The discrete analogue of high-order differential operator and its application to finding coefficients of optimal quadrature formulas

Abstract

The discrete analog of the differential operator plays a significant role in constructing interpolation, quadrature, and cubature formulas. In this work, we consider a discrete analog Dm(hβ)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$D_{m}(h\\beta )$\\end{document} of the differential operator d2mdx2m+1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$\\frac{d^{2m}}{dx^{2m}}+1$\\end{document} designed specifically for even natural numbers m. The operator’s effectiveness in constructing an optimal quadrature formula in the L2(2,0)(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L_{2}^{(2,0)}(0,1)$\\end{document} space is demonstrated. The errors of the optimal quadrature formula in the W2(2,1)(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$W_{2}^{(2,1)}(0,1)$\\end{document} space and in the L2(2,0)(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$L_{2}^{(2,0)}(0,1)$\\end{document} space are compared numerically. The numerical results indicate that the optimal quadrature formula constructed in this work has a smaller error than the one constructed in the W2(2,1)(0,1)\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$W_{2}^{(2,1)}(0,1)$\\end{document} space.