Journal of Inequalities in Pure and
Applied Mathematics
APPROXIMATING THE FINITE HILBERT TRANSFORM VIA AN
OSTROWSKI TYPE INEQUALITY FOR FUNCTIONS OF
BOUNDED VARIATION
volume 3, issue 4, article 51,
2002.
S.S. DRAGOMIR
School of Communications and Informatics
Victoria University of Technology
PO Box 14428
Melbourne City MC
8001 Victoria, Australia
EMail: sever@matilda.vu.edu.au
URL: http://rgmia.vu.edu.au/SSDragomirWeb.html
Received 5 April, 2002;
accepted 20 May, 2002.
Communicated by: B. Mond
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c
2000
Victoria University
ISSN (electronic): 1443-5756
032-01
Quit
Abstract
Using the Ostrowski type inequality for functions of bounded variation, an approximation of the finite Hilbert Transform is given. Some numerical experiments are also provided.
2000 Mathematics Subject Classification: Primary 26D10, 26D15; Secondary
41A55, 47A99.
Key words: Finite Hilbert Transform, Ostrowski’s Inequality.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Some Inequalities on the Interval [a, b] . . . . . . . . . . . . . . . . . . . 4
3
A Quadrature Formula for Equidistant Divisions . . . . . . . . . . 15
4
A More General Quadrature Formula . . . . . . . . . . . . . . . . . . . . 21
5
Numerical Experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
References
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
1.
Introduction
Cauchy principal value integrals of the form
Z b
f (τ )
(T f ) (a, b; t) = P V
(1.1)
dτ
a τ −t
Z t−ε
Z b
f (τ )
f (τ )
dτ +
dτ
:= lim
ε→0+
τ −t
a
t+ε τ − t
play an important role in fields like aerodynamics, the theory of elasticity and
other areas of the engineering sciences. They are also helpful tools in some
methods for the solution of differential equations (cf., e.g. [23]).
For different approaches in approximating the finite Hilbert transform (1.1)
including: interpolatory, noninterpolatory, Gaussian, Chebychevian and spline
methods, see for example the papers [1] – [12], [14] – [22], [24] – [33] and the
references therein.
In contrast with all these methods, we point out here a new method in approximating the finite Hilbert transform by the use of the Ostrowski inequality
for functions of bounded variation established in [13].
For a comprehensive list of papers on Ostrowski’s inequality, visit the site
http://rgmia.vu.edu.au.
Estimates for the error bounds and some numerical examples for the obtained
approximation are also presented.
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
2.
Some Inequalities on the Interval [a, b]
We start with the following lemma proved in [13] dealing with an Ostrowski
type inequality for functions of bounded variation.
Lemma 2.1. Let u : [a, b] → R be a function of bounded variation on [a, b].
Then, for all x ∈ [a, b], we have the inequality:
(2.1)
_
Z b
b
1
a
+
b
u (x) (b − a) −
x −
≤
u
(t)
dt
(b
−
a)
+
(u) ,
2
2 a
a
W
where ba (u) denotes the total variation of u on [a, b].
The constant 12 is the best possible one.
S.S. Dragomir
Proof. For the sake of completeness and since this result will be essentially used
in what follows, we give here a short proof.
Using the integration by parts formula for the Riemann-Stieltjes integral we
have
Z x
Z x
(t − a) du (t) = u (x) (x − a) −
u (t) dt
a
and
Z
a
b
Z
(t − b) du (t) = u (x) (b − x) −
x
u (t) dt.
If we add the above two equalities, we get
Z b
Z x
Z b
(2.2) u (x) (b − a) −
u (t) dt =
(t − a) du (t) +
(t − b) du (t)
a
Title Page
Contents
JJ
J
II
I
Go Back
b
x
a
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
x
Close
Quit
Page 4 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
for any x ∈ [a, b].
If p : [c, d] → R is continuous on [c, d] and v : [c, d] → R is of bounded
variation on [c, d], then:
Z d
d
_
(2.3)
p (x) dv (x) ≤ sup |p (x)| (u) .
x∈[c,d]
c
Using (2.2) and (2.3), we deduce
Z
Z b
u (x) (b − a) −
u (t) dt ≤ a
a
x
c
Z b
(t − a) du (t) + (t − b) du (t)
≤ (x − a)
x
x
_
(u) + (b − x)
a
(u)
S.S. Dragomir
x
"
≤ max {x − a, b − x}
b
_
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
x
_
a
(u) +
b
_
#
(u)
x
b
1
a + b _
=
(b − a) + x −
(u)
2
2 a
and the inequality (2.1) is proved.
Now, assume that the inequality (2.2) holds with a constant c > 0, i.e.,
b
Z b
_
a
+
b
u (x) (b − a) −
≤ c (b − a) + x −
(u)
(2.4)
u
(t)
dt
2 a
a
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 37
for all x ∈ [a, b].
