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Approximating the finite Hilbert transform via a companion of Ostrowski’s inequality for function of bounded variation and applications
Applied Mathematics and Computation 247 (2014) 373–385
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Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Approximating the finite Hilbert transform via a companion of Ostrowski’s inequality for function of bounded variation and applications Wenjun Liu ⇑, Xingyue Gao College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
a r t i c l e
i n f o
a b s t r a c t
Keywords: A companion of Ostrowski’s inequality Finite Hilbert transform Function of bounded variation Numerical experiments
The finite Hilbert transform plays an important role in scientific and engineering computing. By using a companion of Ostrowski’s inequality for function of bounded variation, we give some new approximations of the finite Hilbert transform, which may have the better error bounds than the known results obtained via Ostrowski type inequality. Some numerical experiments are also presented. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Finite Hilbert transform plays an important role in many fields and its mathematics expression is
ðTf Þ ða; b; t Þ ¼
1
p
PV
Z a
b
f ðsÞ 1 ds :¼ lim st p e!0þ
"Z
a
te
f ðsÞ ds þ st
Z
b
tþe
# f ð sÞ ds : st
ð1:1Þ
For different approaches in approximating the finite Hilbert transform (1.1) including: noninterpolatory, interpolatory, Gaussian, Chebychevian, Taylor formula with integral remainder and spline methods, we refer the papers [4,5,7,11– 13,15,18,20–22,24]. In [10], Dragomir proved the following Ostrowski type inequality for function of bounded variation: Theorem 1.1. Let u : ½a; b ! R be a function of bounded variation on ½a; b. Then, for all x 2 ½a; b, we have the inequality:
b Z b 1 a þ b _ uðxÞðb aÞ uðtÞdt 6 ðb aÞ þ x ðuÞ; 2 2 a a
where
b W
ðuÞ denotes the total variation of u on ½a; b. The constant
a
ð1:2Þ 1 2
is the best possible one.
In [6], Dragomir approximated the finite Hilbert transform via the above Ostrowski type inequality for function of bounded variation and proved the following inequality on the interval ½a; b: 0
Theorem 1.2. Let f : ½a; b ! R be a function such that its derivative f : ½a; b ! R is of bounded variation on ½a; b. Then we have the inequality: ⇑ Corresponding author. E-mail addresses: [email protected] (W. Liu), [email protected] (X. Gao). http://dx.doi.org/10.1016/j.amc.2014.08.099 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
ðTf Þða; b; tÞ f ðtÞ ln b t b a ½f ; kt þ ð1 kÞb; kt þ ð1 kÞa ta p p _ b 1 1 1 1 a þ b 0 þ k ðb aÞ þ t ðf Þ 6 2 a p 2 2 2
ð1:3Þ
for any t 2 ½a; b and k 2 ½0; 1Þ, where ½f ; a; b is the divided difference, i.e.,
½f ; a; b :¼
f ðaÞ f ðbÞ : ab
ð1:4Þ
Then, Dragomir proved the following inequality with the derivative of bounded variation on equidistant division of ½a; b: 0
Theorem 1.3. Let f : ½a; b ! R be a differentiable function such that its derivative f is of bounded variation on ½a; b. If k ¼ ðki Þi¼0;n1 ; ki 2 ½0; 1Þ ði ¼ 0; n 1Þ and
Sn ðf ; k; tÞ ¼
n1 b aX bt at f ; ði þ 1 ki Þ þ t; ði þ 1 ki Þ þt ; n n pn i¼0
ð1:5Þ
then we have the estimate:
ðTf Þða; b; tÞ f ðtÞ ln b t Sn ðf ; k; tÞ ta p " # b ba 1 1 1 a þ b _ 0 6 þ max ki ðb aÞ þ t ðf Þ np 2 i¼0;n1 2 2 2 a 6
b b a_ 0 ðf Þ: np a
ð1:6Þ
Recently, some companions of Ostrowski’s inequality were established in [1–3,9,17,19] (see also [14,16,23,25] for other related Ostrowski type inequalities). A companion of Ostrowski’s inequality for functions of bounded variation was established in [8]: Theorem 1.4. Let u : ½a; b ! R be a function of bounded variation on ½a; b. Then, for all x 2 ½a; aþb 2 , we have the inequality:
b Z b uðxÞ þ uða þ b xÞ 1 3a þ b _ ðb aÞ uðtÞdt 6 ðb aÞ þ x ðuÞ; 2 4 4 a a
where
b W ðuÞ denotes the total variation of u on ½a; b. The constant a
1 4
ð1:7Þ
is the best possible one.
In this paper, motivated by [6,8], we shall point out a new method in approximating the finite Hilbert transform by the use of the above inequality (1.7). In Section 2, we will estimate the finite Hilbert transform via a companion of Ostrowski’s inequality for function of bounded variation on the interval ½a; b, and point out some interesting inequalities for the functions for which the finite Hilbert transforms can be expressed in terms of special functions. Our result can give a smaller error bound than the similar result in [6] (see Remarks 1, 2 and 5 below). In Section 3, we will prove a quadrature formula for equidistant division of ½a; b in approximating the finite Hilbert transform of a differentiable function with the derivative of bounded variation. In Section 4, we may get a more general quadrature formula. Some numerical examples for the obtained approximations are presented in Section 5. 2. Some inequalities in estimating the finite Hilbert transform Using the above a companion of Ostrowski’s inequality (1.7), we may point out the following result in estimating the finite Hilbert transform. 0
Theorem 2.1. Let f : ½a; b ! R be a function such that its derivative f : ½a; b ! R is of bounded variation on ½a; b. Then we have the inequality:
ðTf Þða; b; tÞ f ðtÞ ln b t ta p b a ½f ; kt þ ð1 kÞb; kt þ ð1 kÞa þ ½f ; kb þ ð1 kÞt; ka þ ð1 kÞt 2 p _ b 1 1 3 1 a þ b 0 6 þ k ðb aÞ þ t ðf Þ 2 p 4 4 2 a
ð2:1Þ
W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
375
for any t 2 ½a; b and k 2 ½12 ; 1Þ, where ½f ; a; b is the divided difference given in (1.4). 0
Proof. Since f is bounded on ½a; b, it follows that f is Lipschitzian on ½a; b and thus the finite Hilbert transform exists everywhere in ða; bÞ. As in [6], for the function f 0 : ða; bÞ ! R; f 0 ðtÞ ¼ 1; t 2 ða; bÞ, we have
ðTf Þða; b; tÞ ¼
1
p
ln
bt ; ta
t 2 ða; bÞ;
then obviously
ðTf Þða; b; tÞ
f ðtÞ
p
ln
Z b bt 1 f ðsÞ f ðtÞ ¼ PV ds: ta p st a
ð2:2Þ
0
Now, if we choose in (1.7), u ¼ f ; x ¼ kc þ ð1 kÞd; k 2 ½12 ; 1Þ, then we get
0 0 f ðdÞ f ðcÞ f ðkc þ ð1 kÞdÞ þ f ðkd þ ð1 kÞcÞ ðd cÞ 2 _ 1 3c þ d d 0 6 jd cj þ kc þ ð1 kÞd ðf Þ; 4 4 c
where c; d 2 ða; bÞ, which is equivalent to
d f ðdÞ f ðcÞ f 0 ðkc þ ð1 kÞdÞ þ f 0 ðkd þ ð1 kÞcÞ 1 3 _ 0 6 þ k ðf Þ dc 2 4 4 c
ð2:3Þ
for any c; d 2 ða; bÞ; c – d. Using (2.3), we may write
Z b Z b 1 f ðsÞ f ðtÞ 1 0 0 PV ½f ðkt þ ð1 kÞsÞ þ f ðks þ ð1 kÞtÞds ds PV p 2p st a a Z b s 1 1 3 _ 0 6 þ k PV ðf Þds p 4 4 a t # "Z t _ Z b_ t s 1 1 3 0 0 ¼ þ k ðf Þds þ ðf Þds p 4 4 a s t t " # t b _ _ 1 1 3 0 0 ðt aÞ ðf Þ þ ðb tÞ ðf Þ 6 þ k p 4 4 a t b _ 0 1 1 3 1 a þ b 6 þ k ðb aÞ þ t ðf Þ: 2 a p 4 4 2
ð2:4Þ
Since (for k – 1) (see [6])
1 PV 2p
Z
b
0
0
½f ðkt þ ð1 kÞsÞ þ f ðks þ ð1 kÞtÞds "Z Z # a
te b 1 0 0 ½f ðkt þ ð1 kÞsÞ þ f ðks þ ð1 kÞtÞds lim þ 2p e!0þ a tþe 1 f ðkt þ ð1 kÞaÞ þ f ðkt þ ð1 kÞbÞ f ðka þ ð1 kÞtÞ þ f ðkb þ ð1 kÞtÞ ¼ þ 2p 1k k b a ½f ; kt þ ð1 kÞb; kt þ ð1 kÞa þ ½f ; kb þ ð1 kÞt; ka þ ð1 kÞt ; ¼ 2 p
¼
