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On regularization in superreflexive Banach spaces by infimal convolution formulas. (English) Zbl 0918.46014

Summary: We present here a new method for approximating functions defined on superreflexive Banach spaces by differentiable functions with \(\alpha\)-Hölder derivatives (for some \(0<\alpha\leq 1\)). 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 \(\Delta\)-convex \({\mathcal C}^{1,\alpha}\) 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.

MSC:

46B20 Geometry and structure of normed linear spaces
46B10 Duality and reflexivity in normed linear and Banach spaces
41A30 Approximation by other special function classes
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