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

  • SS DragomirEmail author
  • S Abelman
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Research Article
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Abstract

In the present paper, we investigate the problem of approximating the Riemann-Stieltjes integral a b f ( λ ) d u ( λ ) Open image in new window in the case when the integrand f is n-time differentiable and the derivative f ( n ) Open image in new window is either of locally bounded variation, or Lipschitzian on an interval incorporating [ a , b ] Open image in new window. 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 ) Open image in new window, 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 ] Open image in new window, 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 [10, 11, 12, 13, 14, 15, 16] 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 ) Open image in new window, where p , v : [ a , b ] R Open image in new window 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 u ( t ) = t Open image in new window, t [ a , b ] Open image in new window, 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 [10, 11, 12, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32] and [33]. For other results concerning error bounds of quadrature rules related to midpoint and trapezoid rules, see [34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45] 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 ) Open image in new window in the case when the integrand f is n-times differentiable and the derivative f ( n ) Open image in new window is either of locally bounded variation, or Lipschitzian on an interval incorporating [ a , b ] Open image in new window. 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 Open image in new window is n-times differentiable on the interior Open image in new window of the interval I ( n 1 Open image in new window) and the nth derivative f ( n ) Open image in new window is of locally bounded variation on Open image in new window . If Open image in new window with a < b Open image in new window, c [ a , b ] Open image in new window and u : [ a , b ] C Open image in new window is of bounded variation on [ a , b ] Open image in new window, then the Riemann-Stieltjes integral a b f ( λ ) d u ( λ ) Open image in new window 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 ) , Open image in new window
(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 Open image in new window
(2.2)
and the remainder R n ( f , u , a , c , b ) Open image in new window 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 ) . Open image in new window
(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 ) Open image in new window
(2.4)

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

We can prove this equality by induction.

Indeed, for n = 0 Open image in new window, we have
f ( λ ) = f ( c ) + c λ d f ( t ) Open image in new window

that holds for any function of locally bounded variation on Open image in new window .

Now, assume that (2.4) is true for an n 0 Open image in new window and let us prove that it holds for ‘ n + 1 Open image in new window’, 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 ) Open image in new window
(2.5)

provided that the function f : I C Open image in new window is ( n + 1 ) Open image in new window-times differentiable on the interior Open image in new window of the interval I and the ( n + 1 ) Open image in new window-th derivative f ( n + 1 ) Open image in new window is of locally bounded variation on Open image in new window .

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:
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 ! Open image in new window
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 ] . Open image in new window
(2.7)

We observe that, by division with ( n + 1 ) ! Open image in new window, 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 ) . Open image in new window
(2.8)
Utilizing the integration by parts formula, we have for k 1 Open image in new window 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 . Open image in new window
(2.9)

For k = 0 Open image in new window, we have a b d u ( λ ) = u ( b ) u ( a ) Open image in new window.

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 f : I C Open image in new window is n-times differentiable on the interior Open image in new window of the interval I ( n 1 Open image in new window) and the n th derivative f ( n ) Open image in new window is of locally bounded variation on Open image in new window . If Open image in new window with a < b Open image in new window and u : [ a , b ] C Open image in new window is of bounded variation on [ a , b ] Open image in new window, then, by choosing c = a Open image in new window 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 Open image in new window
(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 ) . Open image in new window
(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 ) . Open image in new window
(2.13)
Now, if we choose c = a + b 2 Open image in new window, 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 Open image in new window
(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 ) , Open image in new window
(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 ) . Open image in new window
(2.16)
Finally, if we choose c = b Open image in new window, 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 Open image in new window
(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 ) . Open image in new window
(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 ) . Open image in new window
(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 Open image in new window two bounded functions on the compact interval [ a , b ] Open image in new window.
  1. (i)
    If p is continuous and v is of bounded variation, then the Riemann-Stieltjes integral a b p ( t ) d v ( t ) Open image in new window exists and
    | a b p ( t ) d v ( t ) | max t [ a , b ] | p ( t ) | a b ( v ) , Open image in new window
    (3.1)
     
where a b ( v ) Open image in new window denotes the total variation of v on the interval [ a , b ] Open image in new window.
  1. (ii)
    If p is Riemann integrable and v is Lipschitzian with the constant L > 0 Open image in new window, i.e.,
    | v ( t ) v ( s ) | L | t s | for each t , s [ a , b ] , Open image in new window
     
then the Riemann-Stieltjes integral a b p ( t ) d v ( t ) Open image in new window 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 ) ) . Open image in new window
(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 ) Open image in new window.