using (2.4) and (2.2), we deduce the desired result (2.1). It is obvious that the best inequality we can get from (2.1) is the one for k ¼ 34. Thus, we may state the following corollary. Corollary 2.1. With the assumption of Theorem 2.1, we have
3tþb 3tþa 3bþt 3aþt _ b 0 ðTf Þða; b; tÞ f ðtÞ ln b t b a f ; 4 ; 4 þ f ; 4 ; 4 6 1 1 ðb aÞ þ t a þ b ðf Þ: 2 ta 4p 2 2 a p p
ð2:5Þ
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W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
Remark 1. If we take k ¼ 12 in (1.3), we get
_ b 0 ðTf Þða; b; tÞ f ðtÞ ln b t b a f ; t þ b ; t þ a 6 1 1 ðb aÞ þ t a þ b ðf Þ: ta 2 2 2p 2 2 a p p
ð2:6Þ
We note that inequality (2.5) we obtained here gives a new estimate of the finite Hilbert transform and a smaller error bound than that of (2.6). The above Theorem 2.1 may be used to point out some interesting inequalities for the functions for which the finite Hilbert transforms ðTf Þða; b; tÞ can be expressed in terms of special functions. For instance, we have: (1) Assume that f : ½a; b ð0; 1Þ ! R; f ðxÞ ¼ 1x. Then
ðTf Þða; b; tÞ ¼
1 ðb tÞa ; ln ðt aÞb pt
t 2 ða; bÞ;
b a ½f ; kt þ ð1 kÞb; kt þ ð1 kÞa þ ½f ; kb þ ð1 kÞt; ka þ ð1 kÞt 2 p ba 1 1 ; þ ¼ 2p ½kt þ ð1 kÞb½kt þ ð1 kÞa ½kb þ ð1 kÞt½ka þ ð1 kÞt Z b _ 0 ðf Þ ¼ a
b
2
00
jf ðtÞjdt ¼
a
b a2 2
a2 b
:
Using the inequality (2.1) we may write that
1 ln ðb tÞa 1 ln b t pt ðt aÞb ta pt ba 1 1 þ þ 2p ½kt þ ð1 kÞb½kt þ ð1 kÞa ½kb þ ð1 kÞt½ka þ ð1 kÞt 2 1 1 3 1 a þ b b a2 6 þ k ðb aÞ þ t ; 4 2 2 a2 b2 p 4 which is equivalent to
b a 1 1 1 b þ ln 2 ½kt þ ð1 kÞb½kt þ ð1 kÞa ½kb þ ð1 kÞt½ka þ ð1 kÞt t a 2 1 3 1 a þ b b a2 6 þ k ðb aÞ þ t : 4 4 2 2 a2 b2 If we use the notations
ba ðthe logarithmic meanÞ; ln b ln a Ak ðx; yÞ :¼ kx þ ð1 kÞy ðthe weighted arithmetic meanÞ; pffiffiffiffiffiffi Gða; bÞ :¼ ab ðthe geometric meanÞ;
Lða; bÞ :¼
Aða; bÞ :¼
aþb 2
ðthe arithmetic meanÞ;
then by (2.7), we get the following proposition: Proposition 1. With the above assumption, we have
1 1 1 1 þ 2 Ak ðt; bÞAk ðt; aÞ Ak ðb; tÞAk ða; tÞ tLða; bÞ 1 3 1 2Aða; bÞ 6 þ k ðb aÞ þ jt Aða; bÞj 4 4 4 2 G ða; bÞ for any t 2 ða; bÞ; k 2 ½12 ; 1Þ.
ð2:7Þ
W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
377
In particular, for t ¼ Aða; bÞ and k ¼ 34, we get
1 16 16 1 þ 2 ð3Aða; bÞ þ bÞð3Aða; bÞ þ aÞ ðAða; bÞ þ 3bÞðAða; bÞ þ 3aÞ Aða; bÞLða; bÞ 1 Aða; bÞ 6 ðb aÞ 4 : 4 G ða; bÞ
ð2:8Þ
(2) Assume that f : ½a; b R ! R; f ðxÞ ¼ expðxÞ. Then
ðTf Þða; b; tÞ ¼ where
EiðzÞ :¼ PV
Z
expðtÞ
p z
1
½Eiðb tÞ Eiða tÞ;
expðtÞ dt; z 2 R: t
Also, we have
b a ½exp; kt þ ð1 kÞb; kt þ ð1 kÞa þ ½exp; kb þ ð1 kÞt; ka þ ð1 kÞt 2 p 1 expðkt þ ð1 kÞbÞ expðkt þ ð1 kÞaÞ expðkb þ ð1 kÞtÞ expðka þ ð1 kÞtÞ þ ¼ 2p 1k k and
Z b _ 0 ðf Þ ¼
b
00
jf ðtÞjdt ¼ expðbÞ expðaÞ:
a
a
Using the inequality (2.1) we may write:
expðtÞ Eiðb tÞ Eiða tÞ ln b t ta 1 expðkt þ ð1 kÞbÞ expðkt þ ð1 kÞaÞ expðkb þ ð1 kÞtÞ expðka þ ð1 kÞtÞ þ 2 1k k 1 3 1 þ k ðb aÞjt Aða; bÞj ½expðbÞ expðaÞ 6 4 4 2
ð2:9Þ
for any t 2 ða; bÞ. If in (2.9) we choose k ¼ 34 and t ¼ aþb , we get 2
exp a þ b Ei b a 1 4 exp 3a þ 5b exp 5a þ 3b þ 4 exp a þ 7b exp 7a þ b 2 2 2 8 8 3 8 8 1 6 ðb aÞ½expðbÞ expðaÞ; 8
which is equivalent to
Ei b a 2 exp b a exp b a þ 2 exp 3b 3a exp 3b 3a 2 8 8 3 8 8 1 ba ba 6 ðb aÞ exp exp : 8 2 2 If in the above inequality we set
ba 2
¼ z > 0, then we get
h
i EiðzÞ 2 exp z exp z þ 2 exp 3 z exp 3 z 4 4 3 4 4 6
1 z½expðzÞ expðzÞ 4
ð2:10Þ
for any z > 0. Consequently, we may state the following proposition. Proposition 2. With the above assumptions, we have
EiðzÞ 4 sinh z 4 sinh 3 z 6 1 z sinhðzÞ 4 3 4 2 for any z > 0.
ð2:11Þ
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W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
Remark 2. We note that inequality (2.11) gives a new estimate of EiðzÞ and a smaller error bound than that of inequality (2.16) in [6]. The reader may get other similar inequalities for special functions if appropriate examples of functions f are chosen. 3. A quadrature formula for equidistant divisions The following lemma is of interest in itself. Lemma 3.1. Let u : ½a; b ! R be a function of bounded variation on ½a; b. Then for all n P 1; ki 2 ½12 ; 1Þ ði ¼ 0; . . . ; n 1Þ and t; s 2 ½a; b with t – s, we have the inequality:
1 Z s n1 uðt þ ði þ 1 ki Þ st Þ þ uðt þ ði þ ki Þ st Þ 1X n n uðsÞds s t t n i¼0 2 " # _ s 1 1 3 6 þ max ki ðuÞ: n 4 i¼0;n1 4 t Proof. Consider the equidistant division of ½t; s (if t < s) or ½s; t (if
En : xi ¼ t þ i
st n
;
ð3:1Þ
s < t) given by
i ¼ 0; n:
ð3:2Þ
Then the points ni ¼ ki ½t þ i st þ ð1 ki Þ½t þ ði þ 1Þ st ðki 2 ½0; 1; i ¼ 0; n 1Þ are between xi and xiþ1 . We choose that we n n may write for simplicity ni ¼ t þ ði þ 1 ki Þ st ði ¼ 0; n 1Þ. We also have n
ni
3xi þ xiþ1 s t 3 ¼ ki n 4 4
for any i ¼ 0; n 1. If we apply the inequality (1.7) on the interval ½xi ; xiþ1 and the intermediate point ni ði ¼ 0; n 1Þ, then we may write that
Z xiþ1 s t uðt þ ði þ 1 ki Þ st Þ þ uðt þ ði þ ki Þ st Þ n n uðsÞds n 2 xi x_ iþ1 1 js tj s t 3 6 þ ki ðuÞ: x 4 n n 4
ð3:3Þ
i
Summing, we get
Z s n1 uðt þ ði þ 1 ki Þ st Þ þ uðt þ ði þ ki Þ st Þ stX n n uðsÞds t 2 n i¼0 s n1 _ js tj X 1 3 6 þ ki ðuÞ t n i¼0 4 4 " #_ s js tj 1 3 6 þ max ki ðuÞ; n 4 i¼0;n1 4 t which is equivalent to (3.1). h We may now state the following theorem in approximating the finite Hilbert transform of a differentiable function with the derivative of bounded variation for equidistant division of ½a; b. 0
Theorem 3.1. Let f : ½a; b ! R be a differentiable function such that its derivative f is of bounded variation on ½a; b. If k ¼ ðki Þi¼0;n1 ; ki 2 ½12 ; 1Þ ði ¼ 0; n 1Þ and
Sn ðf ; k; tÞ ¼
n1 b aX bt at f ; ði þ 1 ki Þ þ t; ði þ 1 ki Þ þt ; n n pn i¼0
ð3:4Þ
T n ðf ; k; tÞ ¼
n1 b aX bt at f ; ði þ ki Þ þ t; ði þ ki Þ þt ; n n pn i¼0
ð3:5Þ
then we have the estimate:
W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
379
ðTf Þða; b; tÞ f ðtÞ ln b t Sn ðf ; k; tÞ þ T n ðf ; k; tÞ ta 2 p " # _ b ba 1 3 1 a þ b 0 6 þ max ki ðb aÞ þ t ðf Þ np 4 i¼0;n1 4 2 2 a 6
b b a_ 0 ðf Þ: np a
ð3:6Þ
0
Proof. Applying Lemma 3.1 for the function f , we may write that
f ðsÞ f ðtÞ 1 X n1 st s t 0 0 þ f t þ ði þ ki Þ f t þ ði þ 1 ki Þ st 2n i¼0 n n " #_ 1 1 3 s 0 6 þ max ki ðf Þ n 4 i¼0;n1 4 t
ð3:7Þ
for any t; s 2 ½a; b; t – s. Consequently, we have
Z b Z b 1 n1 f ðsÞ f ðtÞ 1 X st st 0 0 þ f t þ ði þ ki Þ ds ds PV f t þ ði þ 1 ki Þ PV p 2pn i¼0 n n st a a " # Z b _ s 1 1 3 0 6 þ max ki PV ðf Þds np 4 i¼0;n1 4 a t " # b 1 1 3 1 a þ b _ 0 6 þ max ki ðb aÞ þ t ðf Þ: np 4 i¼0;n1 4 2 2 a
ð3:8Þ
On the other hand (see [6])
PV
Z
b
f
0
t þ ði þ 1 ki Þ
st
a
PV
Z
n
"Z ds ¼ limþ e!0
a
te
þ
# st 0 ds f t þ ði þ 1 ki Þ n tþe
Z
b
bt at ; ; t þ ði þ 1 ki Þ ¼ ðb aÞ f ; t þ ði þ 1 ki Þ n n b
f
0
t þ ði þ ki Þ
bt at ds ¼ ðb aÞ f ; t þ ði þ ki Þ : ; t þ ði þ ki Þ n n n
st
a
ð3:9Þ
ð3:10Þ
Since (see for example (2.2))
ðTf Þða; b; tÞ ¼
1
p
PV
Z
b a
f ðsÞ f ðtÞ f ðtÞ bt ds þ ln ta st p
for t 2 ða; bÞ, then by (3.8)–(3.10) we deduce the desired estimate (3.6).