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
Consider the function u0 : [a, b] → R given by

 0 if x ∈ [a, b] a+b
2
u0 (x) =

1 if x = a+b
.
2
Then u0 is of bounded variation on [a, b] and
b
_
Z
(u0 ) = 2,
a
b
u0 (t) dt = 0.
a
If we apply (2.4) for u0 and choose x = a+b
, then we get 2c ≥ 1 which implies
2
1
1
that c ≥ 2 showing that 2 is the best possible constant in (2.1).
The best inequality we can get from (2.1) is the following midpoint inequality.
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
Title Page
Contents
Corollary 2.2. With the assumptions in Lemma 2.1, we have
(2.5)
Z b
b
_
1
a
+
b
u
≤ (b − a) (u) .
(b
−
a)
−
u
(t)
dt
2
2
a
a
The constant
1
2
is best possible.
Using the above Ostrowski type inequality we may point out the following
result in estimating the finite Hilbert transform.
JJ
J
II
I
Go Back
Close
Quit
Page 6 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
Theorem 2.3. Let f : [a, b] → R be a function such that its derivative f 0 :
[a, b] → R is of bounded variation on [a, b]. Then we have the inequality:
(2.6)
(T f ) (a, b; t) − f (t) ln b − t
π
t−a
b−a
−
[f ; λt + (1 − λ) b, λt + (1 − λ) a]
π
b
1 1
a + b _ 0
1 1 ≤
+ λ− (b − a) + t −
(f ) ,
π 2 2
2
2 a
for any t ∈ (a, b) and λ ∈ [0, 1), where [f ; α, β] is the divided difference, i.e.,
[f ; α, β] :=
f (α) − f (β)
.
α−β
then obviously
(T f ) (a, b; t) −
S.S. Dragomir
Title Page
Proof. Since f 0 is bounded on [a, b], it follows that f is Lipschitzian on [a, b]
and thus the finite Hilbert transform exists everywhere in (a, b).
As for the function f0 : (a, b) → R, f0 (t) = 1, t ∈ (a, b), we have
1
b−t
(T f0 ) (a, b; t) = ln
, t ∈ (a, b) ,
π
t−a
(2.7)
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
Contents
JJ
J
II
I
Go Back
Close
Quit
f (t)
ln
π
b−t
t−a
=
1
PV
π
Z
a
b
f (τ ) − f (t)
dτ.
τ −t
Page 7 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
Now, if we choose in (2.1), u = f 0 , x = λc + (1 − λ) d, λ ∈ [0, 1], then we get
|f (d) − f (c) − (d − c) f 0 (λc + (1 − λ) d)|
_
d
1
c
+
d
0 ≤
|d − c| + λc + (1 − λ) d −
(f )
2
2 c
where c, d ∈ (a, b), which is equivalent to
f (d) − f (c)
1
0
− f (λc + (1 − λ) d) ≤
+ λ −
(2.8)
d−c
2
d
1 _ 0 (f )
2 c
for any c, d ∈ (a, b), c 6= d.
Using (2.8), we may write
Z b
Z b
1
f (τ ) − f (t)
1
0
(2.9) P V
dτ − P V
f (λt + (1 − λ) τ ) dτ π
τ −t
π
a
a
Z b _
t
1 1 1 + λ − P V
≤
(f 0 ) dt
π 2
2
a τ
"
!
! #
Z t t
Z b _
τ
_
1 1 1 =
+ λ− (f 0 ) dt +
(f 0 ) dt
π 2 2
a
t
τ
t
#
"
t
b
_
_
1 1 1 ≤
+ λ − (t − a) (f 0 ) + (b − t) (f 0 )
π 2 2
a
t
b
1 1 1 1
a + b _ 0
≤
+ λ − (b − a) + t −
(f ) .
π 2
2
2
2 a
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 8 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
Since (for λ 6= 1)
Z b
1
PV
f 0 (λt + (1 − λ) τ ) dτ
π
a
Z t−ε Z b 1
lim
+
(f 0 (λt + (1 − λ) τ ) dτ )
=
π ε→0+ a
t+ε
"
t−ε
b #
1
1
1
=
lim
f (λt + (1 − λ) τ ) +
f (λt + (1 − λ) τ )
π ε→0+ 1 − λ
1−λ
a
t+ε
1 f (t) − f (λt + (1 − λ) a) + f (λt + (1 − λ) b) − f (t)
·
π
1−λ
b−a
=
[f ; λt + (1 − λ) b, λt + (1 − λ) a] .
π
=
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
Using (2.9) and (2.7), we deduce the desired result (2.6).
Title Page
It is obvious that the best inequality we can get from (2.6) is the one for
λ = 12 . Thus, we may state the following corollary.
Contents
Corollary 2.4. With the assumptions of Theorem 2.3, we have
(2.10)
(T f ) (a, b; t) − f (t) ln b − t − b − a f ; t + b , a + t π
t−a
π
2
2 b
1 1
a + b _ 0
≤
(b − a) + t −
(f ) .
