Abstract
In the present paper, we investigate the problem of approximating the RiemannStieltjes integral in the case when the integrand f is ntime 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 LaplaceStieltjes transform and the finite FourierStieltjes sine and cosine transforms are provided as well.
MSC: 41A51, 26D15, 26D10.
Keywords:
RiemannStieltjes integral; Taylor’s representation; functions of bounded variation; Lipschitzian functions; integral transforms; finite LaplaceStieltjes transform; finite FourierStieltjes sine and cosine transforms1 Introduction
The concept of RiemannStieltjes 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 [1]. Earlier results in this direction, however, were provided by Dubuc and Todor in their 1984 and 1987 papers [2,3] and [4], respectively. For recent results concerning the approximation of the RiemannStieltjes integral, see the work of Diethelm [5], Liu [6], Mercer [7], Munteanu [8], Mozyrska et al.[9] and the references therein. For other recent results obtained in the same direction by the first author and his colleagues from RGMIA, see [1016] and [17]. A comprehensive list of preprints related to this subject may be found at http://rgmia.org webcite.
In order to approximate the RiemannStieltjes integral , where are functions for which the above integral exists, Dragomir established in [18] the following integral identity:
provided that the involved integrals exist. In the particular case when , , the above identity reduces to the celebrated Montgomery identity (see [[19], 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 [20], the papers [1012,2132] and [33]. For other results concerning error bounds of quadrature rules related to midpoint and trapezoid rules, see [3445] and the references therein.
Motivated by the recent results from [18,46,47] (see also [11,27] and [13]) in the present paper we investigate the problem of approximating the RiemannStieltjes integral in the case when the integrand f is ntimes 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 LaplaceStieltjes transform and the finite FourierStieltjes sine and cosine transforms are provided as well.
2 Some representation results
In this section, we establish some representation results for the RiemannStieltjes integral when the integrand is ntimes differentiable and the integrator is of locally bounded variation. Several particular cases of interest are considered as well.
Theorem 1Assume that the functionisntimes differentiable on the interiorof the intervalI () and thenth derivativeis of locally bounded variation on. Ifwith, andis of bounded variation on, then the RiemannStieltjes integralexists, we have the identity
where
and the remaindercan be represented as
Both integrals in (2.3) are taken in the RiemannStieltjes sense.
Proof
Under the assumption of the theorem, we utilize the following Taylor’s representation
that holds for any and . The integral in (2.4) is taken in the RiemannStieltjes sense.
We can prove this equality by induction.
that holds for any function of locally bounded variation on .
Now, assume that (2.4) is true for an and let us prove that it holds for ‘’, namely
provided that the function is times differentiable on the interior of the interval I and the th derivative is of locally bounded variation on .
Utilizing the integration by parts formula for the RiemannStieltjes integral and the reduction of the RiemannStieltjes integral to a Riemann integral (see, for instance, [48]) we have:
From (2.4), we have that
which inserted in the last part of (2.6) provides the equality
We observe that, by division with , the equality (2.7) becomes the desired representation (2.5).
Further on, from the identity (2.4) we obtain
Utilizing the integration by parts formula, we have for that
Therefore, by (2.9) we get
and by (2.8) the representation (2.1) is thus obtained.
This completes the proof. □
Remark 1 Assume that the function is ntimes differentiable on the interior of the interval I () and the nth derivative is of locally bounded variation on . If with and is of bounded variation on , then, by choosing in the formulae above we have
and
This give the representation
Now, if we choose , then we have
and
which provide the representation
Finally, if we choose , then we have
and the remainder
Making use of (2.1) we get
3 Error bounds
In order to provide sharp error bounds in the approximation rules outlined above, we need the following wellknown lemma concerning sharp estimates for the RiemannStieltjes integral for various pairs of integrands and integrators (see, for instance, [48]).
Lemma 1Lettwo bounded functions on the compact interval.
(i) Ifpis continuous andvis of bounded variation, then the RiemannStieltjes integralexists and
wheredenotes the total variation ofvon the interval.
