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.
KeywordsRiemann-Stieltjes integral Taylor’s representation functions of bounded variation Lipschitzian functions integral transforms finite Laplace-Stieltjes transform finite Fourier-Stieltjes sine and cosine transforms
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.
that holds for any function of locally bounded variation on .
provided that the function is -times differentiable on the interior of the interval I and the -th derivative is of locally bounded variation on .
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.
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