h
Remark 3. For n ¼ 1, we recapture the inequality (2.1). Corollary 3.1. With the assumptions of Theorem 3.1, we have
ðTf Þða; b; tÞ ¼
f ðtÞ
p
ln
bt Sn ðf ; k; tÞ þ T n ðf ; k; tÞ þ lim n!1 ta 2
ð3:11Þ
uniformly by rapport of t 2 ða; bÞ and k with ki 2 ½12 ; 1Þ ði 2 NÞ. Remark 4. If one needs to approximate the finite Hilbert Transform ðTf Þða; b; tÞ in terms of
f ðtÞ
p
ln
bt Sn ðf ; k; tÞ þ T n ðf ; k; tÞ þ ; ta 2
with the accuracy
ne ¼
e > 0 (e small), then the theoretical minimal number ne to be chosen is:
" b ba _
ep
a
#
0
ðf Þ þ 1;
ð3:12Þ
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W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
where ½a is the integer part of a. It is obvious that the best inequality we can get in (3.6) is for ki ¼ 34 ði ¼ 0; n 1Þ obtaining the following corollary. Corollary 3.2. Let f be as in Theorem 3.1. Define
Mn ðf ; tÞ ¼
n1 b aX 1 bt 1 at f; i þ þ t; i þ þt ; 4 n 4 n pn i¼0
ð3:13Þ
Nn ðf ; tÞ ¼
n1 b aX 3 bt 3 at f; i þ þ t; i þ þt : 4 n 4 n pn i¼0
ð3:14Þ
Then we have the estimate
ðTf Þða; b; tÞ f ðtÞ ln b t Mn ðf ; tÞ þ Nn ðf ; tÞ ta 2 p _ b ba 1 a þ b 0 6 ðb aÞ þ t ðf Þ 4np 2 2 a
ð3:15Þ
for any t 2 ða; bÞ. This rule will be numerically implemented in Section 5 for different choices of f and n. 4. A more general quadrature formula
Lemma 4.1. Let u : ½a; b ! R be a function of bounded variation on ½a; b; 0 ¼ l0 < l1 < < ln1 < ln ¼ 1 and mi 2 ½li ; liþ1 ; i ¼ 0; n 1, Then for any t; s 2 ½a; b with t – s, we have the inequality:
1 Z s n1 u½ð1 mi Þt þ mi s þ u½ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þs 1X uðsÞds ð l li Þ s t t 2 i¼0 iþ1 2 " #_ 3l þ liþ1 s 1 6 Dn ðlÞ þ max mi i ðuÞ; t 4 4 i¼0;n1
ð4:1Þ
where Dn ðlÞ :¼ maxi¼0;n1 ðliþ1 li Þ. Proof. Consider the division of ½t; s (if t < s) or ½s; t (if
s < t) given by
In : xi :¼ ð1 li Þt þ li s ði ¼ 0; n 1Þ:
ð4:2Þ
Then the points ni :¼ ð1 mi Þt þ mi s ði ¼ 0; n 1Þ are between xi and xiþ1 . We have
xiþ1 xi ¼ ðliþ1 li Þðs tÞ ði ¼ 0; n 1Þ and
ni
3xi þ xiþ1 ¼ 4
mi
3li þ liþ1 ðs tÞ ði ¼ 0; n 1Þ: 4
Applying the inequality (1.7) on ½xi ; xiþ1 with the intermediate points ni ði ¼ 0; n 1Þ, we get
Z
ðliþ1 li Þðs tÞ ½u½ð1 mi Þt þ mi s þ u½ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þs 2 xi xiþ1 3li þ liþ1 _ 1 ðl li Þjs tj þ js tjmi 6 ðuÞ 4 iþ1 4 xiþ1
uðsÞds
xi
for any i ¼ 0; n 1. Summing over i, using the generalized triangle inequality and dividing by js tj > 0, we obtain
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W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
1 Z s n1 u½ð1 mi Þt þ mi s þ u½ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þs 1X uðsÞds ð l l Þ i s t t 2 i¼0 iþ1 2 x n1 iþ1 X 1 3l þ liþ1 _ 6 ðliþ1 li Þ þ mi i ðuÞ 4 4 xi i¼0 " # _ s 3 l þ l 1 i iþ1 6 Dn ðlÞ þ max mi ðuÞ 4 4 i¼0;n1 t and the inequality (4.1) is proved.
h 0
Theorem 4.1. Let f : ½a; b ! R be a differentiable function such that its derivative f is of bounded variation on ½a; b. If 0 ¼ l0 < l1 < < ln1 < ln ¼ 1 and mi 2 ½li ; liþ1 ; ði ¼ 0; n 1Þ, then
ðTf Þða; b; tÞ ¼
f ðtÞ
p
ln
bt 1 þ ½Q ðl; m; tÞ þ P n ðl; m; tÞ þ W n ðl; m; tÞ ta 2p n
ð4:3Þ
for any t 2 ða; bÞ, where 0
Q n ðl; m; tÞ :¼ l1 f ðtÞðb aÞ þ ðb aÞ
n2 X ðliþ1 li Þ ½f ; ð1 mi Þt þ mi b; ð1 mi Þt þ mi a þ ð1 ln1 Þ½f ðbÞ f ðaÞ;
ð4:4Þ
i¼1
if
m0 ¼ 0; mn1 ¼ 1; 0
Q n ðl; m; tÞ :¼ l1 f ðtÞðb aÞ þ ðb aÞ
n1 X ðliþ1 li Þ½f ; ð1 mi Þt þ mi b; ð1 mi Þt þ mi a;
ð4:5Þ
i¼1
if
m0 ¼ 0; mn1 < 1; n2 X Q n ðl; m; tÞ :¼ ðb aÞ ðliþ1 li Þ½f ; ð1 mi Þt þ mi b; ð1 mi Þt þ mi a þ ð1 ln1 Þ½f ðbÞ f ðaÞ;
ð4:6Þ
i¼0
if
m0 > 0; mn1 ¼ 1; n1 X Q n ðl; m; tÞ :¼ ðb aÞ ðliþ1 li Þ½f ; ð1 mi Þt þ mi b; ð1 mi Þt þ mi a;
ð4:7Þ
i¼0
if
m0 > 0; mn1 < 1, Pn : ðl; m; tÞ :¼ ðb aÞ
n1 X ðliþ1 li Þ ½f ; ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þb; ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þa: i¼0
ð4:8Þ In all cases, the remainder satisfies the estimate:
" # b _ 0 3li þ liþ1 1 1 1 t a þ b jW n ðl; m; tÞj 6 Dn ðlÞ þ max mi ðb aÞ þ ðf Þ 2 2 p 4 4 i¼0;n1 a b 1 1 a þ b _ 0 6 Dn ðlÞ ðb aÞ þ t ðf Þ 2 2 p a 6
1
p
b _ 0 Dn ðlÞ ðf Þ:
ð4:9Þ
a
0
Proof. If we apply Lemma 4.1 for the function f , we may write that
f ðsÞ f ðtÞ 1 X n1 0 0 ðl li Þ f ½ð1 mi Þt þ mi s þ f ½ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þs st 2 i¼0 iþ1 " #_ 3l þ liþ1 s 0 1 6 Dn ðlÞ þ max mi i ðf Þ 4 4 i¼0;n1 t for any t; s 2 ½a; b; t – s.
