Approximating the Riemann-Stieltjes integral of smooth integrands and of bounded variation integrators
© Dragomir and Abelman; licensee Springer. 2013
Received: 20 November 2012
Accepted: 20 March 2013
Published: 4 April 2013
In the present paper, we investigate the problem of approximating the Riemann-Stieltjes integral in the case when the integrand f is n-time differentiable and the derivative is either of locally bounded variation, or Lipschitzian on an interval incorporating . A priory error bounds for several classes of integrators u and applications in approximating the finite Laplace-Stieltjes transform and the finite Fourier-Stieltjes sine and cosine transforms are provided as well.
MSC:41A51, 26D15, 26D10.
The concept of Riemann-Stieltjes integral , where f is called the integrand, u is called the integrator, plays an important role in mathematics, for instance in the definition of complex integral, the representation of bounded linear functionals on the Banach space of all continuous functions on an interval , in the spectral representation of selfadjoint operators on complex Hilbert spaces and other classes of operators such as the unitary operators, etc.
However, the numerical analysis of this integral is quite poor as pointed out by the seminal paper due to Michael Tortorella from 1990 . Earlier results in this direction, however, were provided by Dubuc and Todor in their 1984 and 1987 papers [2, 3] and , respectively. For recent results concerning the approximation of the Riemann-Stieltjes integral, see the work of Diethelm , Liu , Mercer , Munteanu , Mozyrska et al.  and the references therein. For other recent results obtained in the same direction by the first author and his colleagues from RGMIA, see [10–16] and . A comprehensive list of preprints related to this subject may be found at http://rgmia.org.
provided that the involved integrals exist. In the particular case when , , the above identity reduces to the celebrated Montgomery identity (see [, p.565]) that has been extensively used by many authors in obtaining various inequalities of Ostrowski type. For a comprehensive recent collection of works related to Ostrowski’s inequality, see the book , the papers [10–12, 21–32] and . For other results concerning error bounds of quadrature rules related to midpoint and trapezoid rules, see [34–45] and the references therein.
Motivated by the recent results from [18, 46, 47] (see also [11, 27] and ) in the present paper we investigate the problem of approximating the Riemann-Stieltjes integral in the case when the integrand f is n-times differentiable and the derivative is either of locally bounded variation, or Lipschitzian on an interval incorporating . A priori error bounds for several classes of integrators u and applications in approximating the finite Laplace-Stieltjes transform and the finite Fourier-Stieltjes sine and cosine transforms are provided as well.
2 Some representation results
In this section, we establish some representation results for the Riemann-Stieltjes integral when the integrand is n-times differentiable and the integrator is of locally bounded variation. Several particular cases of interest are considered as well.
Both integrals in (2.3) are taken in the Riemann-Stieltjes sense.
that holds for any and . The integral in (2.4) is taken in the Riemann-Stieltjes sense.
We can prove this equality by induction.
We observe that, by division with , the equality (2.7) becomes the desired representation (2.5).
For , we have .
and by (2.8) the representation (2.1) is thus obtained.
This completes the proof. □
3 Error bounds
In order to provide sharp error bounds in the approximation rules outlined above, we need the following well-known lemma concerning sharp estimates for the Riemann-Stieltjes integral for various pairs of integrands and integrators (see, for instance, ).
- (i)If p is continuous and v is of bounded variation, then the Riemann-Stieltjes integral exists and(3.1)
- (ii)If p is Riemann integrable and v is Lipschitzian with the constant , i.e.,
All the above inequalities are sharp in the sense that there are examples of functions for which each equality case is realized.
Utilizing this result concerning bounds for the Riemann-Stieltjes integral, we can provide the following error bounds in approximating the integral .
for any .
for any .
for any .
for any .
Utilizing (3.5) and (3.7), we deduce the desired inequality (3.3).
for any .
Utilizing (3.5) and (3.8), we deduce the desired inequality (3.4). □
The best error bounds we can get from Theorem 2 are as follows.
The case of Lipschitzian integrators may be of interest as well and will be considered in the following.
for any .
for any .
for any , where as above , for .
Making use of (3.14) and (3.15) we deduce the desired inequality (3.12).
Utilizing (3.14) and (3.16), we deduce the desired inequality (3.13). □
The following particular case provides the best error bounds.
- 1.We consider the following finite Laplace-Stieltjes transform defined by(4.1)
where are real numbers with , s is a complex number and is a function of bounded variation.
