JJ II

(1)

volume 3, issue 4, article 51, 2002.

Received 5 April, 2002;

accepted 20 May, 2002.

Communicated by:B. Mond

Abstract Contents

JJ II

J I

Home Page Go Back

Close Quit

Journal of Inequalities in Pure and Applied Mathematics

APPROXIMATING THE FINITE HILBERT TRANSFORM VIA AN OSTROWSKI TYPE INEQUALITY FOR FUNCTIONS OF

BOUNDED VARIATION

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

c

2000Victoria University ISSN (electronic): 1443-5756 032-01

(2)

Approximating the Finite Hilbert Transform via an Ostrowski Type Inequality for Functions of

Bounded Variation S.S. Dragomir

Title Page Contents

JJ II

J I

Go Back Close

Quit Page2of37

J. Ineq. Pure and Appl. Math. 3(4) Art. 51, 2002

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.

1. Introduction

Cauchy principal value integrals of the form (T f) (a, b;t) =P V

Z b a

f(τ) τ−tdτ (1.1)

:= lim

ε→0+

Z t−ε a

f(τ) τ −tdτ+

Z b t+ε

f(τ) τ−tdτ

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.

(4)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page4of37

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 allx∈[a, b], we have the inequality:

(2.1)

u(x) (b−a)− Z b

a

u(t)dt

≤ 1

2(b−a) +

x− a+b 2

^b _

a

(u),

whereWb

a(u)denotes the total variation ofuon[a, b].

The constant ¹₂ is the best possible one.

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

a

(t−a)du(t) =u(x) (x−a)− Z x

a

u(t)dt and

Z b x

(t−b)du(t) = u(x) (b−x)− Z b

x

u(t)dt.

If we add the above two equalities, we get (2.2) u(x) (b−a)−

Z b a

u(t)dt= Z x

a

(t−a)du(t) + Z b

x

(t−b)du(t)

(5)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page5of37

for anyx∈[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:

(2.3)

Z d c

p(x)dv(x)

≤ sup

x∈[c,d]

|p(x)|

d

_

c

(u). Using (2.2) and (2.3), we deduce

u(x) (b−a)− Z b

a

u(t)dt

≤

Z x a

(t−a)du(t)

+

Z b x

(t−b)du(t)

≤(x−a)

x

_

a

(u) + (b−x)

b

_

x

(u)

≤max{x−a, b−x}

" _x _

a

(u) +

b

_

x

(u)

#

= 1

2(b−a) +

x− a+b 2

b

_

a

(u) and the inequality (2.1) is proved.

Now, assume that the inequality (2.2) holds with a constantc >0, i.e., (2.4)

u(x) (b−a)− Z b

a

u(t)dt

≤

c(b−a) +

x−a+b 2

^b _

a

(u) for allx∈[a, b].

(6)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page6of37

Consider the functionu₀ : [a, b]→Rgiven by

u₀(x) =







0 if x∈[a, b]_a+b

2

1 if x= ^a+b₂ . Thenu₀is of bounded variation on[a, b]and

b

_

a

(u₀) = 2, Z b

a

u₀(t)dt= 0.

If we apply (2.4) foru₀and choosex= ^a+b₂ , then we get2c≥1which implies thatc≥ ¹₂ showing that ¹₂ is the best possible constant in (2.1).

The best inequality we can get from (2.1) is the following midpoint inequality.

Corollary 2.2. With the assumptions in Lemma2.1, we have

(2.5)

u

a+b 2

(b−a)− Z b

a

u(t)dt

≤ 1

2(b−a)

b

_

a

(u).

The constant ¹₂ is best possible.

Using the above Ostrowski type inequality we may point out the following result in estimating the finite Hilbert transform.

(7)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page7of37

Theorem 2.3. Let f : [a, b] → R be a function such that its derivative f⁰ : [a, b]→Ris of bounded variation on[a, b]. Then we have the inequality:

(2.6)

(T f) (a, b;t)− f(t) π ln

b−t t−a

−b−a

π [f;λt+ (1−λ)b, λt+ (1−λ)a]

≤ 1 π

1 2+

λ− 1 2

1

2(b−a) +

t− a+b 2

^b _

a

(f⁰),

for anyt ∈(a, b)andλ∈[0,1), where[f;α, β]is the divided difference, i.e., [f;α, β] := f(α)−f(β)

α−β .

