Essential supremum and supremum of summable functions. (December 2019)

• Record Type: Journal Article
• Authors: Phu, H. X.; Hoffmann, A.
• Is Part Of: Numerical functional analysis and optimization. Volume 17:Number 1/2(1996)
• Page Start: 161
• Page End: 180
• Publication Date: 2019-12
• Subjects: Essential supremum -- supremum -- global maximum -- summable function -- convex function; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630569608816689
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract : Let D ≌ RN, 0 < μ(D) < +∞ and f : D → R is an arbitary summable function. Then the function [image omitted] is continuous, non-negative, non-increasing, convex, and has almost everywhere the derivative [image omitted] . Further on, it holds ess sup ƒ = sup{α ↦ R : F(α) > 0}, where ess sup ƒ denotes the essential supremum of ƒ. These properties can be used for computing ess sup ƒ. As example, two algorithms are stated. If the function ƒ is dense, or lower semicontinuous, or if -ƒ is robust, then sup ƒ = ess sup ƒ. In this case, the algorithms mentioned can be applied for determining the supremum of ƒ, i.e., also the global maximum of ƒ if it exists.