Summary: Let \(\Phi:X\to\mathbb R^+\) be a kernel bounded on bounded subsets of a Banach space \(X\) and \(f\) be a proper lower semicontinuous convex function defined on \(X\). The inf-convolution approximates of \(f\) of parameters \(\lambda>0\) associated to \(\Phi\) are the functions defined for each \(x\in X\) by \(f_\lambda(x):= \inf\{f(u)+ \Phi(\frac{x-u}{\lambda}): u\in X\}\).
In this article, we express the Attouch-Wets convergence of sequences in \(\Gamma(X)\) by the convergence of their associated sequences of inf-convolution approximates. More precisely, we prove that the Attouch-Wets convergence of sequences in \(\Gamma(X)\) is equivalent to the convergence in the same sense of their associated sequences of inf-convolution approximates of sufficiently small parameters, which is also equivalent to their uniform convergence on bounded subsets of \(X\). If \(\Phi=\|\cdot\|\), then we find the characterization of the Attouch-Wets convergence in terms of the uniform convergence on bounded subsets of the Baire-Wijsman regularizations; which plays a key role in the characterization of the previous convergence by the uniform lower semicontinuity on bounded subsets of a certain differential operator [see G. Beer, Numer. Funct. Anal. Optimization 15, No. 1–2, 31–46 (1994; Zbl 0817.49018)].