Theorem 2 Assume that the function f : I C Open image in new window is n-times differentiable on the interior Open image in new window of the interval I ( n 1 Open image in new window) and the nth derivative f ( n ) Open image in new window is of locally bounded variation on Open image in new window . If Open image in new window with a < b Open image in new window, c [ a , b ] Open image in new window and u : [ a , b ] C Open image in new window is of bounded variation on [ a , b ] Open image in new window, then we have the representation (2.1), where the approximation term T n ( f , u , a , c , b ) Open image in new window is given by (2.2) and the remainder R n ( f , u , a , c , b ) Open image in new window 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 ) , Open image in new window
(3.3)

for any c [ a , b ] Open image in new window.

If the nth derivative f ( n ) Open image in new window is Lipschitzian with the constant L n > 0 Open image in new window on [ a , b ] Open image in new window, 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 ) , Open image in new window
(3.4)

for any c [ a , b ] Open image in new window.

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 ) Open image in new window
(3.5)

for any c [ a , b ] Open image in new window.

For c , λ [ a , b ] Open image in new window, denote
B ( λ , c ) : = | c λ ( λ t ) n d f ( n ) ( t ) | . Open image in new window
(3.6)
By the property (i) from Lemma 1 applied for f ( n ) Open image in new window we have for c < λ Open image in new window 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 ) ) Open image in new window
and for c > λ Open image in new window 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 ) ) . Open image in new window
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 ) ) , Open image in new window
(3.7)

for any c [ a , b ] Open image in new window.

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

By the property (ii) from Lemma 1 applied for f ( n ) Open image in new window, we have that
B ( λ , c ) L n | c λ | λ t | n d t | = L n n + 1 | λ c | n + 1 , Open image in new window
c , λ [ a , b ] Open image in new window
, 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 Open image in new window
(3.8)

for any c [ a , b ] Open image in new window.

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 ) , Open image in new window
(3.9)
where M n ( f , u , a , b ) Open image in new window is defined in (2.14) and the error R n M ( f , u , a , b ) Open image in new window satisfies the bound
| M R n ( f , u , a , b ) | 1 2 n n ! ( b a ) n a b ( f ( n ) ) a b ( u ) . Open image in new window
(3.10)
Moreover, if the nth derivative f ( n ) Open image in new window is Lipschitzian with the constant L n > 0 Open image in new window on [ a , b ] Open image in new window, 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 ) . Open image in new window
(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 Open image in new window is n-times differentiable on the interior Open image in new window of the interval I ( n 1 Open image in new window) and the nth derivative f ( n ) Open image in new window is of locally bounded variation on Open image in new window . If Open image in new window with a < b Open image in new window, c [ a , b ] Open image in new window and u : [ a , b ] C Open image in new window is Lipschitzian on [ a , b ] Open image in new window with the constant K > 0 Open image in new window then we have the representation (2.1), where the approximation term T n ( f , u , a , c , b ) Open image in new window is given by (2.2) and the remainder R n ( f , u , a , c , b ) Open image in new window 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 ) ) Open image in new window
(3.12)

for any c [ a , b ] Open image in new window.

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

for any c [ a , b ] Open image in new window.

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 Open image in new window
(3.14)

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

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

for c , λ [ a , b ] Open image in new window.

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 ) ) Open image in new window
(3.15)

for c [ a , b ] Open image in new window.

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 ) Open image in new window we have that
B ( λ , c ) L n | c λ | λ t | n d t | = L n n + 1 | λ c | n + 1 Open image in new window
c , λ [ a , b ] Open image in new window
, 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 ] Open image in new window
(3.16)

for c [ a , b ] Open image in new window.