2π 2
2 a
JJ
J
II
I
Go Back
Close
Quit
Page 9 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
The above Theorem 2.3 may be used to point out some interesting inequalities for the functions for which the finite Hilbert transforms (T f ) (a, b; t) can
be expressed in terms of special functions.
For instance, we have:
1) Assume that f : [a, b] ⊂ (0, ∞) → R, f (x) = x1 . Then
(b − t) a
1
ln
, t ∈ (a, b) ,
(T f ) (a, b; t) =
πt
(t − a) b
b−a
· [f ; λt + (1 − λ) b, λt + (1 − λ) a]
π
1
b−a
=− ·
,
π [λt + (1 − λ) b] [λt + (1 − λ) a]
b
_
a
0
Z
(f ) =
a
b
b 2 − a2
.
|f (t)| dt =
a2 b 2
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
Title Page
Contents
00
Using the inequality (2.6) we may write that
1
b−a
ln (b − t) a − 1 ln b − t +
πt
(t − a) b
πt
t−a
π [λt + (1 − λ) b] [λt + (1 − λ) a] 2
b − a2
1 1
a
+
b
1 1 ·
+ λ − (b − a) + t −
≤
π 2
2
2
2 a2 b 2
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
which is equivalent to:
b−a
1
b (2.11) − ln
[λt + (1 − λ) b] [λt + (1 − λ) a] t
a 2
b − a2
1 1 1
a
+
b
·
≤
.
+ λ − (b − a) + t −
2
2
2
2 a2 b 2
If we use the notations
L (a, b)
b−a
:=
ln b − ln a
Aλ (x, y) := λx + (1 − λ) y
√
G (a, b) := ab
A (a, b)
a+b
:=
2
(the logarithmic mean)
(the weighted arithmetic mean)
S.S. Dragomir
(the geometric mean)
Title Page
(the arithmetic mean)
then by (2.11) we deduce
1
1
−
Aλ (t, b) Aλ (t, a) tL (a, b) 1 1 1
2A (a, b)
≤
+ λ − (b − a) + |t − A (a, b)|
,
2
2
2
G4 (a, b)
giving the following proposition:
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
Proposition 2.5. With the above assumption, we have
(2.12) |tL (a, b) − Aλ (t, b) Aλ (t, a)|
2A (a, b) 1 1 1
a + b ≤ 4
+ λ− (b − a) + t −
G (a, b) 2 2
2
2 × tAλ (t, b) Aλ (t, a) L (a, b)
for any t ∈ (a, b), λ ∈ [0, 1).
In particular, for t = A (a, b) and λ = 21 , we get
(A
(a,
b)
+
a)
(A
(a,
b)
+
b)
(2.13) A (a, b) L (a, b) −
4
1 A2 (a, b) (A (a, b) + a) (A (a, b) + b)
≤ · 4
·
L (a, b) .
2 G (a, b)
4
2) Assume that f : [a, b] ⊂ R → R, f (x) = exp (x). Then
exp (t)
(T f ) (a, b; t) =
[Ei (b − t) − Ei (a − t)] ,
π
where
Z z
exp (t)
Ei (z) := P V
dt, z ∈ R.
t
−∞
Also, we have:
b−a
[exp; λt + (1 − λ) b, λt + (1 − λ) a]
π
1 exp (λt + (1 − λ) b) − exp (λt + (1 − λ) a)
= ·
,
π
1−λ
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 12 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
b
_
0
b
Z
|f 00 (t)| dt = exp (b) − exp (a) .
(f ) =
a
a
Using the inequality (2.6) we may write:
b−t
(2.14) exp (t) Ei (b − t) − Ei (a − t) − ln
t−a
exp (λt + (1 − λ) b) − exp (λt + (1 − λ) a) −
1−λ
1 1 1
a + b ≤
+ λ − (b − a) + t −
[exp (b) − exp (a)]
2
2
2
2 S.S. Dragomir
for any t ∈ (a, b).
If in (2.14) we make λ =
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
1
2
and t =
a+b
,
2
we get
exp a + b Ei b − a − 2 exp a + 3b − exp 3a + b 2
2
4
4
1
≤ (b − a) [exp (b) − exp (a)] ,
4
which is equivalent to:
Ei b − a − 2 exp b − a − exp − b − a 2
4
4
b−a
b−a
1
≤ (b − a) exp
− exp −
.
4
2
2
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 13 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
If in this inequality we make
b−a
2
= z > 0, then we get
h
z z i
(2.15) Ei (z) − 2 exp
− exp −
2
2
1
≤ z [exp (z) − exp (−z)]
2
for any z > 0.
Consequently, we may state the following proposition.
Proposition 2.6. With the above assumptions, we have
1 (2.16)
Ei
(z)
−
4
sinh
z ≤ z sinh (z)
2 Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
for any z > 0.
Title Page
The reader may get other similar inequalities for special functions if appropriate examples of functions f are chosen.
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
3.
