Open Access

Two-Point Quadrature Rules for Riemann–Stieltjes Integrals with Lp–error estimates

   | May 16, 2019

Cite

[1] M.W. Alomari and A. Guessab, Lp–error bounds of two and three–point quadrature rules for Riemann–Stieltjes inegrals, Moroccan J. Pure & Appl. Anal. (MJPAA), accepted.Search in Google Scholar

[2] M.W. Alomari, Two-point Ostrowski’s inequality, Results in Mathematics, 72 (3) (2017), 1499–1523.10.1007/s00025-017-0720-6Search in Google Scholar

[3] M.W. Alomari, On Beesack–Wirtinger inequality, Results in Mathematics, 72 (3) (2017), 1213–1225.10.1007/s00025-016-0644-6Search in Google Scholar

[4] M.W. Alomari and S.S. Dragomir, Mercer-Trapezoid rule for Riemann–Stieltjes integral with applications, Journal of Advances in Mathematics, 2 (2) (2013), 67–85.Search in Google Scholar

[5] M.W. Alomari, A companion of Ostrowski’s inequality for the Riemann-Stieltjes integral abf(t)du(t)$\int_a^b {f(t)} \,du\,(t)$, where f is of bounded variation and u is of r-H-Hölder type and applications, Appl. Math. Comput., 219 (2013), 4792–4799.Search in Google Scholar

[6] M.W. Alomari, New sharp inequalities of Ostrowski and generalized trapezoid type for the Riemann–Stieltjes integrals and applications, Ukrainian Mathematical Journal, 65 (7) 2013, 895–916.10.1007/s11253-013-0837-zSearch in Google Scholar

[7] M.W. Alomari, Approximating the Riemann-Stieltjes integral by a three-point quadrature rule and applications, Konuralp J. Math., 2 (2) (2014), 22?34.Search in Google Scholar

[8] M.W. Alomari, Two point Gauss-Legendre quadrature rule for Riemann-Stieltjes integrals, Preprint (2014). Avaliable at https://arxiv.org/pdf/1402.4982.pdfSearch in Google Scholar

[9] M.W. Alomari, A sharp companion of Ostrowski’s inequality for the Riemann–Stieltjes integral and applications, Ann. Univ. Paedagog. Crac. Stud. Math., 15 (2016), 69–78.Search in Google Scholar

[10] M.W. Alomari and S.S. Dragomir, A three-point quadrature rule for the Riemann-Stieltjes integral, Southeast Asian Bulletin Journal of Mathematics, in pressSearch in Google Scholar

[11] N.S. Barnett, S.S. Dragomir and I. Gomma, A companion for the Ostrowski and the generalised trapezoid inequalities, Mathematical and Computer Modelling, 50 (2009), 179–187.10.1016/j.mcm.2009.04.005Search in Google Scholar

[12] N.S. Barnett, W.-S. Cheung, S.S. Dragomir, A. Sofo, Ostrowski and trapezoid type inequalities for the Stieltjes integral with Lipschitzian integrands or integrators, Computer & Mathematics with Applications, 57 (2009), 195–201.10.1016/j.camwa.2007.07.021Search in Google Scholar

[13] P. Cerone, S.S. Dragomir, New bounds for the three-point rule involving the Riemann-Stieltjes integrals, in: C. Gulati, et al. (Eds.), Advances in Statistics Combinatorics and Related Areas, World Science Publishing, 2002, pp. 53–62.10.1142/9789812776372_0006Search in Google Scholar

[14] P. Cerone, S.S. Dragomir, Approximating the Riemann–Stieltjes integral via some moments of the integrand, Mathematical and Computer Modelling, 49 (2009), 242–248.10.1016/j.mcm.2008.02.011Search in Google Scholar

[15] S.S. Dragomir, On the Ostrowski inequality for Riemann–Stieltjes integral abf(t)du(t)$\int_a^b {f(t)} \,du\,(t)$ where f is of Hölder type and u is of bounded variation and applications, J. KSIAM, 5 (2001), 35–45.Search in Google Scholar

