Convergences des Fonctions Convexes et Approximations Inf-convolutives Generalisees

D. Mentagui et K. el Hajioui

Publications de l'Institut Mathématique, _N_S_72 (2002) no. 86, p. 123 .

Résumé

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($\Phi=\|\!\cdot\!\|$).

Détail

DOI : 10.2298/PIM0272123M

Classification : 52A41 54B20 40A30