A Quadrature Formula for Equidistant Divisions
The following lemma is of interest in itself.
Lemma 3.1. Let u : [a, b] → R be a function of bounded variation on [a, b].
Then for all n ≥ 1, λi ∈ [0, 1) (i = 0, . . . , n − 1) and t, τ ∈ [a, b] with t 6= τ ,
we have the inequality:
(3.1)
n−1 1 Z τ
X
1
τ −t u (s) ds −
u t + (i + 1 − λi )
τ − t t
n i=0
n τ
_
1 1 1
≤
+ max λi − (u) .
2 t
n 2 i=0,n−1
Proof. Consider the equidistant division of [t, τ ] (if t < τ ) or [τ, t] (if τ < t)
given by
τ −t
(3.2)
En : xi = t + i ·
, i = 0, n.
n
Then the points ξi = λi t+ i · τ n−t + (1 − λi ) t + (i + 1) · τ n−t
λi ∈ [0, 1] , i = 0, n − 1 are between xi and xi+1 . We observe
that we may
write for simplicity ξi = t + (i + 1 − λi ) τ n−t i = 0, n − 1 . We also have
ξi −
xi + xi+1
τ −t
=
(1 − 2λi ) ,
2
2n
τ −t
ξi − xi = (1 − λi )
n
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 15 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
and
xi+1 − ξi = λi ·
τ −t
n
for any i = 0, n − 1.
If we apply the inequality
(2.1) on the interval [xi , xi+1 ] and the intermediate
point ξi i = 0, n − 1 , then we may write that
Z xi+1
τ − t
τ
−
t
(3.3) u t + (i + 1 − λi )
−
u (s) ds
n
n
xi
xi+1
_
1 |τ − t| τ − t
≤
·
+
(1 − 2λi ) (u) .
2
n
2n
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
xi
Summing, we get
Z
n−1 τ
X
τ
−
t
τ
−
t
u t + (i + 1 − λi )
u (s) ds −
t
n i=0
n x
n−1
i+1
X
_
|τ − t|
≤
[1 + |1 − 2λi |] (u)
2n i=0
xi
_
τ
|τ − t| 1
1
=
+ max λi − (u) ,
n
2 i=0,n−1
2 t
which is equivalent to (3.1).
We may now state the following theorem in approximating the finite Hilbert
transform of a differentiable function with the derivative of bounded variation
on [a, b].
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 16 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
Theorem 3.2. Let f : [a, b] → R be a differentiable function such that its
derivative f 0 is of bounded variation on [a, b]. If λ = (λi )i=0,n−1 , λi ∈ [0, 1)
i = 0, n − 1 and
(3.4) Sn (f ; λ, t)
n−1 b−t
a−t
b−aX
f ; (i + 1 − λi )
+ t, (i + 1 − λi )
+t ,
:=
πn i=0
n
n
then we have the estimate:
f (t)
b−t
(3.5) (T f ) (a, b; t) −
ln
− Sn (f ; λ, t)
π
t−a
b
1 1
a + b _ 0
b−a 1
(b − a) + t −
(f )
≤
+ max λi − 2
2
2 a
nπ 2 i=0,n−1 S.S. Dragomir
Title Page
Contents
b
≤
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
b−a_ 0
(f ) .
nπ a
JJ
J
Proof. Applying Lemma 3.1 for the function f 0 , we may write that
II
I
Go Back
(3.6)
n−1
f (τ ) − f (t) 1 X
τ
−
t
−
f 0 t + (i + 1 − λi )
τ −t
n i=0
n 1 1
+ max λi −
≤
n 2 i=0,n−1
Close
Quit
τ
1 _ 0 (f )
2 Page 17 of 37
t
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
for any t, τ ∈ [a, b], t 6= τ .
Consequently, we have
Z b
1
f (τ ) − f (t)
(3.7) P V
dτ
π
τ −t
a
Z b n−1
X
τ −t
1
−
PV
f 0 t + (i + 1 − λi )
dτ πn i=0
n
a
Z b _
τ
1 1 1
0 ≤
+ max λi − P V
(f ) dτ
2
nπ 2 i=0,n−1 a t
b
1
_ 0
1
a
+
b
1 1
(b − a) + t −
(f ) .
+ max λi − ≤
2
2
2 a
nπ 2 i=0,n−1
On the other hand
Z b τ −t
0
(3.8)
dτ
PV
f t + (i + 1 − λi )
n
a
Z t−ε Z b τ −t
0
= lim
+
f t + (i + 1 − λi )
dτ
ε→0+
n
a
t+ε
"
t−ε
τ − t n
= lim
f t + (i + 1 − λi )
ε→0+ i + 1 − λi
n
a
b #
n
τ − t +
f t + (i + 1 − λi )
i + 1 − λi
n
t+ε
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 18 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
n
b−t
=
f t + (i + 1 − λi )
i + 1 − λi
n
a−t
−f t + (i + 1 − λi )
n
b−t
a−t
= (b − a) f ; t + (i + 1 − λi )
, (i + 1 − λi )
+t .
n
n
Since (see for example (2.7)),
(T f ) (a, b; t) =
1
PV
π
Z
a
b
f (τ ) − f (t)
f (t)
dτ +
ln
τ −t
π
b−t
t−a
for t ∈ (a, b) , then by (3.7) and (3.8) we deduce the desired estimate (3.5).
