Open Access

Approximating the Riemann-Stieltjes integral of smooth integrands and of bounded variation integrators

Journal of Inequalities and Applications20132013:154

DOI: 10.1186/1029-242X-2013-154

Received: 20 November 2012

Accepted: 20 March 2013

Published: 4 April 2013

Abstract

In the present paper, we investigate the problem of approximating the Riemann-Stieltjes integral a b f ( λ ) d u ( λ ) in the case when the integrand f is n-time differentiable and the derivative f ( n ) is either of locally bounded variation, or Lipschitzian on an interval incorporating [ a , b ] . 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.

Keywords

Riemann-Stieltjes integral Taylor’s representation functions of bounded variation Lipschitzian functions integral transforms finite Laplace-Stieltjes transform finite Fourier-Stieltjes sine and cosine transforms

1 Introduction

The concept of Riemann-Stieltjes integral a b f ( t ) d u ( t ) , 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 [ a , b ] , 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 Riemann-Stieltjes 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.

In order to approximate the Riemann-Stieltjes integral a b p ( t ) d v ( t ) , where p , v : [ a , b ] R are functions for which the above integral exists, Dragomir established in [18] the following integral identity:
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_Equ1_HTML.gif
(1.1)

provided that the involved integrals exist. In the particular case when u ( t ) = t , t [ a , b ] , 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 Riemann-Stieltjes integral a b f ( λ ) d u ( λ ) in the case when the integrand f is n-times differentiable and the derivative f ( n ) is either of locally bounded variation, or Lipschitzian on an interval incorporating [ a , b ] . 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.

Theorem 1 Assume that the function f : I C is n-times differentiable on the interior https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_IEq11_HTML.gif of the interval I ( n 1 ) and the nth derivative f ( n ) is of locally bounded variation on https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_IEq11_HTML.gif . If https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_IEq13_HTML.gif with a < b , c [ a , b ] and u : [ a , b ] C is of bounded variation on [ a , b ] , then the Riemann-Stieltjes integral a b f ( λ ) d u ( λ ) exists, we have the identity
a b f ( λ ) d u ( λ ) = T n ( f , u , a , c , b ) + R n ( f , u , a , c , b ) ,
(2.1)
where
T n ( f , u , a , c , b ) : = k = 0 n 1 k ! f ( k ) ( c ) [ ( b c ) k u ( b ) + ( 1 ) k + 1 ( c a ) k u ( a ) ] k = 0 n 1 1 k ! f ( k + 1 ) ( c ) a b ( λ c ) k u ( λ ) d λ
(2.2)
and the remainder R n ( f , u , a , c , b ) can be represented as
R n ( f , u , a , c , b ) : = 1 n ! a b ( c λ ( λ t ) n d f ( n ) ( t ) ) d u ( λ ) .
(2.3)

Both integrals in (2.3) are taken in the Riemann-Stieltjes sense.

Proof

Under the assumption of the theorem, we utilize the following Taylor’s representation
f ( λ ) = k = 0 n 1 k ! f ( k ) ( c ) ( λ c ) k + 1 n ! c λ ( λ t ) n d f ( n ) ( t )
(2.4)

that holds for any c [ a , b ] and n 0 . The integral in (2.4) is taken in the Riemann-Stieltjes sense.

We can prove this equality by induction.

Indeed, for n = 0 , we have
f ( λ ) = f ( c ) + c λ d f ( t )

that holds for any function of locally bounded variation on https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_IEq11_HTML.gif .

Now, assume that (2.4) is true for an n 0 and let us prove that it holds for ‘ n + 1 ’, namely
f ( λ ) = k = 0 n + 1 1 k ! f ( k ) ( c ) ( λ c ) k + 1 ( n + 1 ) ! c λ ( λ t ) n + 1 d f ( n + 1 ) ( t )
(2.5)

provided that the function f : I C is ( n + 1 ) -times differentiable on the interior https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_IEq11_HTML.gif of the interval I and the ( n + 1 ) -th derivative f ( n + 1 ) is of locally bounded variation on https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_IEq11_HTML.gif .

