Bernstein–Kantorovich operators, approximation and shape preserving properties

Abstract

Inspired by the construction of Bernstein and Kantorovich operators, we introduce a family of positive linear operators Kn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal{K}_n$$\\end{document} preserving the affine functions. Their approximation properties are investigated and compared with similar properties of other operators. We determine the central moments of all orders of Kn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {K}}}_n$$\\end{document} and use them in order to establish Voronovskaja type formulas. A special attention is paid to the shape preserving properties. The operators Kn\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$${{\\mathcal {K}}}_n$$\\end{document} preserve monotonicity, convexity, strong convexity and approximate concavity. They have also the property of monotonic convergence under convexity. All the established inequalities involving convex functions can be naturally interpreted in the framework of convex stochastic ordering.