  We present here a new method for approximating functions defined on
superreflexive Banach spaces by differentiable functions with $\alpha$-H\"older
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 ${\Cal{C}}^{1,\alpha}$ 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 {\sl extended
inf-convolution} formulas and convexity properties such as the preservation of
smoothness for the convex envelope of certain differentiable functions.
