Publications de l'Institut Mathématique (Beograd) Vol. 72(86), pp. 123136 (2002) 

CONVERGENCES DES FONCTIONS CONVEXES ET APPROXIMATIONS INFCONVOLUTIVES GENERALISEESD. Mentagui et K. el HajiouiLaboratoire d'Analyse Convexe et Variationnelle, Systèmes Dynamiques et Processus Stochastiques, Faculté des Sciences, Kénitra, MarocAbstract: 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 infconvolution approximates of $f$ of parameters $\lambda>0$ associated to $\Phi$ are the functions defined for each $x\in X$ by $f_\lambda(x)=\inf\{f(u)+\Phi(\frac{xu}\lambda):u\in X\}$. 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 infconvolution 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 AttouchWets 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 BaireWijsman regularizations($\Phi=\\!\cdot\!\$). Keywords: convex functions; infconvolution approximates; slice convergence; AttouchWets convergence; pointwise convergence; uniform convergence Classification (MSC2000): 52A41; 54B20; 40A30 Full text of the article:
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