Mentagui, D.; El Hajioui, K. Convergence of convex functions and generalized inf-convolutive approximations. (Convergences des fonctions convexes et approximations inf-convolutives généralisées.) (French) Zbl 1086.49012 Publ. Inst. Math., Nouv. Sér. 72(86), 123-136 (2002). The authors prove that the slice convergence of a sequence \((f^n)_n\) of appropriate convex functions on a normed linear space \(X\) is equivalent to the slice convergence of its sequence of inf-convolution approximates of sufficiently small parameters associated to a kernel \(\Phi:X\to\mathbb R^+\), where the inf-convolution approximation of a convex \(f\) of parameter \(\lambda\) is \(f_\lambda=\inf\{f(u)+\Phi((x-u)/\lambda)\}\). It is equivalent with the pointwise convergence of certain regularized sequences. It is also shown that the Attouch–Wets convergence of \((f^n)_n\) is equivalent to the same type of convergence of its approximate sequences when \(\lambda\) converges to \(0\) and this is equivalent to the uniform convergence on bounded subsets of \(X\). With this, the authors generalize cases known earlier for \(\Phi =\| \cdot\| \). Reviewer: Stevan Pilipović (Novi Sad) Cited in 2 Documents MSC: 49J45 Methods involving semicontinuity and convergence; relaxation 26B25 Convexity of real functions of several variables, generalizations 52A41 Convex functions and convex programs in convex geometry 54B20 Hyperspaces in general topology 40A30 Convergence and divergence of series and sequences of functions 46B20 Geometry and structure of normed linear spaces Keywords:convex functions; Attouch-Wets convergence; slice convergence PDFBibTeX XMLCite \textit{D. Mentagui} and \textit{K. El Hajioui}, Publ. Inst. Math., Nouv. Sér. 72(86), 123--136 (2002; Zbl 1086.49012) Full Text: DOI EuDML