[16] S.S. Dragomir, On the Ostrowski’s inequality for Riemann-Stieltes integral and applications, Korean J. Comput. & Appl. Math., 7 (2000), 611–627.10.1007/BF03012272Search in Google Scholar

[17] S.S. Dragomir, C. Buşe, M.V. Boldea, L. Braescu, A generalisation of the trapezoid rule for the Riemann- Stieltjes integral and applications, Nonlinear Anal. Forum 6 (2) (2001) 33–351.Search in Google Scholar

[18] S.S. Dragomir, Some inequalities of midpoint and trapezoid type for the Riemann-Stieltjes integral, Nonlinear Anal. 47 (4) (2001) 2333–2340.10.1016/S0362-546X(01)00357-1Search in Google Scholar

[19] S.S. Dragomir, Approximating the Riemann-Stieltjes integral in terms of generalised trapezoidal rules, Nonlinear Anal. TMA 71 (2009) e62–e72.10.1016/j.na.2008.10.004Search in Google Scholar

[20] S.S. Dragomir, Approximating the Riemann-Stieltjes integral by a trapezoidal quadrature rule with applications, Mathematical and Computer Modelling 54 (2011) 243–260.10.1016/j.mcm.2011.02.006Search in Google Scholar

[21] W. Gautschi, On generating orthogonal polynomials, SIAM J. Sci. Stat. Comput., 3 (1982), 289–317.10.1137/0903018Search in Google Scholar

[22] A. Guessab and G. Schmeisser, Sharp integral inequalities of the Hermite-Hadamard type, J. Approx. Th., 115 (2002), 260–288.10.1006/jath.2001.3658Search in Google Scholar

[23] R. M. Dudley, Frechet Differentiability, p-variation and uniform Donsker classes, The Annals of Probability, 20 (4) (1992), 1968–1982.10.1214/aop/1176989537Search in Google Scholar

[24] B.I. Golubov, On criteria for the continuity of functions of bounded p-variation, Sibirskii Matematicheskii Zhurnal, 13 (5) (1972), 1002?–1015.10.1007/BF00968383Search in Google Scholar

[25] B.I. Golubov, On functions of bounded p-variation, Mathematics of the USSR-Izvestiya, 2 (4) (1968), 799?–819.10.1070/IM1968v002n04ABEH000669Search in Google Scholar

[26] V.I. Kolyada and M. Lind, On functions of bounded p-variation, J. Math. Anal. Appl., 356 (2009), 582?–604.10.1016/j.jmaa.2009.03.042Search in Google Scholar

[27] P.R. Mercer, Hadamard’s inequality and trapezoid rules for the Riemann?Stieltjes integral, J. Math. Anal. Appl. 344 (2008) 921–926.10.1016/j.jmaa.2008.03.026Search in Google Scholar

[28] M. Munteanu, Quadrature formulas for the generalized Riemann-Stieltjes integral, Bull. Braz. Math. Soc. (N.S.) 38 (1) (2007) 39?-50.10.1007/s00574-007-0034-5Search in Google Scholar

[29] I.P. Natanson, Theory of Functions of a Real Variable, Vol I, Translated from the Russian by L.F. Boron and E. Hewitt, Frederick Ungar Publishing, New York, 1955.Search in Google Scholar

[30] N. Wiener, The quadratic variation of a function and its Fourier coefficients, Massachusetts J. Math., 3 (1924), 72–94.Search in Google Scholar

[31] L.C. Young, An inequality of the Hölder type, connected with Stieltjes integration, Acta Math., 67 (1936), 251–282.10.1007/BF02401743Search in Google Scholar

[32] M. Tortorella, Closed Newton-Cotes quadrature rules for Stieltjes integrals and numerical convolution of life distributions, SIAM J. Sci. Stat. Comput. 11 (1990) 732–748.10.1137/0911043Search in Google Scholar