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W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
Taking the PV in both sides, we may write that
Z ! Z b X 1 b f ðsÞ f ðtÞ n1 1 0 0 ds PV ðliþ1 li Þ½f ½ð1 mi Þt þ mi s þ f ½ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þs ds p a 2p st a i¼0 " # Z b _ s 3l þ liþ1 1 1 0 6 PV Dn ðlÞ þ max mi i ðf Þds: p 4 4 i¼0;n1 a t ð4:10Þ If
m0 ¼ 0; mn1 ¼ 1, then Z
PV
b
a
! Z n1 X 0 ðliþ1 li Þf ½ð1 mi Þt þ mi s ds ¼ PV
b
a
i¼0
l1 f 0 ðtÞds þ
þ ð1 ln1 ÞPV þ ðb aÞ
Z a
Z n2 X ðliþ1 li ÞPV
0
f ½ð1 mi Þt þ mi sds
a
i¼1 b
b
0
0
f ðsÞds ¼ l1 f ðtÞðb aÞ
n2 X ðliþ1 li Þ½f ; ð1 mi Þt þ mi b; ð1 mi Þt þ mi a i¼1
þ ð1 ln1 Þ½f ðbÞ f ðaÞ: If
m0 ¼ 0; mn1 < 1, then Z
PV
b
a
If
! n1 n1 X X 0 0 ðliþ1 li Þf ½ð1 mi Þt þ mi s ds ¼ l1 f ðtÞðb aÞ þ ðb aÞ ðliþ1 li Þ½f ; ð1 mi Þt þ mi b; ð1 mi Þt þ mi a: i¼0
i¼1
m0 > 0; mn1 ¼ 1, then PV
Z
b a
Finally, if
PV
! n1 n2 X X 0 ðliþ1 li Þf ½ð1 mi Þt þ mi s ds ¼ ðb aÞ ðliþ1 li Þ½f ;ð1 mi Þt þ mi b;ð1 mi Þt þ mi a þ ð1 ln1 Þ½f ðbÞ f ðaÞ: i¼0
i¼0
m0 > 0; mn1 < 1, then Z
b
a
! n1 n1 X X 0 ðliþ1 li Þf ½ð1 mi Þt þ mi s ds ¼ ðb aÞ ðliþ1 li Þ½f ; ð1 mi Þt þ mi b; ð1 mi Þt þ mi a: i¼0
i¼0
Similarly
PV
Z
b
a
! n1 X 0 ðliþ1 li Þf ½ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þs ds i¼0
¼ ðb aÞ
n1 X ðliþ1 li Þ ½f ; ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þb; ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þa: i¼0
Since
PV
Z a
b
b _ s 1 a þ b _ 0 0 ðb aÞ þ t ðf Þ ðf Þds 6 t 2 2 a
and
ðTf Þða; b; tÞ ¼
1
p
PV
Z
b
a
f ðsÞ f ðtÞ f ðtÞ bt ; ds þ ln ta st p
then by (4.10) we deduce (4.3).
h
5. Numerical experiments For a function f : ½a; b ! R, we may consider the quadrature formula
En ðf ; a; b; tÞ :¼
f ðtÞ
p
ln
bt M n ðf ; tÞ þ Nn ðf ; tÞ þ ; ta 2
t 2 ½a; b:
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W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
If we consider the function f : ½1; 1 ! R; f ðxÞ ¼ expðxÞ, then the exact finite Hilbert transform of f is
ðTf Þð1; 1; tÞ ¼
expðtÞEið1 tÞ expðtÞEið1 tÞ
p
t 2 ½1; 1
;
and the plot of the finite Hilbert transform is incorporated in Fig. 1. If we implement the quadrature formula provided by En ðf ; a; b; tÞ using Matlab and choose the value of n ¼ 100, then the error Erðf ; a; b; tÞ :¼ ðTf Þða; b; tÞ En ðf ; a; b; tÞ has the variation described in the Fig. 2. For n ¼ 1000, the plot of Erðf ; a; b; tÞ is embodied in the Fig. 3. Now, if we consider another function, f : ½1:1 ! R; f ðxÞ ¼ sin x, then the exact value of the Hilbert transform is
ðTf Þð1; 1; tÞ ¼
Sið1 þ tÞ cosðtÞ þ Cið1 tÞ sinðtÞ
p
þ
Siðt þ 1Þ cosðtÞ Ciðt þ 1Þ sinðtÞ
p
;
t 2 ½1; 1;
where
SiðxÞ ¼
Z 0
x
sinðtÞ dt; t
CiðxÞ ¼ c þ ln x þ
Z 0
x
cosðtÞ 1 dt t
having the plot embodied in the following Fig. 4. If we choose the value of n ¼ 100, then the error Erðf ; a; b; tÞ for the function f ðxÞ ¼ sin x; x 2 ½1; 1 has the variation described in the Fig. 5. If choose the value of n ¼ 1000, the behaviour of Erðf ; a; b; tÞ is plotted in Fig. 6. Remark 5. When n ¼ 100, for functions f ðxÞ ¼ expðxÞ and f ðxÞ ¼ sinðxÞ, the precision of the error is 106 in [6] while the precision obtained here is 107 . When n ¼ 1000, we also have the higher precision. Therefore, our results may have the better error bounds. 2 1 0 1 2 3 4 5 6
1
0.8
0.6
0.4
0.2
0 t
0.2
0.4
0.6
0.8
1
Fig. 1. The plot of the finite Hilbert transform of f ðxÞ ¼ expðxÞ.
−7
14
x 10
12 10 8 6 4 2 −1
−0.8
−0.6
−0.4
−0.2
0 t
0.2
0.4
0.6
0.8
Fig. 2. The plot of the error for f ðxÞ ¼ expðxÞ and n ¼ 100.
1
384
W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385 −9
14
x 10
12
10
8
6
4
2 −1
−0.8
−0.6
−0.4
−0.2
0 t
0.2
0.4
0.6
0.8
1
Fig. 3. The plot of the error for f ðxÞ ¼ expðxÞ and n ¼ 1000.
1 0.5 0 −0.5 −1 −1.5 −2 −1
−0.8
−0.6
−0.4
−0.2
0 t
0.2
0.4
0.6
0.8
1
Fig. 4. The plot of the finite Hilbert transform of f ðxÞ ¼ sin x.
−7
x 10 −2
−3
−4
−5
−6
−7
−8 −1
−0.8
−0.6
−0.4
−0.2
0 t
0.2
0.4
0.6
0.8
Fig. 5. The plot of the error for f ðxÞ ¼ sin x and n ¼ 100.
1
W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
385
−9
x 10 −2
−3
−4
−5
−6
−7
−8 −1
−0.8
−0.6
−0.4
−0.2
0 t
0.2
0.4
0.6
0.8
1
Fig. 6. The plot of the error for f ðxÞ ¼ sin x and n ¼ 1000.
Acknowledgements This work was partly supported by the National Natural Science Foundation of China (Grant No. 11301277), the Qing Lan Project of Jiangsu Province, the Overseas Scholarship of Jiangsu Provincial Government, and the Training Abroad Project of Outstanding Young and Middle-Aged University Teachers and Presidents. References [1] M.W. Alomari, A companion of Ostrowski’s inequality with applications, Transylv. J. Math. Mech. 3 (1) (2011) 9–14. [2] M.W. Alomari, A companion of Ostrowski’s inequality for mappings whose first derivatives are bounded and applications in numerical integration, Transylv. J. Math. Mech. 4 (2) (2012) 103–109. [3] M.W. Alomari, A generalization of companion inequality of Ostrowski’s type for mappings whose first derivatives are bounded and applications in numerical integration, Kragujevac J. Math. 36 (1) (2012) 77–82. [4] G. Criscuolo, G. Mastrotanni, Sulla convergenza di alcune formule di quadratura per il calcolo di integrali a valor principale secondo Cauchy, Anal. Numr. Thor. Approx. 14 (1985) 109–116. [5] N.M. Dragomir, S.S. Dragomir, P. Farrell, Approximating the finite Hilbert transform via trapezoid type inequalities, Comput. Math. Appl. 43 (10–11) (2002) 1359–1369. [6] S.S. Dragomir, Approximating the finite Hilbert transform via an Ostrowski type inequality for functions of bounded variation, JIPAM J. Inequal. Pure Appl. Math. 3 (4) (2002) 19 (Article 51). [7] S.S. Dragomir, Sharp error bounds of a quadrature rule with one multiple node for the finite Hilbert transform in some classes of continuous differentiable functions, Taiwanese J. Math. 9 (1) (2005) 95–109. [8] S.S. Dragomir, A companions of Ostrowski’s inequality for functions of bounded variation and applications, RGMIA Res. Rep. Coll. 5 (Supp.) (2002) (article 28). [9] S.S. Dragomir, Some companions of Ostrowski’s inequality for absolutely continuous functions and applications, Bull. Korean Math. Soc. 42 (2) (2005) 213–230. [10] S.S. Dragomir, On the Ostrowski’s integral inequality for mappings with bounded variation and applications, Math. Inequal. Appl. 4 (1) (2001) 59–66. [11] C. Dagnino, V. Demichelis, E. Santi, Numerical integration based on quasiinterpolating splines, Computing 50 (1993) 149–163. [12] T. Hasegawa, T. Torii, Hilbert and Hadamard transforms by generalised Chebychev expansion, J. Comput. Appl. Math. 51 (1994) 71–83. [13] D.B. Hunter, Some Gauss type formulae for the evaluation of Cauchy principal value integrals, Numer. Math. 19 (1972) 419–424. [14] V.N. Huy, Q.-A. Ngô, New bounds for the Ostrowski-like type inequalities, Bull. Korean Math. Soc. 48 (1) (2011) 95–104. [15] N.I. Ioakimidis, On the uniform convergence of Gaussian quadrature rules for Cauchy principal value integrals and their derivatives, Math. Comput. 44 (1985) 191–198. [16] W.J. Liu, A. Tuna, Weighted Ostrowski, trapezoid and Grüss type inequalities on time scales, J. Math. Inequal. 6 (3) (2012) 381–399. [17] Z. Liu, Some companions of an Ostrowski type inequality and applications, JIPAM J. Inequal. Pure Appl. Math. 10 (2) (2009) 12 (Article 52). [18] G. Monegato, The numerical evaluation of one-dimensional Cauchy principal value integrals, Computing 29 (1982) 337–354. [19] J. Park, Some companions of perturbed Ostrowski type inequalities for functions whose second derivatives are bounded, Int. J. Appl. Math. Stat. 36 (2013) 95–103. [20] P. Rabinowotz, On an interpolatory product rule for evaluating Cauchy principal value integrals, BIT 29 (1989) 347–355. [21] P. Rabinowotz, Numerical integration based on approximating splines, J. Comput. Appl. Math. 33 (1990) 73–83. [22] P. Rabinowotz, E. Santi, On the uniform convergence of Cauchy principle value of quasiinterpolating splines, Bit 35 (1995) 277–290. [23] M.Z. Sarikaya, On the Ostrowski type integral inequality, Acta Math. Univ. Comenian. (N.S.) 79 (1) (2010) 129–134. [24] G. Tsamasphyros, P.S. Theocaris, On the convergence of some quadrature rules for Cauchy principal-value and finite-part integrals, Computing 31 (1983) 105–114. [25] S.W. Vong, A note on some Ostrowski-like type inequalities, Comput. Math. Appl. 62 (1) (2011) 532–535.