Proof. Since f⁰ 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 functionf₀ : (a, b)→R,f₀(t) = 1,t∈(a, b), we have (T f₀) (a, b;t) = 1

πln

b−t t−a

, t∈(a, b), then obviously

(2.7) (T f) (a, b;t)−f(t) π ln

b−t t−a

= 1 πP V

Z b a

f(τ)−f(t) τ −t dτ.

(8)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page8of37

Now, if we choose in (2.1),u=f⁰, x=λc+ (1−λ)d,λ∈[0,1], then we get

|f(d)−f(c)−(d−c)f⁰(λc+ (1−λ)d)|

≤ 1

2|d−c|+

λc+ (1−λ)d− c+d 2

d

_

c

(f⁰) wherec, d∈(a, b), which is equivalent to

(2.8)

f(d)−f(c)

d−c −f⁰(λc+ (1−λ)d)

≤ 1

2 +

λ− 1 2

d

_

c

(f⁰) for anyc, d∈(a, b),c6=d.

Using (2.8), we may write

1 πP V

Z b a

f(τ)−f(t)

τ−t dτ − 1 πP V

Z b a

f⁰(λt+ (1−λ)τ)dτ (2.9)

≤ 1 π

1 2+

λ− 1 2

P V Z b

a

t

_

τ

(f⁰)

dt

= 1 π

1 2 +

λ−1 2

"

Z t a

t

_

τ

(f⁰)

! dt+

Z b t

τ

_

t

(f⁰)

! dt

#

≤ 1 π

1 2+

λ− 1 2

"

(t−a)

t

_

a

(f⁰) + (b−t)

b

_

t

(f⁰)

#

≤ 1 π

1 2+

λ− 1 2

1

2(b−a) +

t−a+b 2

^b _

a

(f⁰).

(9)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page9of37

Since (forλ6= 1) 1

πP V Z b

a

f⁰(λt+ (1−λ)τ)dτ

= 1 π lim

ε→0+

Z t−ε a

+ Z b

t+ε

(f⁰(λt+ (1−λ)τ)dτ)

= 1 π lim

ε→0+

"

1

1−λf(λt+ (1−λ)τ)

t−ε

a

+ 1

1−λf(λt+ (1−λ)τ)

b

t+ε

#

= 1

π · f(t)−f(λt+ (1−λ)a) +f(λt+ (1−λ)b)−f(t) 1−λ

= b−a

π [f;λt+ (1−λ)b, λt+ (1−λ)a]. Using (2.9) and (2.7), we deduce the desired result (2.6).

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.

Corollary 2.4. With the assumptions of Theorem2.3, we have

(2.10)

b−t t−a

− b−a π

f;t+b 2 ,a+t

2

≤ 1 2π

1

2(b−a) +

t− a+b 2

^b _

a

(f⁰).

(10)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page10of37

The above Theorem2.3may 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 thatf : [a, b]⊂(0,∞)→R,f(x) = ¹_x. Then (T f) (a, b;t) = 1

πtln

(b−t)a (t−a)b

, t ∈(a, b),

b−a

π ·[f;λt+ (1−λ)b, λt+ (1−λ)a]

=−1

π · b−a

[λt+ (1−λ)b] [λt+ (1−λ)a],

b

_

a

(f⁰) = Z b

a

|f⁰⁰(t)|dt= b²−a² a²b² . Using the inequality (2.6) we may write that

1 πtln

(b−t)a (t−a)b

− 1 πtln

b−t t−a

+ b−a

π[λt+ (1−λ)b] [λt+ (1−λ)a]

≤ 1 π

1 2+

λ− 1 2

1

2(b−a) +

t− a+b 2

· b²−a² a²b²

(11)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page11of37

which is equivalent to:

(2.11)

b−a

[λt+ (1−λ)b] [λt+ (1−λ)a]− 1 t ln

b a

≤ 1

2+

λ− 1 2

1

2(b−a) +

t− a+b 2

· b² −a² a²b² . If we use the notations

L(a, b) := b−a

lnb−lna (the logarithmic mean)

A_λ(x, y) :=λx+ (1−λ)y (the weighted arithmetic mean) G(a, b) :=√

ab (the geometric mean)