Remark 3.1. For n = 1, we recapture the inequality (2.6).
Corollary 3.3. With the assumptions of Theorem 3.2, we have
b−t
f (t)
ln
+ lim Sn (f ; λ, t)
(3.9)
(T f ) (a, b; t) =
n→∞
π
t−a
uniformly by rapport of t ∈ (a, b) and λ with λi ∈ [0, 1) (i ∈ N).
Remark 3.2. If one needs to approximate the finite Hilbert Transform (T f ) (a, b; t)
in terms of
f (t)
b−t
ln
+ Sn (f ; λ, t)
π
t−a
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 19 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
with the accuracy ε > 0 (ε small), then the theoretical minimal number nε to be
chosen is:
"
#
b
b−a_ 0
(3.10)
nε :=
(f ) + 1
επ a
where [α] is the integer part of α.
It is obvious
that the best inequality we can get in (3.5) is for λi =
i = 0, n − 1 obtaining the following corollary.
1
2
Corollary 3.4. Let f be as in Theorem 3.2. Define
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
n−1 b−aX
1 b−t
1 a−t
(3.11) Mn (f ; t) :=
f; i +
+ t, i +
+t .
πn i=0
2
n
2
n
S.S. Dragomir
Then we have the estimate
b
−
t
f
(t)
(3.12) (T f ) (a, b; t) −
ln
− Mn (f ; t)
π
t−a
b
b−a 1
a + b _ 0
≤
(b − a) + t −
(f )
2nπ 2
2 a
Contents
for any t ∈ (a, b).
This rule will be numerically implemented in Section 5 for different choices
of f and n.
Title Page
JJ
J
II
I
Go Back
Close
Quit
Page 20 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
4.
A More General Quadrature Formula
We may state the following lemma.
Lemma 4.1. Let u : [a, b] → R be a function of bounded variation on [a, b],
0 = µ0 < µ1 < · · · < µn−1 < µn = 1 and νi ∈ [µi , µi+1 ], i = 0, n − 1. Then
for any t, τ ∈ [a, b] with t 6= τ , we have the inequality:
(4.1)
n−1
1 Z τ
X
u (s) ds −
(µi+1 − µi ) u [(1 − νi ) t + νi τ ]
τ − t t
i=0
_
τ
µ
+
µ
1
i
i+1
≤ ∆n (µ) + max νi −
(u) ,
t
2
2
i=0,n−1
where ∆n (µ) := max (µi+1 − µi ) .
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
Title Page
i=0,n−1
Proof. Consider the division of [t, τ ] (if t < τ ) or [τ, t] (if τ < t) given by
(4.2)
In : xi := (1 − µi ) t + µi τ i = 0, n .
Then the points ξi := (1 − νi ) t + νi τ i = 0, n − 1 are between xi and xi+1 .
We have
xi+1 − xi = (µi+1 − µi ) (τ − t) i = 0, n − 1
Contents
JJ
J
II
I
Go Back
Close
Quit
and
xi + xi+1
ξi −
=
2
µi + µi+1
νi −
2
Page 21 of 37
(τ − t)
i = 0, n − 1 .
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
Applying the inequality
(2.1) on [xi , xi+1 ] with the intermediate points
ξi i = 0, n − 1 , we get
Z
xi+1
xi
u (s) ds − (µi+1 − µi ) (τ − t) u [(1 − νi ) t + νi τ ]
xi+1
_
µ
+
µ
1
i
i+1 (u)
(µi+1 − µi ) |τ − t| + |τ − t| νi −
≤
2
2
x
i
for any i = 0, n − 1. Summing over i, using the generalised triangle inequality
and dividing by |t − τ | > 0, we obtain
n−1
1 Z b
X
u (s) ds −
(µi+1 − µi ) u [(1 − νi ) t + νi τ ]
τ − t a
i=0
xi+1
n−1 X
_
µ
+
µ
1
i
i+1
≤
(µi+1 − µi ) + νi −
(u)
2
2
x
i=0
i
τ
_ µ
+
µ
1
i
i+1
(u)
≤ ∆n (µ) + max νi −
2
2
i=0,n−1
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
t
Go Back
and the inequality (4.1) is proved.
The following theorem holds.