Contents lists available at ScienceDirect
Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc
Approximating the finite Hilbert transform via a companion of Ostrowski’s inequality for function of bounded variation and applications Wenjun Liu ⇑, Xingyue Gao College of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing 210044, China
a r t i c l e
i n f o
a b s t r a c t
Keywords: A companion of Ostrowski’s inequality Finite Hilbert transform Function of bounded variation Numerical experiments
The finite Hilbert transform plays an important role in scientific and engineering computing. By using a companion of Ostrowski’s inequality for function of bounded variation, we give some new approximations of the finite Hilbert transform, which may have the better error bounds than the known results obtained via Ostrowski type inequality. Some numerical experiments are also presented. Ó 2014 Elsevier Inc. All rights reserved.
1. Introduction Finite Hilbert transform plays an important role in many fields and its mathematics expression is
ðTf Þ ða; b; t Þ ¼
1
p
PV
Z a
b
f ðsÞ 1 ds :¼ lim st p e!0þ
"Z
a
te
f ðsÞ ds þ st
Z
b
tþe
# f ð sÞ ds : st
ð1:1Þ
For different approaches in approximating the finite Hilbert transform (1.1) including: noninterpolatory, interpolatory, Gaussian, Chebychevian, Taylor formula with integral remainder and spline methods, we refer the papers [4,5,7,11– 13,15,18,20–22,24]. In [10], Dragomir proved the following Ostrowski type inequality for function of bounded variation: Theorem 1.1. Let u : ½a; b ! R be a function of bounded variation on ½a; b. Then, for all x 2 ½a; b, we have the inequality:
b Z b 1 a þ b _ uðxÞðb aÞ uðtÞdt 6 ðb aÞ þ x ðuÞ; 2 2 a a
where
b W
ðuÞ denotes the total variation of u on ½a; b. The constant
a
ð1:2Þ 1 2
is the best possible one.
In [6], Dragomir approximated the finite Hilbert transform via the above Ostrowski type inequality for function of bounded variation and proved the following inequality on the interval ½a; b: 0
Theorem 1.2. Let f : ½a; b ! R be a function such that its derivative f : ½a; b ! R is of bounded variation on ½a; b. Then we have the inequality: ⇑ Corresponding author. E-mail addresses: [email protected] (W. Liu), [email protected] (X. Gao). http://dx.doi.org/10.1016/j.amc.2014.08.099 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.
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W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
ðTf Þða; b; tÞ f ðtÞ ln b t b a ½f ; kt þ ð1 kÞb; kt þ ð1 kÞa ta p p _ b 1 1 1 1 a þ b 0 þ k ðb aÞ þ t ðf Þ 6 2 a p 2 2 2
ð1:3Þ
for any t 2 ½a; b and k 2 ½0; 1Þ, where ½f ; a; b is the divided difference, i.e.,
½f ; a; b :¼
f ðaÞ f ðbÞ : ab
ð1:4Þ
Then, Dragomir proved the following inequality with the derivative of bounded variation on equidistant division of ½a; b: 0
Theorem 1.3. Let f : ½a; b ! R be a differentiable function such that its derivative f is of bounded variation on ½a; b. If k ¼ ðki Þi¼0;n1 ; ki 2 ½0; 1Þ ði ¼ 0; n 1Þ and
Sn ðf ; k; tÞ ¼
n1 b aX bt at f ; ði þ 1 ki Þ þ t; ði þ 1 ki Þ þt ; n n pn i¼0
ð1:5Þ
then we have the estimate:
ðTf Þða; b; tÞ f ðtÞ ln b t Sn ðf ; k; tÞ ta p " # b ba 1 1 1 a þ b _ 0 6 þ max ki ðb aÞ þ t ðf Þ np 2 i¼0;n1 2 2 2 a 6
b b a_ 0 ðf Þ: np a
ð1:6Þ
Recently, some companions of Ostrowski’s inequality were established in [1–3,9,17,19] (see also [14,16,23,25] for other related Ostrowski type inequalities). A companion of Ostrowski’s inequality for functions of bounded variation was established in [8]: Theorem 1.4. Let u : ½a; b ! R be a function of bounded variation on ½a; b. Then, for all x 2 ½a; aþb 2 , we have the inequality:
b Z b uðxÞ þ uða þ b xÞ 1 3a þ b _ ðb aÞ uðtÞdt 6 ðb aÞ þ x ðuÞ; 2 4 4 a a
where
b W ðuÞ denotes the total variation of u on ½a; b. The constant a
1 4
ð1:7Þ
is the best possible one.