A(a, b) := a+b

2 (the arithmetic mean)

then by (2.11) we deduce

1

Aλ(t, b)Aλ(t, a)− 1 tL(a, b)

≤ 1

2 +

λ− 1 2

1

2(b−a) +|t−A(a, b)|

2A(a, b) G⁴(a, b), giving the following proposition:

(12)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page12of37

Proposition 2.5. With the above assumption, we have (2.12) |tL(a, b)−A_λ(t, b)A_λ(t, a)|

≤ 2A(a, b) G⁴(a, b)

1 2 +

λ−1 2

1

2(b−a) +

t− a+b 2

×tA_λ(t, b)A_λ(t, a)L(a, b) for anyt∈(a, b),λ∈[0,1).

In particular, fort=A(a, b)andλ= ¹₂,we get (2.13)

A(a, b)L(a, b)− (A(a, b) +a) (A(a, b) +b) 4

≤ 1

2· A²(a, b)

G⁴(a, b) · (A(a, b) +a) (A(a, b) +b)

4 L(a, b).

2) Assume thatf : [a, b]⊂R→R,f(x) = exp (x). Then (T f) (a, b;t) = exp (t)

π [Ei(b−t)−Ei(a−t)], where

Ei(z) := P V Z z

−∞

exp (t)

t dt, z ∈R. Also, we have:

b−a

π [exp;λt+ (1−λ)b, λt+ (1−λ)a]

= 1

π · exp (λt+ (1−λ)b)−exp (λt+ (1−λ)a)

1−λ ,

(13)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page13of37

b

_

a

(f⁰) = Z b

a

|f⁰⁰(t)|dt = exp (b)−exp (a). Using the inequality (2.6) we may write:

(2.14)

exp (t)

Ei(b−t)−Ei(a−t)−ln

b−t t−a

− exp (λt+ (1−λ)b)−exp (λt+ (1−λ)a) 1−λ

≤ 1

2+

λ− 1 2

1

2(b−a) +

t− a+b 2

[exp (b)−exp (a)]

for anyt ∈(a, b).

If in (2.14) we makeλ= ¹₂ andt= ^a+b₂ , we get

exp

a+b 2

Ei

b−a 2

−2

exp

a+ 3b 4

−exp

3a+b 4

≤ 1

4(b−a) [exp (b)−exp (a)], which is equivalent to:

Ei

b−a 2

−2

exp

b−a 4

−exp

−b−a 4

≤ 1

4(b−a)

exp

b−a 2

−exp

−b−a 2

.

(14)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page14of37

If in this inequality we make ^b−a₂ =z >0, then we get (2.15)

Ei(z)−2h

expz 2

−exp

−z 2

i

≤ 1

2z[exp (z)−exp (−z)]

for anyz >0.

Consequently, we may state the following proposition.

Proposition 2.6. With the above assumptions, we have (2.16)

Ei(z)−4 sinh 1

2z

≤zsinh (z) for anyz >0.

The reader may get other similar inequalities for special functions if appro- priate examples of functionsf are chosen.

(15)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page15of37

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, λ_i ∈ [0,1) (i= 0, . . . , n−1)andt, τ ∈ [a, b]witht 6=τ, we have the inequality:

(3.1)

1 τ −t

Z τ t

u(s)ds− 1 n

n−1

X

i=0

u

t+ (i+ 1−λ_i)τ −t n

≤ 1 n

1

2+ max

i=0,n−1

λ_i− 1 2

τ

_

t

(u) .

Proof. Consider the equidistant division of [t, τ] (ift < τ) or[τ, t] (ifτ < t) given by

(3.2) E_n:x_i =t+i· τ −t

n , i= 0, n.

Then the pointsξ_i =λ_i

t+i·^τ−t_n

+ (1−λ_i)

t+ (i+ 1)· ^τ−t_n λi ∈[0,1], i= 0, n−1

are between xi and xi+1. We observe that we may write for simplicityξ_i =t+ (i+ 1−λ_i)^τ−t_n i= 0, n−1

. We also have ξ_i− x_i+x_i+1

2 = τ −t

2n (1−2λ_i), ξ_i−x_i = (1−λ_i)τ−t

n

(16)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page16of37

and

x_i+1−ξ_i =λ_i· τ−t n for anyi= 0, n−1.