It is important to notice that, in the particular case , (4.1) becomes the finite Laplace transform which has various applications in other fields of Mathematics; see, for instance, [25, 26, 49–51] and  and the references therein. Therefore, any approximation of the more general finite Laplace-Stieltjes transform can be used for the particular case of finite Laplace transform.
Since the function , is continuous for any , the transform (4.1) is well defined for any .
Here, and .
for any and .
for any .
for any and .
- 2.We consider now the finite Fourier-Stieltjes sine and cosine transforms defined by(4.12)
where a, b are real numbers with , u is a real number and is a function of bounded variation.
Since the functions , , are continuous for any , the transforms (4.12) are well defined for any .
for any and .
for any , the closed interval and .
for any and .
for any .
for any .
for any and .
for any .
Similar results may be stated for the finite Fourier-Stieltjes cosine transform, however the details are left to the interested reader.
Dedicated to Professor Hari M Srivastava.
The authors would like to thank the anonymous referees for the valuable suggestions that have been incorporated in the final version of the paper.
- Tortorella M: Closed Newton-Cotes quadrature rules for Stieltjes integrals and numerical convolution of life distributions. SIAM J. Sci. Stat. Comput. 1990, 11(4):732–748. 10.1137/0911043MathSciNetView ArticleGoogle Scholar
- Dubuc S, Todor F: La règle du trapèze pour l’intégrale de Riemann-Stieltjes. I. Ann. Sci. Math. Qué. 1984, 8(2):135–140. (French) [The trapezoid formula for the Riemann-Stieltjes integral. I].MathSciNetGoogle Scholar
- Dubuc S, Todor F: La règle du trapèze pour l’intégrale de Riemann-Stieltjes. II. Ann. Sci. Math. Qué. 1984, 8(2):141–153. (French) [The trapezoid formula for the Riemann-Stieltjes integral. II].MathSciNetGoogle Scholar
- Dubuc S, Todor F: La règle optimale du trapèze pour l’intégrale de Riemann-Stieltjes d’une fonction donnée. C. R. Math. Rep. Acad. Sci. Canada 1987, 9(5):213–218. (French) [The optimal trapezoidal rule for the Riemann-Stieltjes integral of a given function].MathSciNetGoogle Scholar
- Diethelm K: A note on the midpoint rectangle formula for Riemann-Stieltjes integrals. J. Stat. Comput. Simul. 2004, 74(12):920–922.MathSciNetGoogle Scholar
- Liu Z: Refinement of an inequality of Grüss type for Riemann-Stieltjes integral. Soochow J. Math. 2004, 30(4):483–489.MathSciNetGoogle Scholar
- Mercer PR: Hadamard’s inequality and trapezoid rules for the Riemann-Stieltjes integral. J. Math. Anal. Appl. 2008, 344(2):921–926. 10.1016/j.jmaa.2008.03.026MathSciNetView ArticleGoogle Scholar
- Munteanu M: Quadrature formulas for the generalized Riemann-Stieltjes integral. Bull. Braz. Math. Soc. 2007, 38(1):39–50. 10.1007/s00574-007-0034-5MathSciNetView ArticleGoogle Scholar
- Mozyrska D, Pawluszewicz E, Torres DFM: The Riemann-Stieltjes integral on time scales. Aust. J. Math. Anal. Appl. 2010., 7(1): Article ID 10Google Scholar
- Barnett NS, Cerone P, Dragomir SS: Majorisation inequalities for Stieltjes integrals. Appl. Math. Lett. 2009, 22: 416–421. 10.1016/j.aml.2008.06.009MathSciNetView ArticleGoogle Scholar
- Barnett NS, Cheung W-S, Dragomir SS, Sofo A: Ostrowski and trapezoid type inequalities for the Stieltjes integral with Lipschitzian integrands or integrators. Comput. Math. Appl. 2009, 57: 195–201. 10.1016/j.camwa.2007.07.021MathSciNetView ArticleGoogle Scholar
- Barnett NS, Dragomir SS: The Beesack-Darst-Pollard inequalities and approximations of the Riemann-Stieltjes integral. Appl. Math. Lett. 2009, 22: 58–63. 10.1016/j.aml.2008.02.005MathSciNetView ArticleGoogle Scholar