In this paper, motivated by [6,8], we shall point out a new method in approximating the finite Hilbert transform by the use of the above inequality (1.7). In Section 2, we will estimate the finite Hilbert transform via a companion of Ostrowski’s inequality for function of bounded variation on the interval ½a; b, and point out some interesting inequalities for the functions for which the finite Hilbert transforms can be expressed in terms of special functions. Our result can give a smaller error bound than the similar result in [6] (see Remarks 1, 2 and 5 below). In Section 3, we will prove a quadrature formula for equidistant division of ½a; b in approximating the finite Hilbert transform of a differentiable function with the derivative of bounded variation. In Section 4, we may get a more general quadrature formula. Some numerical examples for the obtained approximations are presented in Section 5. 2. Some inequalities in estimating the finite Hilbert transform Using the above a companion of Ostrowski’s inequality (1.7), we may point out the following result in estimating the finite Hilbert transform. 0
Theorem 2.1. Let f : ½a; b ! R be a function such that its derivative f : ½a; b ! R is of bounded variation on ½a; b. Then we have the inequality:
ðTf Þða; b; tÞ f ðtÞ ln b t ta p b a ½f ; kt þ ð1 kÞb; kt þ ð1 kÞa þ ½f ; kb þ ð1 kÞt; ka þ ð1 kÞt 2 p _ b 1 1 3 1 a þ b 0 6 þ k ðb aÞ þ t ðf Þ 2 p 4 4 2 a
ð2:1Þ
W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
375
for any t 2 ½a; b and k 2 ½12 ; 1Þ, where ½f ; a; b is the divided difference given in (1.4). 0
Proof. Since f is bounded on ½a; b, it follows that f is Lipschitzian on ½a; b and thus the finite Hilbert transform exists everywhere in ða; bÞ. As in [6], for the function f 0 : ða; bÞ ! R; f 0 ðtÞ ¼ 1; t 2 ða; bÞ, we have
ðTf Þða; b; tÞ ¼
1
p
ln
bt ; ta
t 2 ða; bÞ;
then obviously
ðTf Þða; b; tÞ
f ðtÞ
p
ln
Z b bt 1 f ðsÞ f ðtÞ ¼ PV ds: ta p st a
ð2:2Þ
0
Now, if we choose in (1.7), u ¼ f ; x ¼ kc þ ð1 kÞd; k 2 ½12 ; 1Þ, then we get
0 0 f ðdÞ f ðcÞ f ðkc þ ð1 kÞdÞ þ f ðkd þ ð1 kÞcÞ ðd cÞ 2 _ 1 3c þ d d 0 6 jd cj þ kc þ ð1 kÞd ðf Þ; 4 4 c
where c; d 2 ða; bÞ, which is equivalent to
d f ðdÞ f ðcÞ f 0 ðkc þ ð1 kÞdÞ þ f 0 ðkd þ ð1 kÞcÞ 1 3 _ 0 6 þ k ðf Þ dc 2 4 4 c
ð2:3Þ
for any c; d 2 ða; bÞ; c – d. Using (2.3), we may write
Z b Z b 1 f ðsÞ f ðtÞ 1 0 0 PV ½f ðkt þ ð1 kÞsÞ þ f ðks þ ð1 kÞtÞds ds PV p 2p st a a Z b s 1 1 3 _ 0 6 þ k PV ðf Þds p 4 4 a t # "Z t _ Z b_ t s 1 1 3 0 0 ¼ þ k ðf Þds þ ðf Þds p 4 4 a s t t " # t b _ _ 1 1 3 0 0 ðt aÞ ðf Þ þ ðb tÞ ðf Þ 6 þ k p 4 4 a t b _ 0 1 1 3 1 a þ b 6 þ k ðb aÞ þ t ðf Þ: 2 a p 4 4 2
ð2:4Þ
Since (for k – 1) (see [6])
1 PV 2p
Z
b
0
0
½f ðkt þ ð1 kÞsÞ þ f ðks þ ð1 kÞtÞds "Z Z # a
te b 1 0 0 ½f ðkt þ ð1 kÞsÞ þ f ðks þ ð1 kÞtÞds lim þ 2p e!0þ a tþe 1 f ðkt þ ð1 kÞaÞ þ f ðkt þ ð1 kÞbÞ f ðka þ ð1 kÞtÞ þ f ðkb þ ð1 kÞtÞ ¼ þ 2p 1k k b a ½f ; kt þ ð1 kÞb; kt þ ð1 kÞa þ ½f ; kb þ ð1 kÞt; ka þ ð1 kÞt ; ¼ 2 p
¼
using (2.4) and (2.2), we deduce the desired result (2.1). It is obvious that the best inequality we can get from (2.1) is the one for k ¼ 34. Thus, we may state the following corollary. Corollary 2.1. With the assumption of Theorem 2.1, we have
3tþb 3tþa 3bþt 3aþt _ b 0 ðTf Þða; b; tÞ f ðtÞ ln b t b a f ; 4 ; 4 þ f ; 4 ; 4 6 1 1 ðb aÞ þ t a þ b ðf Þ: 2 ta 4p 2 2 a p p
ð2:5Þ
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W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
Remark 1. If we take k ¼ 12 in (1.3), we get
_ b 0 ðTf Þða; b; tÞ f ðtÞ ln b t b a f ; t þ b ; t þ a 6 1 1 ðb aÞ þ t a þ b ðf Þ: ta 2 2 2p 2 2 a p p
ð2:6Þ
We note that inequality (2.5) we obtained here gives a new estimate of the finite Hilbert transform and a smaller error bound than that of (2.6). The above Theorem 2.1 may be used to point out some interesting inequalities for the functions for which the finite Hilbert transforms ðTf Þða; b; tÞ can be expressed in terms of special functions. For instance, we have: (1) Assume that f : ½a; b ð0; 1Þ ! R; f ðxÞ ¼ 1x. Then
ðTf Þða; b; tÞ ¼
1 ðb tÞa ; ln ðt aÞb pt
t 2 ða; bÞ;
b a ½f ; kt þ ð1 kÞb; kt þ ð1 kÞa þ ½f ; kb þ ð1 kÞt; ka þ ð1 kÞt 2 p ba 1 1 ; þ ¼ 2p ½kt þ ð1 kÞb½kt þ ð1 kÞa ½kb þ ð1 kÞt½ka þ ð1 kÞt Z b _ 0 ðf Þ ¼ a
b
2
00
jf ðtÞjdt ¼
a
b a2 2
a2 b
:
Using the inequality (2.1) we may write that
1 ln ðb tÞa 1 ln b t pt ðt aÞb ta pt ba 1 1 þ þ 2p ½kt þ ð1 kÞb½kt þ ð1 kÞa ½kb þ ð1 kÞt½ka þ ð1 kÞt 2 1 1 3 1 a þ b b a2 6 þ k ðb aÞ þ t ; 4 2 2 a2 b2 p 4 which is equivalent to
b a 1 1 1 b þ ln 2 ½kt þ ð1 kÞb½kt þ ð1 kÞa ½kb þ ð1 kÞt½ka þ ð1 kÞt t a 2 1 3 1 a þ b b a2 6 þ k ðb aÞ þ t : 4 4 2 2 a2 b2 If we use the notations
ba ðthe logarithmic meanÞ; ln b ln a Ak ðx; yÞ :¼ kx þ ð1 kÞy ðthe weighted arithmetic meanÞ; pffiffiffiffiffiffi Gða; bÞ :¼ ab ðthe geometric meanÞ;
Lða; bÞ :¼
Aða; bÞ :¼
aþb 2
ðthe arithmetic meanÞ;
then by (2.7), we get the following proposition: Proposition 1. With the above assumption, we have
1 1 1 1 þ 2 Ak ðt; bÞAk ðt; aÞ Ak ðb; tÞAk ða; tÞ tLða; bÞ 1 3 1 2Aða; bÞ 6 þ k ðb aÞ þ jt Aða; bÞj 4 4 4 2 G ða; bÞ for any t 2 ða; bÞ; k 2 ½12 ; 1Þ.
ð2:7Þ
W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
377
In particular, for t ¼ Aða; bÞ and k ¼ 34, we get
1 16 16 1 þ 2 ð3Aða; bÞ þ bÞð3Aða; bÞ þ aÞ ðAða; bÞ þ 3bÞðAða; bÞ þ 3aÞ Aða; bÞLða; bÞ 1 Aða; bÞ 6 ðb aÞ 4 : 4 G ða; bÞ
ð2:8Þ
(2) Assume that f : ½a; b R ! R; f ðxÞ ¼ expðxÞ. Then
ðTf Þða; b; tÞ ¼ where
EiðzÞ :¼ PV
Z
expðtÞ
p z
1
½Eiðb tÞ Eiða tÞ;
expðtÞ dt; z 2 R: t
Also, we have
b a ½exp; kt þ ð1 kÞb; kt þ ð1 kÞa þ ½exp; kb þ ð1 kÞt; ka þ ð1 kÞt 2 p 1 expðkt þ ð1 kÞbÞ expðkt þ ð1 kÞaÞ expðkb þ ð1 kÞtÞ expðka þ ð1 kÞtÞ þ ¼ 2p 1k k and
Z b _ 0 ðf Þ ¼
b
00
jf ðtÞjdt ¼ expðbÞ expðaÞ:
a
a
Using the inequality (2.1) we may write:
expðtÞ Eiðb tÞ Eiða tÞ ln b t ta 1 expðkt þ ð1 kÞbÞ expðkt þ ð1 kÞaÞ expðkb þ ð1 kÞtÞ expðka þ ð1 kÞtÞ þ 2 1k k 1 3 1 þ k ðb aÞjt Aða; bÞj ½expðbÞ expðaÞ 6 4 4 2
ð2:9Þ
for any t 2 ða; bÞ. If in (2.9) we choose k ¼ 34 and t ¼ aþb , we get 2
exp a þ b Ei b a 1 4 exp 3a þ 5b exp 5a þ 3b þ 4 exp a þ 7b exp 7a þ b 2 2 2 8 8 3 8 8 1 6 ðb aÞ½expðbÞ expðaÞ; 8
which is equivalent to
Ei b a 2 exp b a exp b a þ 2 exp 3b 3a exp 3b 3a 2 8 8 3 8 8 1 ba ba 6 ðb aÞ exp exp : 8 2 2 If in the above inequality we set
ba 2
¼ z > 0, then we get
h
i EiðzÞ 2 exp z exp z þ 2 exp 3 z exp 3 z 4 4 3 4 4 6
1 z½expðzÞ expðzÞ 4
ð2:10Þ
for any z > 0. Consequently, we may state the following proposition. Proposition 2. With the above assumptions, we have
EiðzÞ 4 sinh z 4 sinh 3 z 6 1 z sinhðzÞ 4 3 4 2 for any z > 0.