(ii) Ifpis Riemann integrable andvis Lipschitzian with the constant, i.e.,
then the RiemannStieltjes integralexists and
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 RiemannStieltjes integral, we can provide the following error bounds in approximating the integral .
Theorem 2Assume that the functionisntimes differentiable on the interiorof the intervalI () and thenth derivativeis of locally bounded variation on. Ifwith, andis of bounded variation on, then we have the representation (2.1), where the approximation termis given by (2.2) and the remaindersatisfies the inequality
If thenth derivativeis Lipschitzian with the constanton, then we have
Proof
Utilizing the property (i) from Lemma 1, we have successively
By the property (i) from Lemma 1 applied for we have for that
Therefore,
Utilizing (3.5) and (3.7), we deduce the desired inequality (3.3).
By the property (ii) from Lemma 1 applied for , we have that
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.
Corollary 1Under the assumptions of Theorem 2 we have the representation
whereis defined in (2.14) and the errorsatisfies the bound
Moreover, if thenth derivativeis Lipschitzian with the constanton, then we have
The case of Lipschitzian integrators may be of interest as well and will be considered in the following.
Theorem 3Assume that the functionisntimes differentiable on the interiorof the intervalI () and thenth derivativeis of locally bounded variation on. Ifwith, andis Lipschitzian onwith the constantthen we have the representation (2.1), where the approximation termis given by (2.2) and the remaindersatisfies the inequality
If thenth derivativeis Lipschitzian with the constanton, then we have
Proof
Utilizing the property (ii) from Lemma 1, we have successively
for any , where as above , for .
By the property (i) from Lemma 1 applied for , we have for that
which gives that
This implies that
Making use of (3.14) and (3.15) we deduce the desired inequality (3.12).
By the property (ii) from Lemma 1 applied for we have that
Utilizing (3.14) and (3.16), we deduce the desired inequality (3.13). □
The following particular case provides the best error bounds.
Corollary 2Under the assumptions of Theorem 3, we have the representation (3.9), whereis defined in (2.14) and the errorsatisfies the bound
Moreover, if thenth derivativeis Lipschitzian with the constanton, then we have
4 Applications
1. We consider the following finite LaplaceStieltjes transform defined by
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, [25,26,4951] and [52] and the references therein. Therefore, any approximation of the more general finite LaplaceStieltjes 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 .
We observe that the function has derivatives of all orders and
We also observe that
To simplify the notations, we denote by
On utilizing Theorem 1, we have the representation
where
and the remainder can be represented as
Since g is of bounded variation on and the derivative is Lipschitzian with the constant
then by Theorem 2 we have the bound
As above, the best approximation we can get from (4.4) is for , namely, we have the representation
where
and the remainder can be represented as
Now, if we restrict the function g to belong to the class of Lipschitzian functions with the constant on the interval , then the error in the representation (4.4) will satisfy the bound
Finally, the error from the representation (4.8) satisfies the inequality
2. We consider now the finite FourierStieltjes sine and cosine transforms defined by
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 .
Utilizing the wellknown formulae for the nth derivatives of sine and cosine functions, namely,
and
then we have
We observe that, in general, we have the bounds
and
for any , the closed interval and .
On utilizing Theorem 1, we have the representation
where
and the remainder can be represented as
Since g is of bounded variation on and the derivative is Lipschitzian with the constant
then by Theorem 2 we have the bound
As above, the best approximation we can get from (4.4) is for , namely, we have the representation
where
and the remainder can be represented as
Here, the error satisfies the bound
Now, if we restrict the function g to belong to the class of Lipschitzian functions with the constant on the interval , then the error in the representation (4.17) will satisfy the bound:
Finally, the error from the representation (4.17) satisfies the inequality
Similar results may be stated for the finite FourierStieltjes cosine transform, however the details are left to the interested reader.
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
The authors SSD and SA have contributed equally in all stages of writing the paper.
Acknowledgements
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.
References