If we apply the inequality (2.1) on the interval[x_i, x_i+1]and the intermediate pointξi i= 0, n−1

,then we may write that (3.3)

τ −t n u

t+ (i+ 1−λ_i)τ −t n

− Z xi+1

xi

u(s)ds

≤ 1

2 · |τ−t|

n +

τ−t

2n (1−2λ_i)

xi+1

_

xi

(u) . Summing, we get

Z τ t

u(s)ds−τ −t n

n−1

X

i=0

u

t+ (i+ 1−λ_i)τ −t n

≤ |τ −t|

2n

n−1

X

i=0

[1 +|1−2λ_i|]

xi+1

_

xi

(u)

= |τ−t|

n 1

2+ max

i=0,n−1

λ_i− 1 2

τ

_

t

(u) , 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].

(17)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page17of37

Theorem 3.2. Let f : [a, b] → R be a differentiable function such that its derivative f⁰ is of bounded variation on [a, b]. Ifλ = (λi)_i=0,n−1, λi ∈ [0,1)

i= 0, n−1 and

(3.4) S_n(f;λ, t) := b−a

πn

n−1

X

i=0

f; (i+ 1−λ_i)b−t

n +t,(i+ 1−λ_i)a−t n +t

,

then we have the estimate:

b−t t−a

−S_n(f;λ, t) (3.5)

≤ b−a nπ

1

2 + max

i=0,n−1

λ_i− 1 2

1

2(b−a) +

t−a+b 2

b

_

a

(f⁰)

≤ b−a nπ

b

_

a

(f⁰).

Proof. Applying Lemma3.1for the functionf⁰, we may write that (3.6)

f(τ)−f(t) τ−t − 1

n

n−1

X

i=0

f⁰

t+ (i+ 1−λ_i)τ−t n

≤ 1 n

1

2 + max

i=0,n−1

λ_i− 1 2

τ

_

t

(f⁰)

(18)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page18of37

for anyt, τ ∈[a, b],t6=τ. Consequently, we have

1 πP V

Z b a

f(τ)−f(t) τ −t dτ (3.7)

− 1 πn

n−1

X

i=0

P V Z b

a

f⁰

t+ (i+ 1−λ_i)τ −t n

dτ

≤ 1 nπ

1

2+ max

i=0,n−1

λ_i− 1 2

P V Z b

a

τ

_

t

(f⁰)

dτ

≤ 1 nπ

1

2+ max

i=0,n−1

λ_i− 1 2

1

2(b−a) +

t− a+b 2

^b _

a

(f⁰).

On the other hand P V

Z b a

f⁰

t+ (i+ 1−λ_i)τ −t n

dτ (3.8)

= lim

ε→0+

Z t−ε a

+ Z b

t+ε

f⁰

t+ (i+ 1−λ_i)τ −t n

dτ

= lim

ε→0+

"

n i+ 1−λ_if

t+ (i+ 1−λ_i)τ −t n

t−ε

a

+ n

i+ 1−λ_if

t+ (i+ 1−λ_i)τ−t n

b

t+ε

#

(19)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page19of37

= n

i+ 1−λ_i

f

t+ (i+ 1−λ_i)b−t n

−f

t+ (i+ 1−λ_i)a−t n

= (b−a)

f;t+ (i+ 1−λ_i)b−t

n ,(i+ 1−λ_i)a−t n +t

.

Since (see for example (2.7)), (T f) (a, b;t) = 1

πP V Z b

a

f(τ)−f(t)

τ −t dτ +f(t) π ln

b−t t−a

fort∈(a, b),then by (3.7) and (3.8) we deduce the desired estimate (3.5).

Remark 3.1. Forn= 1, we recapture the inequality (2.6).

Corollary 3.3. With the assumptions of Theorem3.2, we have (3.9) (T f) (a, b;t) = f(t)

π ln

b−t t−a

+ lim

n→∞S_n(f;λ, t) uniformly by rapport oft ∈(a, b)andλwithλ_i ∈[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) π ln

b−t t−a

+Sn(f;λ, t)

(20)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page20of37

with the accuracyε >0(εsmall), then the theoretical minimal numbern_εto be chosen is:

(3.10) n_ε:=

"

b−a επ

b

_

a

(f⁰)

# + 1

where[α]is the integer part ofα.

It is obvious that the best inequality we can get in (3.5) is for λ_i = ¹₂ i= 0, n−1

obtaining the following corollary.