Utilizing the integration by parts formula for the Riemann-Stieltjes integral and the reduction of the Riemann-Stieltjes integral to a Riemann integral (see, for instance, [48]) we have:
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_Equ7_HTML.gif
(2.6)
From (2.4), we have that
c λ ( λ t ) n d f ( n ) ( t ) = [ f ( λ ) k = 0 n 1 k ! f ( k ) ( c ) ( λ c ) k ] n !
which inserted in the last part of (2.6) provides the equality
c λ ( λ t ) n + 1 d f ( n + 1 ) ( t ) = ( λ c ) n + 1 f ( n + 1 ) ( c ) + ( n + 1 ) ! [ f ( λ ) k = 0 n 1 k ! f ( k ) ( c ) ( λ c ) k ] .
(2.7)

We observe that, by division with ( n + 1 ) ! , the equality (2.7) becomes the desired representation (2.5).

Further on, from the identity (2.4) we obtain
a b f ( λ ) d u ( λ ) = k = 0 n 1 k ! f ( k ) ( c ) a b ( λ c ) k d u ( λ ) + 1 n ! a b ( c λ ( λ t ) n d f ( n ) ( t ) ) d u ( λ ) .
(2.8)
Utilizing the integration by parts formula, we have for k 1 that
a b ( λ c ) k d u ( λ ) = ( λ c ) k u ( λ ) | a b k a b ( λ c ) k 1 u ( λ ) d λ = ( b c ) k u ( b ) + ( 1 ) k + 1 ( c a ) k u ( a ) k a b ( λ c ) k 1 u ( λ ) d λ .
(2.9)

For k = 0 , we have a b d u ( λ ) = u ( b ) u ( a ) .

Therefore, by (2.9) we get
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_Equ11_HTML.gif
(2.10)

and by (2.8) the representation (2.1) is thus obtained.

This completes the proof. □

Remark 1 Assume that the function f : I C is n-times differentiable on the interior https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_IEq11_HTML.gif of the interval I ( n 1 ) and the n th derivative f ( n ) is of locally bounded variation on https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_IEq11_HTML.gif . If https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_IEq13_HTML.gif with a < b and u : [ a , b ] C is of bounded variation on [ a , b ] , then, by choosing c = a in the formulae above we have
D n d ( f , u , a , b ) : = T n ( f , u , a , a , b ) = k = 0 n 1 k ! f ( k ) ( a ) ( b a ) k u ( b ) k = 0 n 1 1 k ! f ( k + 1 ) ( a ) a b ( λ a ) k u ( λ ) d λ
(2.11)
and
R n d ( f , u , a , b ) : = R n ( f , u , a , a , b ) = 1 n ! a b ( a λ ( λ t ) n d f ( n ) ( t ) ) d u ( λ ) .
(2.12)
This give the representation
a b f ( λ ) d u ( λ ) = d D n ( f , u , a , b ) + d R n ( f , u , a , b ) .
(2.13)
Now, if we choose c = a + b 2 , then we have
M n ( f , u , a , b ) : = T n ( f , u , a , a + b 2 , b ) = k = 0 n 1 k ! 2 k f ( k ) ( a + b 2 ) ( b a ) k [ u ( b ) + ( 1 ) k + 1 u ( a ) ] k = 0 n 1 1 k ! f ( k + 1 ) ( a + b 2 ) a b ( λ a + b 2 ) k u ( λ ) d λ
(2.14)
and
R n M ( f , u , a , b ) : = R n ( f , u , a , a + b 2 , b ) = 1 n ! a b ( a + b 2 λ ( λ t ) n d f ( n ) ( t ) ) d u ( λ ) ,
(2.15)
which provide the representation
a b f ( λ ) d u ( λ ) = M n ( f , u , a , b ) + M R n ( f , u , a , b ) .
(2.16)
Finally, if we choose c = b , then we have
D n u ( f , u , a , b ) : = T n ( f , u , a , b , b ) = k = 0 n 1 k ! f ( k ) ( b ) ( 1 ) k + 1 ( b a ) k u ( a ) + k = 0 n 1 ( 1 ) k + 1 k ! f ( k + 1 ) ( b ) a b ( b λ ) k u ( λ ) d λ
(2.17)
and the remainder
R n u ( f , u , a , b ) : = R n ( f , u , a , b , b ) = ( 1 ) n + 1 n ! a b ( λ b ( t λ ) n d f ( n ) ( t ) ) d u ( λ ) .
(2.18)
Making use of (2.1) we get
a b f ( λ ) d u ( λ ) = u D n ( f , u , a , b ) + u R n ( f , u , a , b ) .
(2.19)

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, [48]).

