A generalization of a theorem of Brass and Schmeisser

Abstract

Let n be an odd positive integer. It was proved by Brass and Schmeisser that for every quadrature Q=α1f(x1)+⋯+αmf(xm)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathcal {Q}=\\alpha _1f(x_1)+\\dots +\\alpha _mf(x_m)$$\\end{document}(with positive weights) of order at least n+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n+1$$\\end{document} and for every n-\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n-$$\\end{document}convex function f, the value of Q on f lies between the values of Gauss-Legendre and Gauss-Lobatto quadratures of order n+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$n+1$$\\end{document} calculated for the same function f. We generalize this result in two directions, replacing Q by an integral with respect to a given measure and allowing the number n to any positive integer (for even n Gauss-Radau quadratures replace Gauss-Legendre and Gauss-Lobatto ones).