- Cerone P, Cheung W-S, Dragomir SS: On Ostrowski type inequalities for Stieltjes integrals with absolutely continuous integrands and integrators of bounded variation. Comput. Math. Appl. 2007, 54: 183–191. 10.1016/j.camwa.2006.12.023MathSciNetView ArticleGoogle Scholar
- Cerone P, Dragomir SS: Bounding the Čebyšev functional for the Riemann Stieltjes integral via a Beesack inequality and applications. Comput. Math. Appl. 2009, 58: 1247–1252. 10.1016/j.camwa.2009.07.029MathSciNetView ArticleGoogle Scholar
- Cerone P, Dragomir SS: Approximating the Riemann Stieltjes integral via some moments of the integrand. Math. Comput. Model. 2009, 49: 242–248. 10.1016/j.mcm.2008.02.011MathSciNetView ArticleGoogle Scholar
- Dragomir SS: Approximating the Riemann Stieltjes integral in terms of generalised trapezoidal rules. Nonlinear Anal. 2009, 71: e62-e72. 10.1016/j.na.2008.10.004View ArticleGoogle Scholar
- Dragomir SS: Inequalities for Stieltjes integrals with convex integrators and applications. Appl. Math. Lett. 2007, 20: 123–130. 10.1016/j.aml.2006.02.027MathSciNetView ArticleGoogle Scholar
- Dragomir SS: On the Ostrowski’s inequality for Riemann-Stieltjes integral. Korean J. Comput. Appl. Math. 2000, 7: 477–485.MathSciNetGoogle Scholar
- Mitrinović DS, Pečarić JE, Fink AM: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Dordrecht; 1991.View ArticleGoogle Scholar
- Dragomir SS, Rassias TM (Eds): Ostrowski Type Inequalities and Applications in Numerical Integration. Kluwer Academic, Dordrecht; 2002.Google Scholar
- Anastassiou AG: Univariate Ostrowski inequalities, revisited. Monatshefte Math. 2002, 135(3):175–189. 10.1007/s006050200015MathSciNetView ArticleGoogle Scholar
- Anastassiou AG: Ostrowski type inequalities. Proc. Am. Math. Soc. 1995, 123(12):3775–3781. 10.1090/S0002-9939-1995-1283537-3MathSciNetView ArticleGoogle Scholar
- Aglić-Aljinović A, Pečarić J: On some Ostrowski type inequalities via Montgomery identity and Taylor’s formula. Tamkang J. Math. 2005, 36(3):199–218.MathSciNetGoogle Scholar
- Aglić-Aljinović A, Pečarić J, Vukelić A: On some Ostrowski type inequalities via Montgomery identity and Taylor’s formula II. Tamkang J. Math. 2005, 36(4):279–301.MathSciNetGoogle Scholar
- Bertero M, Grünbaum FA: Commuting differential operators for the finite Laplace transform. Inverse Probl. 1985, 1(3):181–192. 10.1088/0266-5611/1/3/004View ArticleGoogle Scholar
- Bertero M, Grünbaum FA, Rebolia L: Spectral properties of a differential operator related to the inversion of the finite Laplace transform. Inverse Probl. 1986, 2(2):131–139. 10.1088/0266-5611/2/2/006View ArticleGoogle Scholar
- Cheung W-S, Dragomir SS: Two Ostrowski type inequalities for the Stieltjes integral of monotonic functions. Bull. Aust. Math. Soc. 2007, 75(2):299–311. 10.1017/S0004972700039228MathSciNetView ArticleGoogle Scholar
- Cerone P: Approximate multidimensional integration through dimension reduction via the Ostrowski functional. Nonlinear Funct. Anal. Appl. 2003, 8(3):313–333.MathSciNetGoogle Scholar
- Cerone P, Dragomir SS: On some inequalities arising from Montgomery’s identity. J. Comput. Anal. Appl. 2003, 5(4):341–367.MathSciNetGoogle Scholar
- Kumar P: The Ostrowski type moment integral inequalities and moment-bounds for continuous random variables. Comput. Math. Appl. 2005, 49(11–12):1929–1940. 10.1016/j.camwa.2003.11.002MathSciNetView ArticleGoogle Scholar
- Pachpatte BG: A note on Ostrowski like inequalities. J. Inequal. Pure Appl. Math. 2005., 6(4): Article ID 114Google Scholar
- Sofo A: Integral inequalities for N -times differentiable mappings. In Ostrowski Type Inequalities and Applications in Numerical Integration. Kluwer Academic, Dordrecht; 2002:65–139.View ArticleGoogle Scholar
- Ujević N: Sharp inequalities of Simpson type and Ostrowski type. Comput. Math. Appl. 2004, 48(1–2):145–151. 10.1016/j.camwa.2003.09.026MathSciNetView ArticleGoogle Scholar