Lemma 1 Let p , v : [ a , b ] C two bounded functions on the compact interval [ a , b ] .
  1. (i)
    If p is continuous and v is of bounded variation, then the Riemann-Stieltjes integral a b p ( t ) d v ( t ) exists and
    | a b p ( t ) d v ( t ) | max t [ a , b ] | p ( t ) | a b ( v ) ,
    (3.1)
     
where a b ( v ) denotes the total variation of v on the interval [ a , b ] .
  1. (ii)
    If p is Riemann integrable and v is Lipschitzian with the constant L > 0 , i.e.,
    | v ( t ) v ( s ) | L | t s | for each t , s [ a , b ] ,
     
then the Riemann-Stieltjes integral a b p ( t ) d v ( t ) exists and
| a b p ( t ) d v ( t ) | L a b | p ( t ) | d t ( L sup t [ a , b ] | p ( t ) | ( b a ) ) .
(3.2)

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 a b f ( λ ) d u ( λ ) .

Theorem 2 Assume that the function f : I C is n-times differentiable on the interior https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_IEq11_HTML.gif of the interval I ( n 1 ) and the nth derivative f ( n ) is of locally bounded variation on https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_IEq11_HTML.gif . If https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_IEq13_HTML.gif with a < b , c [ a , b ] and u : [ a , b ] C is of bounded variation on [ a , b ] , then we have the representation (2.1), where the approximation term T n ( f , u , a , c , b ) is given by (2.2) and the remainder R n ( f , u , a , c , b ) satisfies the inequality
| R n ( f , u , a , c , b ) | 1 n ! [ 1 2 ( b a ) + | c a + b 2 | ] n a b ( f ( n ) ) a b ( u ) ,
(3.3)

for any c [ a , b ] .

If the nth derivative f ( n ) is Lipschitzian with the constant L n > 0 on [ a , b ] , then we have
| R n ( f , u , a , c , b ) | 1 ( n + 1 ) ! L n [ 1 2 ( b a ) + | c a + b 2 | ] n + 1 a b ( u ) ,
(3.4)

for any c [ a , b ] .

Proof

Utilizing the property (i) from Lemma 1, we have successively
| R n ( f , u , a , c , b ) | = 1 n ! | a b ( c λ ( λ t ) n d f ( n ) ( t ) ) d u ( λ ) | 1 n ! max λ [ a , b ] | c λ ( λ t ) n d f ( n ) ( t ) | a b ( u )
(3.5)

for any c [ a , b ] .

For c , λ [ a , b ] , denote
B ( λ , c ) : = | c λ ( λ t ) n d f ( n ) ( t ) | .
(3.6)
By the property (i) from Lemma 1 applied for f ( n ) we have for c < λ that
B ( λ , c ) max t [ c , λ ] | λ t | n c λ ( f ( n ) ) = ( λ c ) n c λ ( f ( n ) ) ( λ c ) n a b ( f ( n ) ) ( b c ) n a b ( f ( n ) )
and for c > λ that
B ( λ , c ) max t [ λ , c ] | λ t | n λ c ( f ( n ) ) = ( c λ ) n λ c ( f ( n ) ) ( c λ ) n a b ( f ( n ) ) ( c a ) n a b ( f ( n ) ) .
Therefore,
max λ [ a , b ] B ( λ , c ) max { ( b c ) n , ( c a ) n } a b ( f ( n ) ) = [ max { b c , c a } ] n a b ( f ( n ) ) = [ 1 2 ( b a ) + | c a + b 2 | ] n a b ( f ( n ) ) ,
(3.7)

for any c [ a , b ] .

Utilizing (3.5) and (3.7), we deduce the desired inequality (3.3).