Tortorella, M: Closed NewtonCotes quadrature rules for Stieltjes integrals and numerical convolution of life distributions. SIAM J. Sci. Stat. Comput.. 11(4), 732–748 (1990). Publisher Full Text

Dubuc, S, Todor, F: La règle du trapèze pour l’intégrale de RiemannStieltjes. I. Ann. Sci. Math. Qué.. 8(2), 135–140 (French) [The trapezoid formula for the RiemannStieltjes integral. I]. (1984)

Dubuc, S, Todor, F: La règle du trapèze pour l’intégrale de RiemannStieltjes. II. Ann. Sci. Math. Qué.. 8(2), 141–153 (French) [The trapezoid formula for the RiemannStieltjes integral. II]. (1984)

Dubuc, S, Todor, F: La règle optimale du trapèze pour l’intégrale de RiemannStieltjes d’une fonction donnée. C. R. Math. Rep. Acad. Sci. Canada. 9(5), 213–218 (French) [The optimal trapezoidal rule for the RiemannStieltjes integral of a given function]. (1987)

Diethelm, K: A note on the midpoint rectangle formula for RiemannStieltjes integrals. J. Stat. Comput. Simul.. 74(12), 920–922 (2004)

Liu, Z: Refinement of an inequality of Grüss type for RiemannStieltjes integral. Soochow J. Math.. 30(4), 483–489 (2004)

Mercer, PR: Hadamard’s inequality and trapezoid rules for the RiemannStieltjes integral. J. Math. Anal. Appl.. 344(2), 921–926 (2008). Publisher Full Text

Munteanu, M: Quadrature formulas for the generalized RiemannStieltjes integral. Bull. Braz. Math. Soc.. 38(1), 39–50 (2007). Publisher Full Text

Mozyrska, D, Pawluszewicz, E, Torres, DFM: The RiemannStieltjes integral on time scales. Aust. J. Math. Anal. Appl.. 7(1), Article ID 10 (2010)

Barnett, NS, Cerone, P, Dragomir, SS: Majorisation inequalities for Stieltjes integrals. Appl. Math. Lett.. 22, 416–421 (2009). Publisher Full Text

Barnett, NS, Cheung, WS, Dragomir, SS, Sofo, A: Ostrowski and trapezoid type inequalities for the Stieltjes integral with Lipschitzian integrands or integrators. Comput. Math. Appl.. 57, 195–201 (2009). Publisher Full Text

Barnett, NS, Dragomir, SS: The BeesackDarstPollard inequalities and approximations of the RiemannStieltjes integral. Appl. Math. Lett.. 22, 58–63 (2009). Publisher Full Text

Cerone, P, Cheung, WS, Dragomir, SS: On Ostrowski type inequalities for Stieltjes integrals with absolutely continuous integrands and integrators of bounded variation. Comput. Math. Appl.. 54, 183–191 (2007). Publisher Full Text

Cerone, P, Dragomir, SS: Bounding the Čebyšev functional for the Riemann Stieltjes integral via a Beesack inequality and applications. Comput. Math. Appl.. 58, 1247–1252 (2009). Publisher Full Text

Cerone, P, Dragomir, SS: Approximating the Riemann Stieltjes integral via some moments of the integrand. Math. Comput. Model.. 49, 242–248 (2009). Publisher Full Text

Dragomir, SS: Approximating the Riemann Stieltjes integral in terms of generalised trapezoidal rules. Nonlinear Anal.. 71, e62–e72 (2009). Publisher Full Text

Dragomir, SS: Inequalities for Stieltjes integrals with convex integrators and applications. Appl. Math. Lett.. 20, 123–130 (2007). Publisher Full Text

Dragomir, SS: On the Ostrowski’s inequality for RiemannStieltjes integral. Korean J. Comput. Appl. Math.. 7, 477–485 (2000)

Mitrinović, DS, Pečarić, JE, Fink, AM: Inequalities Involving Functions and Their Integrals and Derivatives, Kluwer Academic, Dordrecht (1991)

Dragomir SS, Rassias TM (eds.): Ostrowski Type Inequalities and Applications in Numerical Integration, Kluwer Academic, Dordrecht (2002)

Anastassiou, AG: Univariate Ostrowski inequalities, revisited. Monatshefte Math.. 135(3), 175–189 (2002). Publisher Full Text

Anastassiou, AG: Ostrowski type inequalities. Proc. Am. Math. Soc.. 123(12), 3775–3781 (1995). Publisher Full Text

AglićAljinović, A, Pečarić, J: On some Ostrowski type inequalities via Montgomery identity and Taylor’s formula. Tamkang J. Math.. 36(3), 199–218 (2005)

AglićAljinović, A, Pečarić, J, Vukelić, A: On some Ostrowski type inequalities via Montgomery identity and Taylor’s formula II. Tamkang J. Math.. 36(4), 279–301 (2005)

Bertero, M, Grünbaum, FA: Commuting differential operators for the finite Laplace transform. Inverse Probl.. 1(3), 181–192 (1985). Publisher Full Text

Bertero, M, Grünbaum, FA, Rebolia, L: Spectral properties of a differential operator related to the inversion of the finite Laplace transform. Inverse Probl.. 2(2), 131–139 (1986). Publisher Full Text

Cheung, WS, Dragomir, SS: Two Ostrowski type inequalities for the Stieltjes integral of monotonic functions. Bull. Aust. Math. Soc.. 75(2), 299–311 (2007). Publisher Full Text