Theorem 4.2. Let f : [a, b] → R be a differentiable function such that its
derivative f 0 is of bounded variation on [a, b]. If 0 = µ0 < µ1 < · · · < µn−1 <
Close
Quit
Page 22 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
µn = 1 and νi ∈ [µi , µi+1 ], i = 0, n − 1 , then
b−t
f (t)
1
+ Qn (µ, ν, t) + Wn (µ, ν, t)
(4.3)
(T f ) (a, b; t) =
ln
π
t−a
π
for any t ∈ (a, b), where
0
(4.4) Qn (µ, ν, t) := µ1 f (t) (b − a) + (b − a)
n−2 X
(µi+1 − µi )
i=1
× [f ; (1 − νi ) t + νi b, (1 − νi ) t + νi a]
+ (1 − µn−1 ) [f (b) − f (a)]
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
if ν0 = 0, νn−1 = 1,
Title Page
(4.5) Qn (µ, ν, t) := µ1 f 0 (t) (b − a) + (b − a)
n−1
X
(µi+1 − µi )
i=1
× [f ; (1 − νi ) t + νi b, (1 − νi ) t + νi a]
if ν0 = 0, νn−1 < 1,
Contents
JJ
J
II
I
Go Back
(4.6) Qn (µ, ν, t) := (b − a)
n−2
X
(µi+1 − µi )
i=1
× [f ; (1 − νi ) t + νi b, (1 − νi ) t + νi a]
+ (1 − µn−1 ) [f (b) − f (a)]
Close
Quit
Page 23 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
if ν0 > 0, νn−1 = 1 and
(4.7) Qn (µ, ν, t)
:= (b − a)
n−1
X
(µi+1 − µi ) [f ; (1 − νi ) t + νi b, (1 − νi ) t + νi a]
i=1
if ν0 > 0, νn−1 < 1.
In all cases, the remainder satisfies the estimate:
µ
+
µ
1 1
i
i+1
(4.8)
∆n (µ) + max νi −
|Wn (µ, ν, t)| ≤
2
π 2
i=0,n−1
b
a + b _ 0
1
×
(b − a) + t −
(f )
2
2 a
b
_ 0 1
1
a
+
b
(f )
≤ ∆n (µ)
(b − a) + t −
π
2
2 a
≤
1
∆n (µ) (b − a)
π
b
_
(f 0 ) .
a
Proof. If we apply Lemma 4.1 for the function f 0 , we may write that
n−1
f (τ ) − f (t) X
0
−
(µ
−
µ
)
f
[(1
−
ν
)
t
+
ν
τ
]
i+1
i
i
i τ −t
i=0
τ
_
µ
+
µ
1
i
i+1 0 (f
)
≤ ∆n (µ) + max νi −
2
2
i=0,n−1
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 24 of 37
t
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
for any t, τ ∈ [a, b] , t 6= τ .
Taking the P V in both sides, we may write that
Z b
1
f (τ ) − f (t)
PV
dτ
π
τ −t
a
! Z b X
n−1
1
(µi+1 − µi ) f 0 [(1 − νi ) t + νi τ ] dτ − PV
π
a
i=0
Z b _
τ
µ
+
µ
1 1
i
i+1 0 P
V
(f
)
∆n (µ) + max νi −
≤
dτ.
2
π 2
i=0,n−1
a (4.9)
t
If ν0 = 0, νn−1 = 1, then
Z
PV
a
b
S.S. Dragomir
!
n−1
X
(µi+1 − µi ) f 0 [(1 − νi ) t + νi τ ] dτ
Title Page
i=0
Z
= PV
b
µ1 f 0 (t) dτ +
n−2
X
a
Z
Contents
Z
(µi+1 − µi ) P V
b
f 0 [(1 − νi ) t + νi τ ] dτ
a
i=1
b
+ (1 − µn−1 ) P V
f 0 (τ ) dτ
a
0
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
= µ1 f (t) (b − a) + (b − a)
n−2
X
JJ
J
II
I
Go Back
Close
(µi+1 − µi )
i=1
× [f ; (1 − νi ) t + νi b, (1 − νi ) t + νi a] + (1 − µn−1 ) [f (b) − f (a)] .
Quit
Page 25 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
If ν0 = 0, νn−1 < 1, then
b
Z
PV
a
n−1
X
!
(µi+1 − µi ) f 0 [(1 − νi ) t + νi τ ] dτ
i=0
0
= µ1 f (t) (b − a) + (b − a)
n−1
X
(µi+1 − µi )
i=1
× [f ; (1 − νi ) t + νi b, (1 − νi ) t + νi a] .
If ν0 > 0, νn−1 = 1, then
b
Z
PV
a
n−1
X
!
0
(µi+1 − µi ) f [(1 − νi ) t + νi τ ] dτ
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
i=0
= (b − a)
n−2
X
Title Page
(µi+1 − µi ) [f ; (1 − νi ) t + νi b, (1 − νi ) t + νi a]
Contents
i=1
+ (1 − µn−1 ) [f (b) − f (a)] .
and, finally, if ν0 > 0, νn−1 < 1, then
JJ
J
II
I
Go Back
Z
PV
a
b
n−1
X
!