ð2:11Þ
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W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
Remark 2. We note that inequality (2.11) gives a new estimate of EiðzÞ and a smaller error bound than that of inequality (2.16) in [6]. The reader may get other similar inequalities for special functions if appropriate examples of functions f are chosen. 3. A quadrature formula for equidistant divisions The following lemma is of interest in itself. Lemma 3.1. Let u : ½a; b ! R be a function of bounded variation on ½a; b. Then for all n P 1; ki 2 ½12 ; 1Þ ði ¼ 0; . . . ; n 1Þ and t; s 2 ½a; b with t – s, we have the inequality:
1 Z s n1 uðt þ ði þ 1 ki Þ st Þ þ uðt þ ði þ ki Þ st Þ 1X n n uðsÞds s t t n i¼0 2 " # _ s 1 1 3 6 þ max ki ðuÞ: n 4 i¼0;n1 4 t Proof. Consider the equidistant division of ½t; s (if t < s) or ½s; t (if
En : xi ¼ t þ i
st n
;
ð3:1Þ
s < t) given by
i ¼ 0; n:
ð3:2Þ
Then the points ni ¼ ki ½t þ i st þ ð1 ki Þ½t þ ði þ 1Þ st ðki 2 ½0; 1; i ¼ 0; n 1Þ are between xi and xiþ1 . We choose that we n n may write for simplicity ni ¼ t þ ði þ 1 ki Þ st ði ¼ 0; n 1Þ. We also have n
ni
3xi þ xiþ1 s t 3 ¼ ki n 4 4
for any i ¼ 0; n 1. If we apply the inequality (1.7) on the interval ½xi ; xiþ1 and the intermediate point ni ði ¼ 0; n 1Þ, then we may write that
Z xiþ1 s t uðt þ ði þ 1 ki Þ st Þ þ uðt þ ði þ ki Þ st Þ n n uðsÞds n 2 xi x_ iþ1 1 js tj s t 3 6 þ ki ðuÞ: x 4 n n 4
ð3:3Þ
i
Summing, we get
Z s n1 uðt þ ði þ 1 ki Þ st Þ þ uðt þ ði þ ki Þ st Þ stX n n uðsÞds t 2 n i¼0 s n1 _ js tj X 1 3 6 þ ki ðuÞ t n i¼0 4 4 " #_ s js tj 1 3 6 þ max ki ðuÞ; n 4 i¼0;n1 4 t which is equivalent to (3.1). h We may now state the following theorem in approximating the finite Hilbert transform of a differentiable function with the derivative of bounded variation for equidistant division of ½a; b. 0
Theorem 3.1. Let f : ½a; b ! R be a differentiable function such that its derivative f is of bounded variation on ½a; b. If k ¼ ðki Þi¼0;n1 ; ki 2 ½12 ; 1Þ ði ¼ 0; n 1Þ and
Sn ðf ; k; tÞ ¼
n1 b aX bt at f ; ði þ 1 ki Þ þ t; ði þ 1 ki Þ þt ; n n pn i¼0
ð3:4Þ
T n ðf ; k; tÞ ¼
n1 b aX bt at f ; ði þ ki Þ þ t; ði þ ki Þ þt ; n n pn i¼0
ð3:5Þ
then we have the estimate:
W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
379
ðTf Þða; b; tÞ f ðtÞ ln b t Sn ðf ; k; tÞ þ T n ðf ; k; tÞ ta 2 p " # _ b ba 1 3 1 a þ b 0 6 þ max ki ðb aÞ þ t ðf Þ np 4 i¼0;n1 4 2 2 a 6
b b a_ 0 ðf Þ: np a
ð3:6Þ
0
Proof. Applying Lemma 3.1 for the function f , we may write that
f ðsÞ f ðtÞ 1 X n1 st s t 0 0 þ f t þ ði þ ki Þ f t þ ði þ 1 ki Þ st 2n i¼0 n n " #_ 1 1 3 s 0 6 þ max ki ðf Þ n 4 i¼0;n1 4 t
ð3:7Þ
for any t; s 2 ½a; b; t – s. Consequently, we have
Z b Z b 1 n1 f ðsÞ f ðtÞ 1 X st st 0 0 þ f t þ ði þ ki Þ ds ds PV f t þ ði þ 1 ki Þ PV p 2pn i¼0 n n st a a " # Z b _ s 1 1 3 0 6 þ max ki PV ðf Þds np 4 i¼0;n1 4 a t " # b 1 1 3 1 a þ b _ 0 6 þ max ki ðb aÞ þ t ðf Þ: np 4 i¼0;n1 4 2 2 a
ð3:8Þ
On the other hand (see [6])
PV
Z
b
f
0
t þ ði þ 1 ki Þ
st
a
PV
Z
n
"Z ds ¼ limþ e!0
a
te
þ
# st 0 ds f t þ ði þ 1 ki Þ n tþe
Z
b
bt at ; ; t þ ði þ 1 ki Þ ¼ ðb aÞ f ; t þ ði þ 1 ki Þ n n b
f
0
t þ ði þ ki Þ
bt at ds ¼ ðb aÞ f ; t þ ði þ ki Þ : ; t þ ði þ ki Þ n n n
st
a
ð3:9Þ
ð3:10Þ
Since (see for example (2.2))
ðTf Þða; b; tÞ ¼
1
p
PV
Z
b a
f ðsÞ f ðtÞ f ðtÞ bt ds þ ln ta st p
for t 2 ða; bÞ, then by (3.8)–(3.10) we deduce the desired estimate (3.6).
h
Remark 3. For n ¼ 1, we recapture the inequality (2.1). Corollary 3.1. With the assumptions of Theorem 3.1, we have
ðTf Þða; b; tÞ ¼
f ðtÞ
p
ln
bt Sn ðf ; k; tÞ þ T n ðf ; k; tÞ þ lim n!1 ta 2
ð3:11Þ
uniformly by rapport of t 2 ða; bÞ and k with ki 2 ½12 ; 1Þ ði 2 NÞ. Remark 4. If one needs to approximate the finite Hilbert Transform ðTf Þða; b; tÞ in terms of
f ðtÞ
p
ln
bt Sn ðf ; k; tÞ þ T n ðf ; k; tÞ þ ; ta 2
with the accuracy
ne ¼
e > 0 (e small), then the theoretical minimal number ne to be chosen is:
" b ba _
ep
a
#
0
ðf Þ þ 1;
ð3:12Þ
380
W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
where ½a is the integer part of a. It is obvious that the best inequality we can get in (3.6) is for ki ¼ 34 ði ¼ 0; n 1Þ obtaining the following corollary. Corollary 3.2. Let f be as in Theorem 3.1. Define
Mn ðf ; tÞ ¼
n1 b aX 1 bt 1 at f; i þ þ t; i þ þt ; 4 n 4 n pn i¼0
ð3:13Þ
Nn ðf ; tÞ ¼
n1 b aX 3 bt 3 at f; i þ þ t; i þ þt : 4 n 4 n pn i¼0
ð3:14Þ
Then we have the estimate
ðTf Þða; b; tÞ f ðtÞ ln b t Mn ðf ; tÞ þ Nn ðf ; tÞ ta 2 p _ b ba 1 a þ b 0 6 ðb aÞ þ t ðf Þ 4np 2 2 a
ð3:15Þ
for any t 2 ða; bÞ. This rule will be numerically implemented in Section 5 for different choices of f and n. 4. A more general quadrature formula
Lemma 4.1. Let u : ½a; b ! R be a function of bounded variation on ½a; b; 0 ¼ l0 < l1 < < ln1 < ln ¼ 1 and mi 2 ½li ; liþ1 ; i ¼ 0; n 1, Then for any t; s 2 ½a; b with t – s, we have the inequality:
1 Z s n1 u½ð1 mi Þt þ mi s þ u½ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þs 1X uðsÞds ð l li Þ s t t 2 i¼0 iþ1 2 " #_ 3l þ liþ1 s 1 6 Dn ðlÞ þ max mi i ðuÞ; t 4 4 i¼0;n1
ð4:1Þ
where Dn ðlÞ :¼ maxi¼0;n1 ðliþ1 li Þ. Proof. Consider the division of ½t; s (if t < s) or ½s; t (if
s < t) given by
In : xi :¼ ð1 li Þt þ li s ði ¼ 0; n 1Þ:
ð4:2Þ
Then the points ni :¼ ð1 mi Þt þ mi s ði ¼ 0; n 1Þ are between xi and xiþ1 . We have
xiþ1 xi ¼ ðliþ1 li Þðs tÞ ði ¼ 0; n 1Þ and
ni
3xi þ xiþ1 ¼ 4
mi
3li þ liþ1 ðs tÞ ði ¼ 0; n 1Þ: 4
Applying the inequality (1.7) on ½xi ; xiþ1 with the intermediate points ni ði ¼ 0; n 1Þ, we get
Z
ðliþ1 li Þðs tÞ ½u½ð1 mi Þt þ mi s þ u½ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þs 2 xi xiþ1 3li þ liþ1 _ 1 ðl li Þjs tj þ js tjmi 6 ðuÞ 4 iþ1 4 xiþ1
uðsÞds
xi
for any i ¼ 0; n 1. Summing over i, using the generalized triangle inequality and dividing by js tj > 0, we obtain
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W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
1 Z s n1 u½ð1 mi Þt þ mi s þ u½ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þs 1X uðsÞds ð l l Þ i s t t 2 i¼0 iþ1 2 x n1 iþ1 X 1 3l þ liþ1 _ 6 ðliþ1 li Þ þ mi i ðuÞ 4 4 xi i¼0 " # _ s 3 l þ l 1 i iþ1 6 Dn ðlÞ þ max mi ðuÞ 4 4 i¼0;n1 t and the inequality (4.1) is proved.
