Abstract

We show that if XX is a Banach space whose dual X∗X^{*} has an equivalent locally uniformly rotund (LUR) norm, then for every open convex U⊆XU\subseteq X, for every real number ε>0\varepsilon >0, and for every continuous and convex function f:U→Rf:U \rightarrow \mathbb{R} (not necessarily bounded on bounded sets) there exists a convex function g:U→Rg:U \rightarrow \mathbb{R} of class C1(U)C^1(U) such that f−ε≤g≤ff-\varepsilon\leq g\leq f on U.U. We also show how the problem of global approximation of continuous (not necessarily bounded on bounded sets) convex functions by CkC^k smooth convex functions can be reduced to the problem of global approximation of Lipschitz convex functions by CkC^k smooth convex functions.

Author information

Contact details are reproduced from the original publication and may be historical.

Daniel Azagra

ICMAT, Dep. de Análisis Matemático, Facultad Ciencias Matemáticas, Universidad Complutense, 28040 Madrid, Spain
and: CSIC

azagra@mat.ucm.es

Carlos Mudarra

ICMAT, Calle Nicolás Cabrera 13-15, Campus de Cantoblanco, 28049 Madrid, Spain
and: CSIC

carlos.mudarra@icmat.es

Keywords

  • Approximation
  • convex function
  • differentiable function
  • Banach space

Mathematics Subject Classification

46B20, 52A99, 26B25, 41A30

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Suggested citation

D. Azagra, C. Mudarra. “Global Approximation of Convex Functions by Differentiable Convex Functions on Banach Spaces.” Journal of Convex Analysis 22 (2015), No. 4, 1197–1205. https://doi.org/10.68381/jca22062

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Published by Heldermann Verlag, 2015. Rights now held by Banach Press LLC.