W] provides essentially a complete an-
swer by showing: for every Ck function f : Rn Rm and every contin-
uous : Rn (0, +) there exists a real analytic function g such that
Djg(x) - Djf(x) (x) for all x Rn and j = 1, ..., k. This is the
so-called Ck fine approximation of f.
The first results for X infinite dimensional concern the smooth, non-
analytic case, and are due to Moulis [M]. She proves, in particular, a C1
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