h 0
Theorem 4.1. Let f : ½a; b ! R be a differentiable function such that its derivative f is of bounded variation on ½a; b. If 0 ¼ l0 < l1 < < ln1 < ln ¼ 1 and mi 2 ½li ; liþ1 ; ði ¼ 0; n 1Þ, then
ðTf Þða; b; tÞ ¼
f ðtÞ
p
ln
bt 1 þ ½Q ðl; m; tÞ þ P n ðl; m; tÞ þ W n ðl; m; tÞ ta 2p n
ð4:3Þ
for any t 2 ða; bÞ, where 0
Q n ðl; m; tÞ :¼ l1 f ðtÞðb aÞ þ ðb aÞ
n2 X ðliþ1 li Þ ½f ; ð1 mi Þt þ mi b; ð1 mi Þt þ mi a þ ð1 ln1 Þ½f ðbÞ f ðaÞ;
ð4:4Þ
i¼1
if
m0 ¼ 0; mn1 ¼ 1; 0
Q n ðl; m; tÞ :¼ l1 f ðtÞðb aÞ þ ðb aÞ
n1 X ðliþ1 li Þ½f ; ð1 mi Þt þ mi b; ð1 mi Þt þ mi a;
ð4:5Þ
i¼1
if
m0 ¼ 0; mn1 < 1; n2 X Q n ðl; m; tÞ :¼ ðb aÞ ðliþ1 li Þ½f ; ð1 mi Þt þ mi b; ð1 mi Þt þ mi a þ ð1 ln1 Þ½f ðbÞ f ðaÞ;
ð4:6Þ
i¼0
if
m0 > 0; mn1 ¼ 1; n1 X Q n ðl; m; tÞ :¼ ðb aÞ ðliþ1 li Þ½f ; ð1 mi Þt þ mi b; ð1 mi Þt þ mi a;
ð4:7Þ
i¼0
if
m0 > 0; mn1 < 1, Pn : ðl; m; tÞ :¼ ðb aÞ
n1 X ðliþ1 li Þ ½f ; ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þb; ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þa: i¼0
ð4:8Þ In all cases, the remainder satisfies the estimate:
" # b _ 0 3li þ liþ1 1 1 1 t a þ b jW n ðl; m; tÞj 6 Dn ðlÞ þ max mi ðb aÞ þ ðf Þ 2 2 p 4 4 i¼0;n1 a b 1 1 a þ b _ 0 6 Dn ðlÞ ðb aÞ þ t ðf Þ 2 2 p a 6
1
p
b _ 0 Dn ðlÞ ðf Þ:
ð4:9Þ
a
0
Proof. If we apply Lemma 4.1 for the function f , we may write that
f ðsÞ f ðtÞ 1 X n1 0 0 ðl li Þ f ½ð1 mi Þt þ mi s þ f ½ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þs st 2 i¼0 iþ1 " #_ 3l þ liþ1 s 0 1 6 Dn ðlÞ þ max mi i ðf Þ 4 4 i¼0;n1 t for any t; s 2 ½a; b; t – s.
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W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
Taking the PV in both sides, we may write that
Z ! Z b X 1 b f ðsÞ f ðtÞ n1 1 0 0 ds PV ðliþ1 li Þ½f ½ð1 mi Þt þ mi s þ f ½ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þs ds p a 2p st a i¼0 " # Z b _ s 3l þ liþ1 1 1 0 6 PV Dn ðlÞ þ max mi i ðf Þds: p 4 4 i¼0;n1 a t ð4:10Þ If
m0 ¼ 0; mn1 ¼ 1, then Z
PV
b
a
! Z n1 X 0 ðliþ1 li Þf ½ð1 mi Þt þ mi s ds ¼ PV
b
a
i¼0
l1 f 0 ðtÞds þ
þ ð1 ln1 ÞPV þ ðb aÞ
Z a
Z n2 X ðliþ1 li ÞPV
0
f ½ð1 mi Þt þ mi sds
a
i¼1 b
b
0
0
f ðsÞds ¼ l1 f ðtÞðb aÞ
n2 X ðliþ1 li Þ½f ; ð1 mi Þt þ mi b; ð1 mi Þt þ mi a i¼1
þ ð1 ln1 Þ½f ðbÞ f ðaÞ: If
m0 ¼ 0; mn1 < 1, then Z
PV
b
a
If
! n1 n1 X X 0 0 ðliþ1 li Þf ½ð1 mi Þt þ mi s ds ¼ l1 f ðtÞðb aÞ þ ðb aÞ ðliþ1 li Þ½f ; ð1 mi Þt þ mi b; ð1 mi Þt þ mi a: i¼0
i¼1
m0 > 0; mn1 ¼ 1, then PV
Z
b a
Finally, if
PV
! n1 n2 X X 0 ðliþ1 li Þf ½ð1 mi Þt þ mi s ds ¼ ðb aÞ ðliþ1 li Þ½f ;ð1 mi Þt þ mi b;ð1 mi Þt þ mi a þ ð1 ln1 Þ½f ðbÞ f ðaÞ: i¼0
i¼0
m0 > 0; mn1 < 1, then Z
b
a
! n1 n1 X X 0 ðliþ1 li Þf ½ð1 mi Þt þ mi s ds ¼ ðb aÞ ðliþ1 li Þ½f ; ð1 mi Þt þ mi b; ð1 mi Þt þ mi a: i¼0
i¼0
Similarly
PV
Z
b
a
! n1 X 0 ðliþ1 li Þf ½ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þs ds i¼0
¼ ðb aÞ
n1 X ðliþ1 li Þ ½f ; ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þb; ð1 li liþ1 þ mi Þt þ ðli þ liþ1 mi Þa: i¼0
Since
PV
Z a
b
b _ s 1 a þ b _ 0 0 ðb aÞ þ t ðf Þ ðf Þds 6 t 2 2 a
and
ðTf Þða; b; tÞ ¼
1
p
PV
Z
b
a
f ðsÞ f ðtÞ f ðtÞ bt ; ds þ ln ta st p
then by (4.10) we deduce (4.3).
h
5. Numerical experiments For a function f : ½a; b ! R, we may consider the quadrature formula
En ðf ; a; b; tÞ :¼
f ðtÞ
p
ln
bt M n ðf ; tÞ þ Nn ðf ; tÞ þ ; ta 2
t 2 ½a; b:
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W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
If we consider the function f : ½1; 1 ! R; f ðxÞ ¼ expðxÞ, then the exact finite Hilbert transform of f is
ðTf Þð1; 1; tÞ ¼
expðtÞEið1 tÞ expðtÞEið1 tÞ
p
t 2 ½1; 1
;
and the plot of the finite Hilbert transform is incorporated in Fig. 1. If we implement the quadrature formula provided by En ðf ; a; b; tÞ using Matlab and choose the value of n ¼ 100, then the error Erðf ; a; b; tÞ :¼ ðTf Þða; b; tÞ En ðf ; a; b; tÞ has the variation described in the Fig. 2. For n ¼ 1000, the plot of Erðf ; a; b; tÞ is embodied in the Fig. 3. Now, if we consider another function, f : ½1:1 ! R; f ðxÞ ¼ sin x, then the exact value of the Hilbert transform is
ðTf Þð1; 1; tÞ ¼
Sið1 þ tÞ cosðtÞ þ Cið1 tÞ sinðtÞ
p
þ
Siðt þ 1Þ cosðtÞ Ciðt þ 1Þ sinðtÞ
p
;
t 2 ½1; 1;
where
SiðxÞ ¼
Z 0
x
sinðtÞ dt; t
CiðxÞ ¼ c þ ln x þ
Z 0
x
cosðtÞ 1 dt t
having the plot embodied in the following Fig. 4. If we choose the value of n ¼ 100, then the error Erðf ; a; b; tÞ for the function f ðxÞ ¼ sin x; x 2 ½1; 1 has the variation described in the Fig. 5. If choose the value of n ¼ 1000, the behaviour of Erðf ; a; b; tÞ is plotted in Fig. 6. Remark 5. When n ¼ 100, for functions f ðxÞ ¼ expðxÞ and f ðxÞ ¼ sinðxÞ, the precision of the error is 106 in [6] while the precision obtained here is 107 . When n ¼ 1000, we also have the higher precision. Therefore, our results may have the better error bounds. 2 1 0 1 2 3 4 5 6
1
0.8
0.6
0.4
0.2
0 t
0.2
0.4
0.6
0.8
1
Fig. 1. The plot of the finite Hilbert transform of f ðxÞ ¼ expðxÞ.
−7
14
x 10
12 10 8 6 4 2 −1
−0.8
−0.6
−0.4
−0.2
0 t
0.2
0.4
0.6
0.8
Fig. 2. The plot of the error for f ðxÞ ¼ expðxÞ and n ¼ 100.
1
384
W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385 −9
14
x 10
12
10
8
6
4
2 −1
−0.8
−0.6
−0.4
−0.2
0 t
0.2
0.4
0.6
0.8
1
Fig. 3. The plot of the error for f ðxÞ ¼ expðxÞ and n ¼ 1000.
1 0.5 0 −0.5 −1 −1.5 −2 −1
−0.8
−0.6
−0.4
−0.2
0 t
0.2
0.4
0.6
0.8
1
Fig. 4. The plot of the finite Hilbert transform of f ðxÞ ¼ sin x.
−7
x 10 −2
−3
−4
−5
−6
−7
−8 −1
−0.8
−0.6
−0.4
−0.2
0 t
0.2
0.4
0.6
0.8
Fig. 5. The plot of the error for f ðxÞ ¼ sin x and n ¼ 100.
1
W. Liu, X. Gao / Applied Mathematics and Computation 247 (2014) 373–385
385
−9
x 10 −2
−3
−4
−5
−6
−7
−8 −1
−0.8
−0.6
−0.4
−0.2
0 t
0.2
0.4
0.6
0.8
1
Fig. 6. The plot of the error for f ðxÞ ¼ sin x and n ¼ 1000.
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