By the property (ii) from Lemma 1 applied for f ( n ) , we have that
B ( λ , c ) L n | c λ | λ t | n d t | = L n n + 1 | λ c | n + 1 ,
c , λ [ a , b ]
, which produces the bound
max λ [ a , b ] B ( λ , c ) L n n + 1 max λ [ a , b ] | λ c | n + 1 = L n n + 1 max { ( b c ) n + 1 , ( c a ) n + 1 } = L n n + 1 [ max { b c , c a } ] n + 1 = L n n + 1 [ 1 2 ( b a ) + | c a + b 2 | ] n + 1
(3.8)

for any c [ a , b ] .

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 1 Under the assumptions of Theorem 2 we have the representation
a b f ( λ ) d u ( λ ) = M n ( f , u , a , b ) + M R n ( f , u , a , b ) ,
(3.9)
where M n ( f , u , a , b ) is defined in (2.14) and the error R n M ( f , u , a , b ) satisfies the bound
| M R n ( f , u , a , b ) | 1 2 n n ! ( b a ) n a b ( f ( n ) ) a b ( u ) .
(3.10)
Moreover, if the nth derivative f ( n ) is Lipschitzian with the constant L n > 0 on [ a , b ] , then we have
| M R n ( f , u , a , b ) | 1 2 n + 1 ( n + 1 ) ! L n ( b a ) n + 1 a b ( u ) .
(3.11)

The case of Lipschitzian integrators may be of interest as well and will be considered in the following.

Theorem 3 Assume that the function f : I C is n-times differentiable on the interior https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_IEq11_HTML.gif of the interval I ( n 1 ) and the nth derivative f ( n ) is of locally bounded variation on https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_IEq11_HTML.gif . If https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_IEq13_HTML.gif with a < b , c [ a , b ] and u : [ a , b ] C is Lipschitzian on [ a , b ] with the constant K > 0 then we have the representation (2.1), where the approximation term T n ( f , u , a , c , b ) is given by (2.2) and the remainder R n ( f , u , a , c , b ) satisfies the inequality
| R n ( f , u , a , c , b ) | 1 n ! K a b | λ c | n | c λ ( f ( n ) ) | d λ 1 ( n + 1 ) ! K [ ( b c ) n + 1 + ( c a ) n + 1 ] a b ( f ( n ) )
(3.12)

for any c [ a , b ] .

If the nth derivative f ( n ) is Lipschitzian with the constant L n > 0 on [ a , b ] , then we have
| R n ( f , u , a , c , b ) | 1 ( n + 2 ) ! K L n [ ( b c ) n + 2 + ( c a ) n + 2 ]
(3.13)

for any c [ a , b ] .

Proof

Utilizing the property (ii) from Lemma 1, we have successively
| R n ( f , u , a , c , b ) | = 1 n ! | a b ( c λ ( λ t ) n d f ( n ) ( t ) ) d u ( λ ) | 1 n ! K a b | c λ ( λ t ) n d f ( n ) ( t ) | d λ = 1 n ! K a b B ( λ , c ) d λ
(3.14)

for any c [ a , b ] , where as above B ( λ , c ) : = | c λ ( λ t ) n d f ( n ) ( t ) | , for c , λ [ a , b ] .

By the property (i) from Lemma 1 applied for f ( n ) , we have for c < λ that
B ( λ , c ) max t [ c , λ ] | λ t | n c λ ( f ( n ) ) = ( λ c ) n c λ ( f ( n ) )
and for c > λ that
B ( λ , c ) max t [ λ , c ] | λ t | n λ c ( f ( n ) ) = ( c λ ) n λ c ( f ( n ) )
which gives that
B ( λ , c ) | λ c | n | c λ ( f ( n ) ) | | λ c | n a b ( f ( n ) )

for c , λ [ a , b ] .

This implies that
a b B ( λ , c ) d λ a b | λ c | n | c λ ( f ( n ) ) | d λ a b ( f ( n ) ) a b | λ c | n d λ = 1 n + 1 [ ( b c ) n + 1 + ( c a ) n + 1 ] a b ( f ( n ) )
(3.15)

for c [ a , b ] .

Making use of (3.14) and (3.15) we deduce the desired inequality (3.12).

By the property (ii) from Lemma 1 applied for f ( n ) we have that
B ( λ , c ) L n | c λ | λ t | n d t | = L n n + 1 | λ c | n + 1
c , λ [ a , b ]
, which produces the bound
a b B ( λ , c ) d λ L n n + 1 a b | λ c | n + 1 d λ = L n ( n + 1 ) ( n + 2 ) [ ( b c ) n + 2 + ( c a ) n + 2 ]
(3.16)

for c [ a , b ] .