(µi+1 − µi ) f 0 [(1 − νi ) t + νi τ ] dτ
Quit
i=0
= (b − a)
Close
n−1
X
(µi+1 − µi ) [f ; (1 − νi ) t + νi b, (1 − νi ) t + νi a] .
Page 26 of 37
i=1
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
Since
_
Z b _
τ
b
a
+
b
1
0
(b − a) + t −
(f 0 )
PV
(f ) dτ ≤
2
2 a
a t
and
1
(T f ) (a, b; t) = P V
π
Z
then by (4.9) we deduce (4.3).
a
b
f (τ ) − f (t)
f (t)
dτ +
ln
τ −t
π
b−t
t−a
,
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 27 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
5.
Numerical Experiments
For a function f : [a, b] → R, we may consider the quadrature formula
f (t)
b−t
En (f ; a, b, t) :=
ln
+ Mn (f ; t) , t ∈ [a, b] .
π
t−a
As shown above in Section 4, En (f ; a, b, t) provides an approximation for the
Finite Hilbert Transform (T f ) (a, b; t) and the error estimate fulfils the bound
described in (2.3).
If we consider the function f : [−1, 1] → R, f (x) = exp(x), then the exact
value of the Hilbert transform is
exp(t)Ei(1 − t) − exp(t)Ei(−1 − t)
(T f ) (a, b; t) =
, t ∈ [−1, 1]
π
and the plot of this function is embodied in Figure 1.
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
Title Page
Contents
If we implement the quadrature formula provided by En (f ; a, b, t) using
Maple 6 and chose the value of n = 100, then the error Er (f ; a, b, t) :=
(T f ) (a, b; t) − En (f ; a, b, t) has the variation described in the Figure 2.
For n =1,000, the plot of Er (f ; a, b, t) is embodied in the following Figure
3.
Now, if we consider another function, f : [−1, 1] → R, f (x) = sin x, then
JJ
J
II
I
Go Back
Close
Quit
Page 28 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
Figure 1:
S.S. Dragomir
the exact value of the Hilbert transform is
−Si(−1 + t) cos(t) + Ci(1 − t) sin(t)
(T f ) (a, b; t) =
π
Si(t + 1) cos(t) − sin(t)Ci(t + 1))
+
, t ∈ [−1, 1] ;
π
where
Z x
Z x
cos(t) − 1
sin(t)
Si(x) =
dt,
Ci(x) = γ + ln x +
dt;
t
t
0
0
having the plot embodied in the following Figure 4.
If we choose the value of n = 100, then the error Er (f ; a, b, t) for the
function f (x) = sin x, x ∈ [−1, 1] has the variation described in the Figure 5
below.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 29 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
Figure 2:
S.S. Dragomir
Moreover, for n =100,000, the behaviour of Er (f ; a, b, t) is plotted in Figure 6.
Finally, if we choose the function f : [−1, 1] → R, f (x) = sin (x2 ) , the
Maple 6 is unable to produce an exact value of the finite Hilbert transform. If
we use our formula
f (t)
b−t
En (f ; a, b, t) :=
ln
+ Mn (f ; t) , t ∈ [a, b]
π
t−a
for n =1,000, then we can produce the plot in Figure 7.
Taking into account the bound (3.12) we know that the accuracy of the plot
in Figure 7 is at least of order 10−5 .
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 30 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
Figure 3:
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 31 of 37
Figure 4:
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
Figure 5:
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 32 of 37
Figure 6:
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
Title Page
Contents
Figure 7:
JJ
J
II
I
Go Back
Close
Quit
Page 33 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
References
[1] G. CRISCUOLO AND G. MASTROIANNI, Formule gaussiane per il calcolo di integrali a valor principale secondo e Cauchy loro convergenza,
Calcolo, 22 (1985), 391–411.
[2] G. CRISCUOLO AND G. MASTROIANNI, Sulla convergenza di alcune
formule di quadratura per il calcolo di integrali a valor principale secondo
Cauchy, Anal. Numér. Théor. Approx., 14 (1985), 109–116.
[3] G. CRISCUOLO AND G. MASTROIANNI, On the convergence of an interpolatory product rule for evaluating Cauchy principal value integrals,
Math. Comp., 48 (1987), 725–735.
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
[4] G. CRISCUOLO AND G. MASTROIANNI, Mean and uniform convergence of quadrature rules for evaluating the finite Hilbert transform, in: A.
Pinkus and P. Nevai, Eds., Progress in Approximation Theory (Academic
Press, Orlando, 1991), 141–175.
[5] C. DAGNINO, V. DEMICHELIS AND E. SANTI, Numerical integration
based on quasi-interpolating splines, Computing, 50 (1993), 149–163.
[6] C. DAGNINO, V. DEMICHELIS AND E. SANTI, An algorithm for numerical integration based on quasi-interpolating splines, Numer. Algorithms, 5 (1993), 443–452.
