On regularization in superreflexive Banach spaces by infimal convolution formulas

Volume 129 / 1998

Manuel Cepedello-Boiso Studia Mathematica 129 (1998), 265-284 DOI: 10.4064/sm-129-3-265-284


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.


  • Manuel Cepedello-Boiso

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