Corollary 3.4. Letf be as in Theorem3.2. Define

(3.11) M_n(f;t) := b−a πn

n−1

X

i=0

f;

i+ 1

2

b−t n +t,

i+1

2

a−t n +t

.

Then we have the estimate

(3.12)

b−t t−a

−M_n(f;t)

≤ b−a 2nπ

1

2(b−a) +

t− a+b 2

^b _

a

(f⁰)

for anyt ∈(a, b).

This rule will be numerically implemented in Section5for different choices off andn.

(21)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page21of37

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 < µ_n = 1 andν_i ∈ [µ_i, µ_i+1], i = 0, n−1.Then for anyt, τ ∈[a, b]witht6=τ, we have the inequality:

(4.1)

1 τ −t

Z τ t

u(s)ds−

n−1

X

i=0

(µ_i+1−µ_i)u[(1−ν_i)t+ν_iτ]

≤ 1

2∆_n(µ) + max

i=0,n−1

ν_i− µ_i+µ_i+1 2

τ

_

t

(u) ,

where∆_n(µ) := max

i=0,n−1

(µ_i+1−µ_i).

Proof. Consider the division of[t, τ](ift < τ) or[τ, t](ifτ < t) given by (4.2) I_n :x_i := (1−µ_i)t+µ_iτ i= 0, n

. Then the pointsξi := (1−νi)t+νiτ i= 0, n−1

are betweenxi andxi+1. We have

x_i+1−x_i = (µ_i+1−µ_i) (τ−t) i= 0, n−1 and

ξ_i−x_i+x_i+1

2 =

ν_i−µ_i+µ_i+1 2

(τ −t) i= 0, n−1 .

(22)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page22of37

Applying the inequality (2.1) on [x_i, x_i+1] with the intermediate points ξi i= 0, n−1

, we get

Z xi+1

xi

u(s)ds−(µ_i+1−µ_i) (τ−t)u[(1−ν_i)t+ν_iτ]

≤ 1

2(µ_i+1−µ_i)|τ −t|+|τ −t|

ν_i− µ_i+µ_i+1 2

xi+1

_

xi

(u) for anyi= 0, n−1. Summing overi, using the generalised triangle inequality and dividing by|t−τ|>0,we obtain

1 τ −t

Z b a

u(s)ds−

n−1

X

i=0

(µ_i+1−µ_i)u[(1−ν_i)t+ν_iτ]

≤

n−1

X

i=0

1

2(µ_i+1−µ_i) +

ν_i− µ_i+µ_i+1 2

xi+1

_

xi

(u)

≤ 1

2∆_n(µ) + max

i=0,n−1

ν_i− µ_i+µ_i+1 2

τ

_

t

(u) 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 derivativef⁰ is of bounded variation on[a, b]. If0 =µ₀ < µ₁ <· · ·< µn−1 <

(23)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page23of37

µ_n= 1andν_i ∈[µ_i, µ_i+1], i= 0, n−1 ,then (4.3) (T f) (a, b;t) = f(t)

π ln

b−t t−a

+ 1

πQ_n(µ, ν, t) +W_n(µ, ν, t) for anyt ∈(a, b), where

(4.4) Qn(µ, ν, t) := µ1f⁰(t) (b−a) + (b−a)

n−2

X

i=1

(µi+1−µi)

×[f; (1−ν_i)t+ν_ib,(1−ν_i)t+ν_ia]

+ (1−µn−1) [f(b)−f(a)]

ifν₀ = 0, νn−1 = 1,

(4.5) Q_n(µ, ν, t) := µ₁f⁰(t) (b−a) + (b−a)

n−1

X

i=1

(µ_i+1−µ_i)

×[f; (1−νi)t+νib,(1−νi)t+νia]

ifν₀ = 0, νn−1 <1,

(4.6) Q_n(µ, ν, t) := (b−a)

n−2

X

i=1

(µ_i+1−µ_i)

×[f; (1−ν_i)t+ν_ib,(1−ν_i)t+ν_ia]

+ (1−µn−1) [f(b)−f(a)]

(24)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page24of37

ifν₀ >0, νn−1 = 1and (4.7) Q_n(µ, ν, t)

:= (b−a)

n−1

X

i=1

(µ_i+1−µ_i) [f; (1−ν_i)t+ν_ib,(1−ν_i)t+ν_ia]

ifν₀ >0, νn−1 <1.