- Dragomir SS: Ostrowski’s inequality for montonous mappings and applications. J. KSIAM 1999, 3(1):127–135.Google Scholar
- Dragomir SS: Some inequalities for Riemann-Stieltjes integral and applications. In Optimization and Related Topics. Edited by: Rubinov A, Glover B. Kluwer Academic, Dordrecht; 2001:197–235.View ArticleGoogle Scholar
- Dragomir, SS: Accurate approximations of the Riemann-Stieltjes integral with ( l , L ) -Lipschitzian integrators. In: Simos, TH, et al. (eds.) AIP Conf. Proc. 939, Numerical Anal. & Appl. Math., pp. 686–690. Preprint RGMIA Res. Rep. Coll. 10(3), Article ID 5 (2007). Online http://rgmia.vu.edu.au/v10n3.htmlGoogle Scholar
- Dragomir SS: Accurate approximations for the Riemann-Stieltjes integral via theory of inequalities. J. Math. Inequal. 2009, 3(4):663–681.MathSciNetView ArticleGoogle Scholar
- Dragomir, SS: Approximating the Riemann-Stieltjes integral by a trapezoidal quadrature rule with applications. Math. Comput. Model. (in Press). Corrected Proof, Available online 18 February 2011Google Scholar
- Dragomir SS, Buşe C, Boldea MV, Brăescu L: A generalisation of the trapezoidal rule for the Riemann-Stieltjes integral and applications. Nonlinear Anal. Forum 2001, 6(2):337–351.MathSciNetGoogle Scholar
- Dragomir SS, Cerone P, Roumeliotis J, Wang S: A weighted version of Ostrowski inequality for mappings of Hölder type and applications in numerical analysis. Bull. Math. Soc. Sci. Math. Roum. 1999, 42(90)(4):301–314.MathSciNetGoogle Scholar
- Dragomir SS, Fedotov I: An inequality of Grüss type for the Riemann-Stieltjes integral and applications for special means. Tamkang J. Math. 1998, 29(4):287–292.MathSciNetGoogle Scholar
- Dragomir SS, Fedotov I: A Grüss type inequality for mappings of bounded variation and applications to numerical analysis. Nonlinear Funct. Anal. Appl. 2001, 6(3):425–433.MathSciNetGoogle Scholar
- Pachpatte BG: A note on a trapezoid type integral inequality. Bull. Greek Math. Soc. 2004, 49: 85–90.MathSciNetGoogle Scholar
- Ujević N: Error inequalities for a generalized trapezoid rule. Appl. Math. Lett. 2006, 19(1):32–37. 10.1016/j.aml.2005.03.005MathSciNetView ArticleGoogle Scholar
- Wu Q, Yang S: A note to Ujević’s generalization of Ostrowski’s inequality. Appl. Math. Lett. 2005, 18(6):657–665. 10.1016/j.aml.2004.08.010MathSciNetView ArticleGoogle Scholar
- Dragomir SS: On the Ostrowski inequality for Riemann-Stieltjes integral , where f is of Hölder type and u is of bounded variation and applications. J. KSIAM 2001, 5(1):35–45.Google Scholar
- Cerone P, Dragomir SS, et al.: New bounds for the three-point rule involving the Riemann-Stieltjes integral. In Advances in Statistics, Combinatorics and Related Areas. Edited by: Gulati C. World Scientific, Singapore; 2002:53–62.View ArticleGoogle Scholar
- Apostol TM: Mathematical Analysis. 2nd edition. Addison-Wesley, Reading; 1975.Google Scholar
- Miletic J: A finite Laplace transform method for the solution of a mixed boundary value problem in the theory of elasticity. J. M éc. Appl. 1980, 4(4):407–419.MathSciNetGoogle Scholar
- Rutily B, Chevallier L: The finite Laplace transform for solving a weakly singular integral equation occurring in transfer theory. J. Integral Equ. Appl. 2004, 16(4):389–409. 10.1216/jiea/1181075298MathSciNetView ArticleGoogle Scholar
- Valbuena M, Galue L, Ali I: Some properties of the finite Laplace transform. In Transform Methods & Special Functions, Varna ’96. Bulgarian Acad. Sci., Sofia; 1998:517–522.Google Scholar
- Watanabe K, Ito M: A necessary condition for spectral controllability of delay systems on the basis of finite Laplace transforms. Int. J. Control 1984, 39(2):363–374. 10.1080/00207178408933171MathSciNetView ArticleGoogle Scholar
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