Utilizing (3.14) and (3.16), we deduce the desired inequality (3.13). □

The following particular case provides the best error bounds.

Corollary 2 Under the assumptions of Theorem 3, we have the representation (3.9), where M n ( f , u , a , b ) is defined in (2.14) and the error R n M ( f , u , a , b ) satisfies the bound
| M R n ( f , u , a , b ) | 1 n ! K a b | λ a + b 2 | n | a + b 2 λ ( f ( n ) ) | d λ 1 2 n ( n + 1 ) ! K ( b a ) n + 1 a b ( f ( n ) ) .
(3.17)
Moreover, if the nth derivative f ( n ) is Lipschitzian with the constant L n > 0 on [ a , b ] , then we have
| M R n ( f , u , a , b ) | 1 2 n + 1 ( n + 2 ) ! K L n ( b a ) n + 2 .
(3.18)

4 Applications

  1. 1.
    We consider the following finite Laplace-Stieltjes transform defined by
    ( L [ a , b ] g ) ( s ) : = a b e s t d g ( t ) ,
    (4.1)
     

where a , b are real numbers with a < b , s is a complex number and g : [ a , b ] C is a function of bounded variation.

It is important to notice that, in the particular case g ( t ) = t , t [ a , b ] , (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 Laplace-Stieltjes transform can be used for the particular case of finite Laplace transform.

Since the function f s : [ a , b ] C , f s ( t ) : = e s t is continuous for any s C , the transform (4.1) is well defined for any s C .

We observe that the function f s has derivatives of all orders and
f s ( k ) ( t ) = ( 1 ) k s k e s t for any  s C , t [ a , b ]  and  k 0 .
(4.2)
We also observe that
f s ( n + 1 ) [ a , b ] , : = sup t [ a , b ] | f s ( n + 1 ) ( t ) | = | s | n + 1 sup t [ a , b ] | e s t | = | s | n + 1 sup t [ a , b ] e t Re s = | s | n + 1 × { e a Re s if  Re s 0 , e b Re s if  Re s < 0 .
To simplify the notations, we denote by
β [ a , b ] ( s ) : = { e a Re s if  Re s 0 , e b Re s if  Re s < 0 .
(4.3)
On utilizing Theorem 1, we have the representation
( L [ a , b ] g ) ( s ) = G n ( g , a , c , b ) ( s ) + Z n ( g , a , c , b ) ( s ) ,
(4.4)
where
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_Equ43_HTML.gif
(4.5)
and the remainder Z n ( g , a , c , b ) ( s ) can be represented as
Z n ( g , a , c , b ) ( s ) : = ( 1 ) n + 1 n ! s n + 1 a b ( c λ ( λ t ) n e s t d t ) d g ( λ ) .
(4.6)

Here, s C and c [ a , b ] .

Since g is of bounded variation on [ a , b ] and the derivative f s ( n ) is Lipschitzian with the constant
L n : = f s ( n + 1 ) [ a , b ] , = | s | n + 1 β [ a , b ] ( s )
then by Theorem 2 we have the bound
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_Equ45_HTML.gif
(4.7)

for any s C and c [ a , b ] .

As above, the best approximation we can get from (4.4) is for c = a + b 2 , namely, we have the representation
( L [ a , b ] g ) ( s ) = M G n ( g , a , b ) ( s ) + M Z n ( g , a , b ) ( s ) ,
(4.8)
where
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_Equ47_HTML.gif
(4.9)
and the remainder Z n M ( g , a , b ) ( s ) can be represented as
Z n M ( g , a , b ) ( s ) : = ( 1 ) n + 1 n ! s n + 1 a b ( a + b 2 λ ( λ t ) n e s t d t ) d g ( λ ) .
(4.10)
The error Z n M ( g , a , b ) ( s ) satisfies the bound
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_Equ49_HTML.gif
(4.11)

for any s C .

