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Approximating the finite Hilbert transform via some companions of Ostrowski’s inequalities. (English) Zbl 1355.26030

Summary: The finite Hilbert transform is a helpful tool in fields like aerodynamics, the theory of elasticity, and other areas of the engineering sciences. In this paper, by using some companions of Ostrowski’s inequalities for absolutely continuous functions due to Dragomir, we give some new inequalities and approximations for the finite Hilbert transform, which may have the better error bounds than the known results.

MSC:

26D15 Inequalities for sums, series and integrals
26A46 Absolutely continuous real functions in one variable
41A80 Remainders in approximation formulas
Full Text: DOI

References:

[1] Alomari, M.W.: A companion of Ostrowski’s inequality with applications. Transylv. J. Math. Mech. 3(1), 9-14 (2011) · Zbl 1233.26006
[2] Alomari, M.W.: A companion of Ostrowski’s inequality for mappings whose first derivatives are bounded and applications in numerical integration. Transylv. J. Math. Mech. 4(2), 103-109 (2012) · Zbl 1289.26037
[3] Alomari, M.W.: A generalization of companion inequality of Ostrowski’s type for mappings whose first derivatives are bounded and applications in numerical integration. Kragujev. J. Math. 36(1), 77-82 (2012) · Zbl 1289.26037
[4] Anastassiou, G.A.: Basic and \[s\] s-convexity Ostrowski and Grüss type inequalities involving several functions. Commun. Appl. Anal. 17(2), 189-212 (2013) · Zbl 1296.26063
[5] Anastassiou, G.A.: General Grüss and Ostrowski type inequalities involving \[SS\]-convexity. Bull. Allahabad Math. Soc. 28(1), 101-129 (2013) · Zbl 1292.26044
[6] Criscuolo, G., Mastroianni, G.: Formule gaussiane per il calcolo di integrali a valor principale secondo e Cauchy loro convergenza. Calcolo 22, 391-411 (1985) · Zbl 0611.65012 · doi:10.1007/BF02600384
[7] Dagnion, C., Demichelis, V., Santi, E.: Numerical integration based on quasi-interpolating splines. Computing 50, 149-163 (1993) · Zbl 0768.41026 · doi:10.1007/BF02238611
[8] Diethelm, K.: Non-optimality of certain quadrature rules for Cauchy principal value integrals. Z. Angew. Math. Mech. 74, 689-690 (1994) · Zbl 0802.65017
[9] Dragomir, S.S.: Applications of Ostrowski’s inequality to the estimation of error bounds for some special means and to some numerical quadrature rules. Comp. Math. Appl. 33, 15-20 (1997) · Zbl 0880.41025 · doi:10.1016/S0898-1221(97)00084-9
[10] Dragomir, S.S.: Approximating the finite Hilbert transform via Ostrowski type inequalities for absolutely continuous functions. Bull. Korean Math. Soc. 39(4), 543-559 (2002) · Zbl 1016.26014 · doi:10.4134/BKMS.2002.39.4.543
[11] Dragomir, S.S.: Some companions of Ostrowski’s inequality for absolutely continuous functions and applications. Bull. Korean Math. Soc. 42(2), 213-230 (2005) · Zbl 1099.26015 · doi:10.4134/BKMS.2005.42.2.213
[12] Elliott, D., Paget, D.F.: On the convergence of a quadrature rule for evaluating certain Cauchy principal value integrals. Numer. Math. 23, 311-319 (1975) · Zbl 0313.65019 · doi:10.1007/BF01438258
[13] Hasegawa, T., Torii, T.: An automatic quadrature for Cauchy principal value integrals. Math. Comput. 56, 741-754 (1991) · Zbl 0725.65025 · doi:10.1090/S0025-5718-1991-1068816-1
[14] Hasegawa, T., Sugiura, H.: Algorithms for approximating finite Hilbert transform with end-point singularities and its derivatives. J. Comput. Appl. Math. 236(2), 243-252 (2011) · Zbl 1236.65158 · doi:10.1016/j.cam.2011.06.027