In all cases, the remainder satisfies the estimate:

|W_n(µ, ν, t)| ≤ 1 π

1

2∆_n(µ) + max

i=0,n−1

ν_i− µ_i+µ_i+1 2

(4.8)

× 1

2(b−a) +

t− a+b 2

^b _

a

(f⁰)

≤ 1

π∆_n(µ) 1

2(b−a) +

t− a+b 2

b

_

a

(f⁰)

≤ 1

π∆_n(µ) (b−a)

b

_

a

(f⁰).

Proof. If we apply Lemma4.1for the functionf⁰, we may write that

f(τ)−f(t)

τ −t −

n−1

X

i=0

(µ_i+1−µ_i)f⁰[(1−ν_i)t+ν_iτ]

≤ 1

2∆_n(µ) + max

i=0,n−1

ν_i− µ_i+µ_i+1 2

τ

_

t

(f⁰)

(25)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page25of37

for anyt, τ ∈[a, b], t6=τ.

Taking theP V in both sides, we may write that (4.9)

1 πP V

Z b a

f(τ)−f(t) τ−t dτ

−1 πP V

Z b a

n−1

X

i=0

(µ_i+1−µ_i)f⁰[(1−ν_i)t+ν_iτ]

! dτ

≤ 1 π

1

2∆_n(µ) + max

i=0,n−1

ν_i− µi+µi+1

2

P V Z b

a

τ

_

t

(f⁰)

dτ.

Ifν₀ = 0, νn−1 = 1,then P V

Z b a

n−1

X

i=0

(µ_i+1−µ_i)f⁰[(1−ν_i)t+ν_iτ]

! dτ

=P V Z b

a

µ₁f⁰(t)dτ +

n−2

X

i=1

(µ_i+1−µ_i)P V Z b

a

f⁰[(1−ν_i)t+ν_iτ]dτ

+ (1−µn−1)P V Z b

a

f⁰(τ)dτ

=µ₁f⁰(t) (b−a) + (b−a)

n−2

X

i=1

(µ_i+1−µ_i)

×[f; (1−νi)t+νib,(1−νi)t+νia] + (1−µn−1) [f(b)−f(a)].

(26)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page26of37

Ifν₀ = 0, νn−1 <1,then P V

Z b a

n−1

X

i=0

(µ_i+1−µ_i)f⁰[(1−ν_i)t+ν_iτ]

! dτ

=µ₁f⁰(t) (b−a) + (b−a)

n−1

X

i=1

(µ_i+1−µ_i)

×[f; (1−ν_i)t+ν_ib,(1−ν_i)t+ν_ia]. Ifν₀ >0, νn−1 = 1,then

P V Z b

a n−1

X

i=0

(µ_i+1−µ_i)f⁰[(1−ν_i)t+ν_iτ]

! dτ

= (b−a)

n−2

X

i=1

(µ_i+1−µ_i) [f; (1−ν_i)t+ν_ib,(1−ν_i)t+ν_ia]

+ (1−µ_n−1) [f(b)−f(a)]. and, finally, ifν₀ >0, ν_n−1 <1,then

P V Z b

a n−1

X

i=0

(µi+1−µi)f⁰[(1−νi)t+νiτ]

! dτ

= (b−a)

n−1

X

i=1

(µ_i+1−µ_i) [f; (1−ν_i)t+ν_ib,(1−ν_i)t+ν_ia].

(27)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page27of37

Since

P V Z b

a

τ

_

t

(f⁰)

dτ ≤ 1

2(b−a) +

t− a+b 2

^b _

a

(f⁰) and

(T f) (a, b;t) = 1 πP V

Z b a

f(τ)−f(t)

τ −t dτ +f(t) π ln

b−t t−a

, then by (4.9) we deduce (4.3).

(28)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page28of37

5. Numerical Experiments

For a functionf : [a, b]→R,we may consider the quadrature formula E_n(f;a, b, t) := f(t)

π ln

b−t t−a

+M_n(f;t), t∈[a, b].

As shown above in Section 4, E_n(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 functionf : [−1,1]→R, f(x) = exp(x),then the exact value of the Hilbert transform is

(T f) (a, b;t) = exp(t)Ei(1−t)−exp(t)Ei(−1−t)

π , t∈[−1,1]

and the plot of this function is embodied in Figure1.

