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-FINE APPROXIMATION OF FUNCTIONS ON BANACH SPACES WITH UNCONDITIONAL BASIS
 

Summary: C1
-FINE APPROXIMATION OF FUNCTIONS ON BANACH
SPACES WITH UNCONDITIONAL BASIS
by DANIEL AZAGRA, JAVIER G ´OMEZ GIL§, JES ´US A. JARAMILLO¶,
MAURICIO LOVO
(Departamento de An´alisis Matem´atico, Facultad de Ciencias Matem´aticas, Universidad
Complutense, 28040 Madrid, Spain)
and ROBB FRY
(Department of Mathematics and Computer Science, St Francis Xavier University, PO Box
5000, Antigonish, Nova Scotia B2G 2W5, Canada)
[Received 16 September 2003]
Abstract
We show that if X is a Banach space having an unconditional basis and a C p-smooth Lipschitz
bump function, then for every C1-smooth function f from X into a Banach space Y, and for
every continuous function : X (0, ), there exists a C p-smooth function g : X Y such
that f - g and f - g .
1. Introduction
Given a Fr´echet smooth function f between Banach spaces, we consider in this note the problem
of uniformly approximating both f and its derivative by functions with a higher order of
differentiability. More generally, if f : X Y is a Ck-smooth function between Banach spaces,

  

Source: Azagra Rueda, Daniel - Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid

 

Collections: Mathematics