Connections between the Approximation Orders of Positive Linear Operators and Their Max-Product Counterparts. (18th August 2021)

• Record Type: Journal Article
• Authors: Coroianu, Lucian; Costarelli, Danilo; Gal, Sorin G.; Vinti, Gianluca
• Is Part Of: Numerical functional analysis and optimization. Volume 42:Number 11(2021)
• Page Start: 1263
• Page End: 1286
• Publication Date: 2021-08-18
• Subjects: K -functional -- Lp-approximation -- max-product Bernstein-type operators -- max-product interpolation operators -- max-product Kantorovich-type operators -- max-product operators -- modulus of continuity -- positive linear operators; 41A35 -- 41A25 -- 41A20; Functional analysis -- Periodicals; Numerical analysis -- Periodicals; Mathematical optimization -- Periodicals; Numerical Analysis, Computer-Assisted; 515.705
• DOI: 10.1080/01630563.2021.1954018
• Languages: English
• ISSNs: 0163-0563

Abstract

Abstract: In this study we establish some direct connections between arbitrary positive linear operators and their corresponding nonlinear (more exactly sublinear) max-product versions, with respect to uniform and L p convergence. There are numerous concrete examples of approximation operators, such as Bernstein-type operators, neural network operators, sampling operators and others, where the linear and the max-product versions converge both uniformly. Here, from the quantitative uniform approximation result for an arbitrary sequence of positive linear operators, we deduce by a simple general method a quantitative uniform approximation result for its max-product counterpart. We also establish convergence with respect to the L p -norm involving the well-known K -functionals, when the supremum of the kernel is bounded from below. Our results cover the cases of bounded and unbounded domains and the case of the Kantorovich variants of the considered operators. Applications to some max-product operators are presented.