Cerone, P: Approximate multidimensional integration through dimension reduction via the Ostrowski functional. Nonlinear Funct. Anal. Appl.. 8(3), 313–333 (2003)

Cerone, P, Dragomir, SS: On some inequalities arising from Montgomery’s identity. J. Comput. Anal. Appl.. 5(4), 341–367 (2003)

Kumar, P: The Ostrowski type moment integral inequalities and momentbounds for continuous random variables. Comput. Math. Appl.. 49(1112), 1929–1940 (2005). Publisher Full Text

Pachpatte, BG: A note on Ostrowski like inequalities. J. Inequal. Pure Appl. Math.. 6(4), Article ID 114 (2005)

Sofo, A: Integral inequalities for Ntimes differentiable mappings. Ostrowski Type Inequalities and Applications in Numerical Integration, pp. 65–139. Kluwer Academic, Dordrecht (2002)

Ujević, N: Sharp inequalities of Simpson type and Ostrowski type. Comput. Math. Appl.. 48(12), 145–151 (2004). Publisher Full Text

Dragomir, SS: Ostrowski’s inequality for montonous mappings and applications. J. KSIAM. 3(1), 127–135 (1999)

Dragomir, SS: Some inequalities for RiemannStieltjes integral and applications. In: Rubinov A, Glover B (eds.) Optimization and Related Topics, pp. 197–235. Kluwer Academic, Dordrecht (2001)

Dragomir, SS: Accurate approximations of the RiemannStieltjes integral with (l,L)Lipschitzian integrators. In: Simos, TH, et al. (eds.) AIP Conf. Proc. 939, Numerical Anal. & Appl. Math., pp. 686690. Preprint RGMIA Res. Rep. Coll. 10(3), Article ID 5 (2007). Online http://rgmia.vu.edu.au/v10n3.html

Dragomir, SS: Accurate approximations for the RiemannStieltjes integral via theory of inequalities. J. Math. Inequal.. 3(4), 663–681 (2009)

Dragomir, SS: Approximating the RiemannStieltjes integral by a trapezoidal quadrature rule with applications. Math. Comput. Model. (in Press). Corrected Proof, Available online 18 February 2011

Dragomir, SS, Buşe, C, Boldea, MV, Brăescu, L: A generalisation of the trapezoidal rule for the RiemannStieltjes integral and applications. Nonlinear Anal. Forum. 6(2), 337–351 (2001)

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.. 42(90)(4), 301–314 (1999)

Dragomir, SS, Fedotov, I: An inequality of Grüss type for the RiemannStieltjes integral and applications for special means. Tamkang J. Math.. 29(4), 287–292 (1998)

Dragomir, SS, Fedotov, I: A Grüss type inequality for mappings of bounded variation and applications to numerical analysis. Nonlinear Funct. Anal. Appl.. 6(3), 425–433 (2001)

Pachpatte, BG: A note on a trapezoid type integral inequality. Bull. Greek Math. Soc.. 49, 85–90 (2004)

Ujević, N: Error inequalities for a generalized trapezoid rule. Appl. Math. Lett.. 19(1), 32–37 (2006). Publisher Full Text

Wu, Q, Yang, S: A note to Ujević’s generalization of Ostrowski’s inequality. Appl. Math. Lett.. 18(6), 657–665 (2005). Publisher Full Text

Dragomir, SS: On the Ostrowski inequality for RiemannStieltjes integral , where f is of Hölder type and u is of bounded variation and applications. J. KSIAM. 5(1), 35–45 (2001)

Cerone, P, Dragomir, SS: New bounds for the threepoint rule involving the RiemannStieltjes integral. In: Gulati C (ed.) Advances in Statistics, Combinatorics and Related Areas, pp. 53–62. World Scientific, Singapore (2002)

Apostol, TM: Mathematical Analysis, AddisonWesley, Reading (1975)

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.. 4(4), 407–419 (1980). PubMed Abstract  Publisher Full Text

Rutily, B, Chevallier, L: The finite Laplace transform for solving a weakly singular integral equation occurring in transfer theory. J. Integral Equ. Appl.. 16(4), 389–409 (2004). Publisher Full Text

Valbuena, M, Galue, L, Ali, I: Some properties of the finite Laplace transform. Transform Methods & Special Functions, Varna ’96, pp. 517–522. Bulgarian Acad. Sci., Sofia (1998).

Watanabe, K, Ito, M: A necessary condition for spectral controllability of delay systems on the basis of finite Laplace transforms. Int. J. Control. 39(2), 363–374 (1984). Publisher Full Text