If we implement the quadrature formula provided by E_n(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 Figure2.

Forn =1,000, the plot ofEr(f;a, b, t)is embodied in the following Figure 3.

Now, if we consider another function,f : [−1,1] →R, f(x) = sinx,then

(29)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page29of37

Figure 1:

the exact value of the Hilbert transform is

(T f) (a, b;t) = −Si(−1 +t) cos(t) +Ci(1−t) sin(t) π

+ Si(t+ 1) cos(t)−sin(t)Ci(t+ 1))

π , t∈[−1,1] ;

where Si(x) =

Z x 0

sin(t)

t dt, Ci(x) = γ+ lnx+ Z x

0

cos(t)−1

t dt;

having the plot embodied in the following Figure4.

If we choose the value of n = 100, then the error Er(f;a, b, t) for the function f(x) = sinx, x ∈ [−1,1]has the variation described in the Figure5 below.

(30)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page30of37

Figure 2:

Moreover, forn =100,000, the behaviour ofEr(f;a, b, t)is plotted in Fig- ure6.

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

E_n(f;a, b, t) := f(t) π ln

b−t t−a

+M_n(f;t), t ∈[a, b]

forn=1,000, then we can produce the plot in Figure7.

Taking into account the bound(3.12)we know that the accuracy of the plot in Figure7is at least of order10⁻⁵.

(31)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page31of37

Figure 3:

Figure 4:

(32)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page32of37

Figure 5:

Figure 6:

(33)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page33of37

Figure 7:

(34)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page34of37

References

[1] G. CRISCUOLOANDG. 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. CRISCUOLOANDG. MASTROIANNI, On the convergence of an interpolatory product rule for evaluating Cauchy principal value integrals, Math. Comp., 48 (1987), 725–735.

[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 ANDE. SANTI, Numerical integration based on quasi-interpolating splines, Computing, 50 (1993), 149–163.

[6] C. DAGNINO, V. DEMICHELIS AND E. SANTI, An algorithm for nu- merical integration based on quasi-interpolating splines, Numer. Algo- rithms, 5 (1993), 443–452.

[7] C. DAGNINO, V. DEMICHELIS AND E. SANTI, Local spline approxi- mation methods for singular product integration, Approx. Theory Appl., 12 (1996), 1–14.

(35)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page35of37

[8] C. DAGNINOANDP. RABINOWITZ, Product integration of singular in- tegrands 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 Gaus- sian and related quadrature rules for Cauchy principal value integrals, Nu- mer. 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 ANDD.F. PAGET, On the convergence of a quadrature rule for evaluating certain Cauchy principal value integrals, Numer. Math., 23 (1975), 311–319.

(36)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page36of37

[16] D. ELLIOTT ANDD.F. PAGET, On the convergence of a quadrature rule for evaluating certain Cauchy principal value integrals: An addendum, Nu- mer. Math., 25 (1976), 287–289.

[17] D. ELLIOTT ANDD.F. PAGET, Gauss type quadrature rules for Cauchy principal value integrals, Math. Comp., 33 (1979), 301–309.

[18] T. HASEGAWAANDT. TORII, An automatic quadrature for Cauchy prin- cipal value integrals, Math. Comp., 56 (1991), 741–754.

[19] T. HASEGAWAANDT. TORII, Hilbert and Hadamard transforms by gen- eralised Chebychev expansion, J. Comput. Appl. Math., 51 (1994), 71–

83.

[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.

[23] A.I. KALANDIYA, Mathematical Methods of Two-Dimensional Elastic- ity, (Mir, Moscow, 1st English ed., 1978).

[24] G. MONEGATO, The numerical evaluation of one-dimensional Cauchy principal value integrals, Computing, 29 (1982), 337–354.

(37)

Title Page Contents

JJ II

J I

Go Back Close

Quit Page37of37

[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. PAGETANDD. 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.

[29] P. RABINOWITZ, Application of approximating splines for the solution of Cauchy singular integral equations, Appl. Numer. Math., 15 (1994), 285–297.

[30] P. RABINOWITZ, Numerical evaluation of Cauchy principal value inte- grals with singular integrands, Math. Comput., 55 (1990), 265–276.

[31] P. RABINOWITZANDE. 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 singu- lar 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.