Now, if we restrict the function g to belong to the class of Lipschitzian functions with the constant K > 0 on the interval [ a , b ] , then the error in the representation (4.4) will satisfy the bound
| Z n ( g , a , c , b ) ( s ) | 1 ( n + 2 ) ! K | s | n + 1 β [ a , b ] ( s ) [ ( b c ) n + 2 + ( c a ) n + 2 ]

for any s C and c [ a , b ] .

Finally, the error Z n M ( g , a , b ) ( s ) from the representation (4.8) satisfies the inequality
| M Z n ( g , a , b ) ( s ) | 1 2 n + 1 ( n + 2 ) ! K | s | n + 1 β [ a , b ] ( s ) ( b a ) n + 2
for any s C .
  1. 2.
    We consider now the finite Fourier-Stieltjes sine and cosine transforms defined by
    ( F s , [ a , b ] g ) ( u ) : = a b sin ( u t ) d g ( t ) , ( F c , [ a , b ] g ) ( u ) : = a b cos ( u t ) d g ( t ) ,
    (4.12)
     

where a, b are real numbers with a < b , u is a real number and g : [ a , b ] C is a function of bounded variation.

Since the functions f s ; u , f c ; u : [ a , b ] R , f s ; u ( t ) : = sin ( u t ) , f c ; u ( t ) : = cos ( u t ) are continuous for any u R , the transforms (4.12) are well defined for any u R .

Utilizing the well-known formulae for the n th derivatives of sine and cosine functions, namely,
if  y = sin ( A x + B )  then  d n y d x n = A n sin ( A x + B n π 2 )
and
if  y = cos ( A x + B )  then  d n y d x n = A n cos ( A x + B n π 2 ) ,
then we have
f s ; u ( k ) ( t ) = u k sin ( u t k π 2 ) and f c ; u ( k ) ( t ) = u k cos ( u t k π 2 )

for any u R and k 0 .

We observe that, in general, we have the bounds
f s ; u ( n + 1 ) [ a , b ] , = sup t [ a , b ] | u n + 1 sin ( u t ( n + 1 ) π 2 ) | | u | n + 1
and
f c ; u ( n + 1 ) [ a , b ] , = sup t [ a , b ] | u n + 1 cos ( u t ( n + 1 ) π 2 ) | | u | n + 1

for any u R , the closed interval [ a , b ] and n 0 .

On utilizing Theorem 1, we have the representation
( F s , [ a , b ] g ) ( u ) = K s , n ( g , a , c , b ) ( u ) + W s , n ( g , a , c , b ) ( u ) ,
(4.13)
where
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_Equ52_HTML.gif
(4.14)
and the remainder W s , n ( g , a , c , b ) ( u ) can be represented as
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_Equ53_HTML.gif
(4.15)
Since g is of bounded variation on [ a , b ] and the derivative f s ( n ) is Lipschitzian with the constant
L n : = f s ( n + 1 ) [ a , b ] , | u | n + 1
then by Theorem 2 we have the bound
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_Equ54_HTML.gif
(4.16)

for any u R and c [ a , b ] .

As above, the best approximation we can get from (4.4) is for c = a + b 2 , namely, we have the representation
( F s , [ a , b ] g ) ( u ) = M K s , n ( g , a , b ) ( u ) + M W s , n ( g , a , b ) ( u ) ,
(4.17)
where
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_Equ56_HTML.gif
(4.18)
and the remainder W s , n M ( g , a , b ) ( u ) can be represented as
https://static-content.springer.com/image/art%3A10.1186%2F1029-242X-2013-154/MediaObjects/13660_2012_Article_639_Equ57_HTML.gif
(4.19)

for any u R .

Here, the error satisfies the bound
| M W s , n ( g , a , b ) ( u ) | 1 2 n + 1 ( n + 1 ) ! | u | n + 1 ( b a ) n + 1 a b ( g )
(4.20)

for any u R .

Now, if we restrict the function g to belong to the class of Lipschitzian functions with the constant K > 0 on the interval [ a , b ] , then the error in the representation (4.17) will satisfy the bound:
| W s , n ( g , a , c , b ) ( u ) | 1 ( n + 2 ) ! K | u | n + 1 [ ( b c ) n + 2 + ( c a ) n + 2 ] ,
(4.21)

for any u R and c [ a , b ] .

