On regularization in superreflexive Banach spaces by infimal convolution formulas

Manuel Cepedello-Boiso

Studia Mathematica (1998)

  • Volume: 129, Issue: 3, page 265-284
  • ISSN: 0039-3223

Abstract

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We present here a new method for approximating functions defined on superreflexive Banach spaces by differentiable functions with α-Hölder derivatives (for some 0 < α≤ 1). The smooth approximation is given by means of an explicit formula enjoying good properties from the minimization point of view. For instance, for any function f which is bounded below and uniformly continuous on bounded sets this formula gives a sequence of Δ-convex C 1 , α functions converging to f uniformly on bounded sets and preserving the infimum and the set of minimizers of f. The techniques we develop are based on the use of extended inf-convolution formulas and convexity properties such as the preservation of smoothness for the convex envelope of certain differentiable functions.

How to cite

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Cepedello-Boiso, Manuel. "On regularization in superreflexive Banach spaces by infimal convolution formulas." Studia Mathematica 129.3 (1998): 265-284. <http://eudml.org/doc/216504>.

@article{Cepedello1998,
abstract = {We present here a new method for approximating functions defined on superreflexive Banach spaces by differentiable functions with α-Hölder derivatives (for some 0 < α≤ 1). The smooth approximation is given by means of an explicit formula enjoying good properties from the minimization point of view. For instance, for any function f which is bounded below and uniformly continuous on bounded sets this formula gives a sequence of Δ-convex $C^\{1,α\}$ functions converging to f uniformly on bounded sets and preserving the infimum and the set of minimizers of f. The techniques we develop are based on the use of extended inf-convolution formulas and convexity properties such as the preservation of smoothness for the convex envelope of certain differentiable functions.},
author = {Cepedello-Boiso, Manuel},
journal = {Studia Mathematica},
keywords = {regularization in Banach spaces; convex functions; approximating functions; superreflexive Banach spaces; -Hölder derivatives; extended inf-convolution formulas},
language = {eng},
number = {3},
pages = {265-284},
title = {On regularization in superreflexive Banach spaces by infimal convolution formulas},
url = {http://eudml.org/doc/216504},
volume = {129},
year = {1998},
}

TY - JOUR
AU - Cepedello-Boiso, Manuel
TI - On regularization in superreflexive Banach spaces by infimal convolution formulas
JO - Studia Mathematica
PY - 1998
VL - 129
IS - 3
SP - 265
EP - 284
AB - We present here a new method for approximating functions defined on superreflexive Banach spaces by differentiable functions with α-Hölder derivatives (for some 0 < α≤ 1). The smooth approximation is given by means of an explicit formula enjoying good properties from the minimization point of view. For instance, for any function f which is bounded below and uniformly continuous on bounded sets this formula gives a sequence of Δ-convex $C^{1,α}$ functions converging to f uniformly on bounded sets and preserving the infimum and the set of minimizers of f. The techniques we develop are based on the use of extended inf-convolution formulas and convexity properties such as the preservation of smoothness for the convex envelope of certain differentiable functions.
LA - eng
KW - regularization in Banach spaces; convex functions; approximating functions; superreflexive Banach spaces; -Hölder derivatives; extended inf-convolution formulas
UR - http://eudml.org/doc/216504
ER -

References

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