On regularization in superreflexive Banach spaces by infimal convolution formulas

Abstract

We present here a new method for approximating functions defined on superreflexive Banach spaces by differentiable functions with α-H¨older 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 ∆-convex C1,α functions converging uniformly on bounded sets to f 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.