[7] C. DAGNINO, V. DEMICHELIS AND E. SANTI, Local spline approximation methods for singular product integration, Approx. Theory Appl., 12
(1996), 1–14.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 34 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
[8] C. DAGNINO AND P. RABINOWITZ, Product integration of singular integrands based on quasi-interpolating splines, Comput. Math., to appear,
special issue dedicated to 100th birthday of Cornelius Lanczos.
[9] P.J. DAVIS AND P. RABINOWITZ, Methods of Numerical Integration,
(Academic Press, Orlando, 2nd Ed., 1984).
[10] K. DIETHELM, Non-optimality of certain quadrature rules for Cauchy
principal value integrals, Z. Angew. Math. Mech, 74(6) (1994), T689–
T690.
[11] K. DIETHELM, Peano kernels and bounds for the error constants of Gaussian and related quadrature rules for Cauchy principal value integrals, Numer. Math., to appear.
[12] K. DIETHELM, Uniform convergence to optimal order quadrature rules
for Cauchy principal value integrals, J. Comput. Appl. Math., 56(3) (1994),
321–329.
[13] S.S. DRAGOMIR, On the Ostrowski integral inequality for mappings of
bounded variation and applications, Math. Ineq.& Appl., 3(1) (2001), 59–
66.
[14] D. ELLIOTT, On the convergence of Hunter’s quadrature rule for Cauchy
principal value integrals, BIT, 19 (1979), 457–462.
[15] D. ELLIOTT AND D.F. PAGET, On the convergence of a quadrature rule
for evaluating certain Cauchy principal value integrals, Numer. Math., 23
(1975), 311–319.
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 35 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
[16] D. ELLIOTT AND D.F. PAGET, On the convergence of a quadrature rule
for evaluating certain Cauchy principal value integrals: An addendum, Numer. Math., 25 (1976), 287–289.
[17] D. ELLIOTT AND D.F. PAGET, Gauss type quadrature rules for Cauchy
principal value integrals, Math. Comp., 33 (1979), 301–309.
[18] T. HASEGAWA AND T. TORII, An automatic quadrature for Cauchy principal value integrals, Math. Comp., 56 (1991), 741–754.
[19] T. HASEGAWA AND T. TORII, Hilbert and Hadamard transforms by generalised Chebychev expansion, J. Comput. Appl. Math., 51 (1994), 71–
83.
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
S.S. Dragomir
[20] D.B. HUNTER, Some Gauss type formulae for the evaluation of Cauchy
principal value integrals, Numer. Math., 19 (1972), 419–424.
[21] N.I. IOAKIMIDIS, Further convergence results for two quadrature rules
for Cauchy type principal value integrals, Apl. Mat., 27 (1982), 457–466.
[22] N.I. IOAKIMIDIS, On the uniform convergence of Gaussian quadrature
rules for Cauchy principal value integrals and their derivatives, Math.
Comp., 44 (1985), 191–198.
Title Page
Contents
JJ
J
II
I
Go Back
[23] A.I. KALANDIYA, Mathematical Methods of Two-Dimensional Elasticity, (Mir, Moscow, 1st English ed., 1978).
Close
[24] G. MONEGATO, The numerical evaluation of one-dimensional Cauchy
principal value integrals, Computing, 29 (1982), 337–354.
Page 36 of 37
Quit
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au
[25] G. MONEGATO, On the weights of certain quadratures for the numerical
evaluation of Cauchy principal value integrals and their derivatives, Numer.
Math., 50 (1987), 273–281.
[26] D.F. PAGET AND D. ELLIOTT, An algorithm for the numerical evaluation
of certain Cauchy principal value integrals, Numer. Math., 19 (1972), 373–
385.
[27] P. RABINOWITZ, On an interpolatory product rule for evaluating Cauchy
principal value integrals, BIT, 29 (1989), 347–355.
[28] P. RABINOWITZ, Numerical integration based on approximating splines,
J. Comput. Appl. Math., 33 (1990), 73–83.
Approximating the Finite Hilbert
Transform via an Ostrowski
Type Inequality for Functions of
Bounded Variation
[29] P. RABINOWITZ, Application of approximating splines for the solution
of Cauchy singular integral equations, Appl. Numer. Math., 15 (1994),
285–297.
S.S. Dragomir
[30] P. RABINOWITZ, Numerical evaluation of Cauchy principal value integrals with singular integrands, Math. Comput., 55 (1990), 265–276.
Contents
[31] P. RABINOWITZ AND E. SANTI, On the uniform convergence of Cauchy
principle value of quasi-interpolating splines, Bit, 35 (1995), 277–290.
[32] M.S. SHESHKO, On the convergence of quadrature processes for a singular integral, Izv, Vyssh. Uohebn. Zaved Mat., 20 (1976), 108–118.
[33] G. TSAMASPHYROS AND P.S. THEOCARIS, On the convergence of
some quadrature rules for Cauchy principal-value and finite-part integrals,
Computing, 31 (1983), 105–114.
Title Page
JJ
J
II
I
Go Back
Close
Quit
Page 37 of 37
J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002
http://jipam.vu.edu.au