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A note on the range of the derivatives of analytic approximations of uniformly continuous functions on co

Jiménez Sevilla, María del Mar (2008) A note on the range of the derivatives of analytic approximations of uniformly continuous functions on co. Journal of Mathematical Analysis and Applications, 348 (2). pp. 573-580. ISSN 0022-247X

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Abstract

This paper is a contribution to the body of results concerning the size of the set of derivatives of differentiable functions on a Banach space. The results so far have consisted of examples of highly differentiable bump functions (or functions approximating a given continuous function) whose set of derivatives either is surprisingly small or has a given shape. The paper under review treats the case of analytic smoothness. The main result states that every uniformly continuous function on c0 (more generally on a space with property (K)) can be approximated by a real-analytic function whose set of derivatives is contained in T
p>0 lp. This is a significant step forward, as analytic functions are substantially harder to deal with than C1 smooth ones. Indeed, a local perturbation of an analytic function necessarily changes the values of the function everywhere. Also of particular value is the quite precise and elegant description of the set of derivatives of the
approximating function.

Item Type:Article
Uncontrolled Keywords:Approximation by analytic functions; Range of the derivatives
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:13813
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Deposited On:10 Nov 2011 11:16
Last Modified:06 Feb 2014 09:54

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