CONVERGENCES DES FONCTIONS CONVEXES ET APPROXIMATIONS INF-CONVOLUTIVES GENERALISEES

D. Mentagui et K. el Hajioui

Abstract: Let $\Phi:X\to \mathbb R^+$ be a kernel bounded on bounded subsets of a normed linear space $X$ and $f$ be a function in $\Gamma(X)$. 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 prove that the slice convergence of a sequence $(f^n)_n$ in $\Gamma(X)$ is equivalent on the one hand to the convergence in the same sense of its sequences of inf-convolution approximates of sufficiently small parameters associated to $\Phi$, and on the other hand to the pointwise convergence of the regularized sequences defined in the theorem 3.10 of this paper. As well, we show that the Attouch-Wets convergence of $(f^n)_n$ is equivalent to the convergence in the same sense of its approximate sequences when the parameters $\lambda$ converge to $0$; which is also equivalent to their uniform convergence on bounded subsets of $X$. Then, we generalize in particular the main results of G. Beer [12] established in the case of Baire-Wijsman regularizations($\Phi=\|\!\cdot\!\|$).

Keywords: convex functions; inf-convolution approximates; slice convergence; Attouch-Wets convergence; pointwise convergence; uniform convergence

Classification (MSC2000): 52A41; 54B20; 40A30

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