[15] Hunter, D.B.: Some Gauss type formulae for the evaluation of Cauchy principal value integrals. Numer. Math. 19, 419-424 (1972) · Zbl 0231.65028 · doi:10.1007/BF01404924
[16] Huy, V.N., Ngô, Q.-A.: New bounds for the Ostrowski-like type inequalities. Bull. Korean Math. Soc. 48(1), 95-104 (2011) · Zbl 1222.26022 · doi:10.4134/BKMS.2011.48.1.095
[17] Hwang, S.-R., Hsu, K.-C., Tseng, K.-L.: Weighted Ostrowski integral inequalities for mappings whose derivatives belong to \[L_P(a, b)\] LP(a,b). Appl. Math. Comput. 219(17), 9516-9523 (2013) · Zbl 1296.26074 · doi:10.1016/j.amc.2013.03.094
[18] Li, X.-J.: An explicit formula for finite Hilbert transforms associated with a polynomial. Indiana Univ. Math. J. 53(1), 185-203 (2004) · Zbl 1059.44001 · doi:10.1512/iumj.2004.53.2361
[19] Liu, W.: Some weighted integral inequalities with a parameter and applications. Acta Appl. Math. 109(2), 389-400 (2010) · Zbl 1196.26030 · doi:10.1007/s10440-008-9323-2
[20] Liu, W., Ngô, Q.A., Chen, W.: Ostrowski type inequalities on time scales for double integrals. Acta Appl. Math. 110(1), 477-497 (2010) · Zbl 1194.26030 · doi:10.1007/s10440-009-9456-y
[21] Liu, W., Sun, Y., Zhang, Q.: Some new error inequalities for a generalized quadrature rule of open type. Comput. Math. Appl. 62(5), 2218-2224 (2011) · Zbl 1231.65058 · doi:10.1016/j.camwa.2011.07.007
[22] Liu, W., Tuna, A.: Weighted Ostrowski, trapezoid and Grüss type inequalities on time scales. J. Math. Inequal 6(3), 381-399 (2012) · Zbl 1251.26016 · doi:10.7153/jmi-06-37
[23] Liu, W., Tuna, A., Jiang, Y.: On weighted Ostrowski type, Trapezoid type, Grüss type and Ostrowski-Grüss like inequalities on time scales. Appl. Anal. 93(3), 551-571 (2014) · Zbl 1294.26026 · doi:10.1080/00036811.2013.786045
[24] Liu, W., Zhu, Y., Park, J.: Some companions of perturbed Ostrowski-type inequalities based on the quadratic kernel function with three sections and applications. J. Inequal. Appl. 2013, 14 (2013) · Zbl 1282.26035 · doi:10.1186/1029-242X-2013-14
[25] Liu, Z.: Some companions of an Ostrowski type inequality and applications, J. Inequal. Pure Appl. Math. 10(2), Article 52, 12 pp (2009) · Zbl 1168.26310
[26] Masjed-Jamei, M., Dragomir, S.S.: A generalization of the Ostrowski-Grüss Inequality. Anal. Appl. 12(2), 117-130 (2014) · Zbl 1291.26027 · doi:10.1142/S0219530513500309
[27] Nandagopal, M., Arunajadai, N.: On the evaluation of finite Hilbert transforms. Comput. Sci. Eng. 9(6), 90-95 (2007) · doi:10.1109/MCSE.2007.116
[28] Park, J.: Some companions of perturbed ostrowski type inequalities for functions whose second derivatives are bounded. Int. J. Appl. Math. Stat. 36, 95-103 (2013)
[29] Qayyum, A., Hussain, S.: A new generalized Ostrowski Grüss type inequality and applications. Appl. Math. Lett. 25(11), 1875-1880 (2012) · Zbl 1254.26034 · doi:10.1016/j.aml.2012.02.052
[30] Sarikaya, M.Z.: On the Ostrowski type integral inequality. Acta Math. Univ. Comenian. (N.S.) 79(1), 129-134 (2010) · Zbl 1212.26058
[31] Wang, Z., Vong, S.: On some generalizations of an Ostrowski-Grüss type integral inequality. Appl. Math. Comput. 229, 239-244 (2014) · Zbl 1364.26032 · doi:10.1016/j.amc.2013.12.051
[32] Vong, S.W.: A note on some Ostrowski-like type inequalities. Comput. Math. Appl. 62(1), 532-535 (2011) · Zbl 1228.26034 · doi:10.1016/j.camwa.2011.05.037
[33] Zeng, G.: Image reconstruction via the finite Hilbert transform of the derivative of the backprojection. Med. Phys. 34, 2837-2843 (2007) · doi:10.1118/1.2739813
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