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Global Approximation of Convex Functions by Differentiable Convex Functions on Banach Spaces.

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Azagra Rueda, Daniel and Mudarra, C. (2015) Global Approximation of Convex Functions by Differentiable Convex Functions on Banach Spaces. Journal of Convex Analysis, 22 (4). pp. 1197-1205. ISSN 0944-6532

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Official URL: http://www.heldermann-verlag.de/jca/jca22/jca1499_b.pdf


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Abstract

We show that if X is a Banach space whose dual X* has an equivalent locally uniformly rotund (LUR) norm, then for every open convex U subset of X, for every real number epsilon > 0, and for every continuous and convex function f : U -> R (not necessarily bounded on bounded sets) there exists a convex function g : U -> R of class C-1 (U) such that f - epsilon <= g <= f on U. We also show how the problem of global approximation of continuous (not necessarily bounded on bounded sets) convex functions by C-k smooth convex functions can be reduced to the problem of global approximation of Lipschitz convex functions by C-k smooth convex functions.


Item Type:Article
Uncontrolled Keywords:Approximation; Convex function; Differentiable function; Banach space
Subjects:Sciences > Mathematics > Functional analysis and Operator theory
ID Code:36175
Deposited On:01 Apr 2016 11:54
Last Modified:01 Apr 2016 11:54

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