Finally, the error from the representation (4.17) satisfies the inequality
| M W s , n ( g , a , b ) ( u ) | 1 2 n + 1 ( n + 2 ) ! K | u | n + 1 ( b a ) n + 2
(4.22)

for any u R .

Similar results may be stated for the finite Fourier-Stieltjes cosine transform, however the details are left to the interested reader.

Declarations

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.

Authors’ Affiliations

(1)
School of Computational & Applied Mathematics, University of the Witwatersrand
(2)
Mathematics, College of Engineering & Science, Victoria University

References

  1. 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
  2. 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
  3. 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
  4. 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
  5. Diethelm K: A note on the midpoint rectangle formula for Riemann-Stieltjes integrals. J. Stat. Comput. Simul. 2004, 74(12):920–922.MathSciNetGoogle Scholar
  6. Liu Z: Refinement of an inequality of Grüss type for Riemann-Stieltjes integral. Soochow J. Math. 2004, 30(4):483–489.MathSciNetGoogle Scholar
  7. 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
  8. 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
  9. Mozyrska D, Pawluszewicz E, Torres DFM: The Riemann-Stieltjes integral on time scales. Aust. J. Math. Anal. Appl. 2010., 7(1): Article ID 10
  10. 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
  11. 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
  12. 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
  13. 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
  14. 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
  15. 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
  16. 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
  17. 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
  18. Dragomir SS: On the Ostrowski’s inequality for Riemann-Stieltjes integral. Korean J. Comput. Appl. Math. 2000, 7: 477–485.MathSciNetGoogle Scholar
  19. Mitrinović DS, Pečarić JE, Fink AM: Inequalities Involving Functions and Their Integrals and Derivatives. Kluwer Academic, Dordrecht; 1991.View ArticleGoogle Scholar
  20. Dragomir SS, Rassias TM (Eds): Ostrowski Type Inequalities and Applications in Numerical Integration. Kluwer Academic, Dordrecht; 2002.Google Scholar
  21. Anastassiou AG: Univariate Ostrowski inequalities, revisited. Monatshefte Math. 2002, 135(3):175–189. 10.1007/s006050200015MathSciNetView ArticleGoogle Scholar
  22. Anastassiou AG: Ostrowski type inequalities. Proc. Am. Math. Soc. 1995, 123(12):3775–3781. 10.1090/S0002-9939-1995-1283537-3MathSciNetView ArticleGoogle Scholar
  23. 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
  24. 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
  25. 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
  26. 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
  27. 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
  28. Cerone P: Approximate multidimensional integration through dimension reduction via the Ostrowski functional. Nonlinear Funct. Anal. Appl. 2003, 8(3):313–333.MathSciNetGoogle Scholar
  29. Cerone P, Dragomir SS: On some inequalities arising from Montgomery’s identity. J. Comput. Anal. Appl. 2003, 5(4):341–367.MathSciNetGoogle Scholar
  30. 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
  31. Pachpatte BG: A note on Ostrowski like inequalities. J. Inequal. Pure Appl. Math. 2005., 6(4): Article ID 114
  32. 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
  33. 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
  34. Dragomir SS: Ostrowski’s inequality for montonous mappings and applications. J. KSIAM 1999, 3(1):127–135.Google Scholar
  35. 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
  36. 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.html
  37. Dragomir SS: Accurate approximations for the Riemann-Stieltjes integral via theory of inequalities. J. Math. Inequal. 2009, 3(4):663–681.MathSciNetView ArticleGoogle Scholar
  38. 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 2011
  39. 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
  40. 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
  41. 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
  42. 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
  43. Pachpatte BG: A note on a trapezoid type integral inequality. Bull. Greek Math. Soc. 2004, 49: 85–90.MathSciNetGoogle Scholar
  44. 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
  45. 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
  46. Dragomir SS: On the Ostrowski inequality for Riemann-Stieltjes integral a b f ( t ) d u ( t ) , where f is of Hölder type and u is of bounded variation and applications. J. KSIAM 2001, 5(1):35–45.Google Scholar
  47. 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
  48. Apostol TM: Mathematical Analysis. 2nd edition. Addison-Wesley, Reading; 1975.Google Scholar
  49. 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
  50. 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
  51. 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
  52. 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

Copyright

© Dragomir and Abelman; licensee Springer. 2013

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.