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 ) Open image in new window is defined in (2.14) and the error R n M ( f , u , a , b ) Open image in new window 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 ) ) . Open image in new window
(3.17)
Moreover, if the nth derivative f ( n ) Open image in new window is Lipschitzian with the constant L n > 0 Open image in new window on [ a , b ] Open image in new window, then we have
| M R n ( f , u , a , b ) | 1 2 n + 1 ( n + 2 ) ! K L n ( b a ) n + 2 . Open image in new window
(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 ) , Open image in new window
    (4.1)
     

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

It is important to notice that, in the particular case g ( t ) = t , t [ a , b ] Open image in new window, (4.1) becomes the finite Laplace transform which has various applications in other fields of Mathematics; see, for instance, [25, 26, 49, 50, 51] 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 Open image in new window, f s ( t ) : = e s t Open image in new window is continuous for any s C Open image in new window, the transform (4.1) is well defined for any s C Open image in new window.

We observe that the function f s Open image in new window 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 . Open image in new window
(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 . Open image in new window
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 . Open image in new window
(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 ) , Open image in new window
(4.4)
and the remainder Z n ( g , a , c , b ) ( s ) Open image in new window 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 ) . Open image in new window
(4.6)

Here, s C Open image in new window and c [ a , b ] Open image in new window.

Since g is of bounded variation on [ a , b ] Open image in new window and the derivative f s ( n ) Open image in new window is Lipschitzian with the constant
L n : = f s ( n + 1 ) [ a , b ] , = | s | n + 1 β [ a , b ] ( s ) Open image in new window
then by Theorem 2 we have the bound

for any s C Open image in new window and c [ a , b ] Open image in new window.

As above, the best approximation we can get from (4.4) is for c = a + b 2 Open image in new window, namely, we have the representation
( L [ a , b ] g ) ( s ) = M G n ( g , a , b ) ( s ) + M Z n ( g , a , b ) ( s ) , Open image in new window
(4.8)
and the remainder Z n M ( g , a , b ) ( s ) Open image in new window 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 ) . Open image in new window
(4.10)
The error Z n M ( g , a , b ) ( s ) Open image in new window satisfies the bound

for any s C Open image in new window.

Now, if we restrict the function g to belong to the class of Lipschitzian functions with the constant K > 0 Open image in new window on the interval [ a , b ] Open image in new window, 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 ] Open image in new window

for any s C Open image in new window and c [ a , b ] Open image in new window.

Finally, the error Z n M ( g , a , b ) ( s ) Open image in new window 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 Open image in new window
for any s C Open image in new window.
  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 ) , Open image in new window
    (4.12)
     

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

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

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 ) Open image in new window
and
if  y = cos ( A x + B )  then  d n y d x n = A n cos ( A x + B n π 2 ) , Open image in new window
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 ) Open image in new window

for any u R Open image in new window and k 0 Open image in new window.

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 Open image in new window
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 Open image in new window

for any u R Open image in new window, the closed interval [ a , b ] Open image in new window and n 0 Open image in new window.

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 ) , Open image in new window
(4.13)
and the remainder W s , n ( g , a , c , b ) ( u ) Open image in new window can be represented as
Since g is of bounded variation on [ a , b ] Open image in new window and the derivative f s ( n ) Open image in new window is Lipschitzian with the constant
L n : = f s ( n + 1 ) [ a , b ] , | u | n + 1 Open image in new window
then by Theorem 2 we have the bound

for any u R Open image in new window and c [ a , b ] Open image in new window.

As above, the best approximation we can get from (4.4) is for c = a + b 2 Open image in new window, 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 ) , Open image in new window
(4.17)
and the remainder W s , n M ( g , a , b ) ( u ) Open image in new window can be represented as

for any u R Open image in new window.

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 ) Open image in new window
(4.20)

for any u R Open image in new window.

Now, if we restrict the function g to belong to the class of Lipschitzian functions with the constant K > 0 Open image in new window on the interval [ a , b ] Open image in new window, 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 ] , Open image in new window
(4.21)

for any u R Open image in new window and c [ a , b ] Open image in new window.

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 Open image in new window
(4.22)

for any u R Open image in new window.

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

Notes

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.

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© 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.

Authors and Affiliations

  1. 1.School of Computational & Applied MathematicsUniversity of the WitwatersrandJohannesburgSouth Africa
  2. 2.Mathematics, College of Engineering & ScienceVictoria UniversityMelbourneAustralia

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