APPROXIMATING THE FINITE HILBERT TRANSFORM VIA

OSTROWSKI TYPE INEQUALITIES FOR ABSOLUTELY CONTINUOUS FUNCTIONS

S.S. DRAGOMIR

Abstract. Some inequalities and approximations for the finite Hilbert transform by the use of Ostrowski type inequalities for absolutely continuous functions are given.

1. Introduction

Cauchy principal value integrals of the form

(1.1) (T f) (a, b; t) = P V ∫_{a}^{b} \frac{f (τ)}{τ − t}dτ := lim_{ε→0+} [∫_{a}^{t−ε} \frac{f (τ)}{τ − t}dτ + ∫_{t+ε}^{b} \frac{f (τ)}{τ − t}dτ]

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 finding the solutions of differential equations (cf., e.g. [25]).

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], [16] – [24], [26] – [35] 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 inequalities for absolutely continuous functions established in [13], [14] and [15].

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.

2. The Results

For the sake of completeness, we state and prove the following lemma providing some Ostrowski type inequalities for absolutely continuous functions (see [13], [14] and [15]). Lemma 1. Let u : [a, b] → ℝ be an absolutely continuous function on [a, b] . Then we have:

(2.1)  |u (x) (b − a) − ∫_{a}^{b} u (t) dt|

Date: May 28, 2001.

1991 Mathematics Subject Classification. Primary 26D15; Secondary 26D20.

Key words and phrases. Hilbert Transform, Cauchy principal value integrals, Ostrowski Inequality.

≤ \{ \begin{matrix}[\frac{1}{4} (b − a)^{2} + (x − \frac{a + b}{2})^{2}] ∥u′∥_{[a,b],∞} & if u′ ∈ L_{∞} [a, b] ; \\ _{\frac{1}{(q+1)^{\frac{1}{q}}}} [(x − a)^{1+ \frac{1}{q}} + (b − x)^{1+ \frac{1}{q}} ] ∥u′∥_{[a,b],p} & if u′ ∈ L_{p} [a, b] , \\ & p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ [\frac{1}{2} (b − a) + |x − \frac{a + b}{2}|] ∥u′∥_{[a,b],1} , & if u′ ∈ L [a, b] \end{matrix}

where ∥·∥_{r} (r ∈ [1, ∞]) are the usual Lebesgue norms, i.e., for c < d

∥h∥_{[c,d],∞} := ess sup_{t∈[c,d]} |h (t)|

and

∥h∥_{[c,d],r} := (∫_{c}^{d} |h (t)|^{r} dt)^{\frac{1}{r}} , r ≥ 1.

Proof. Using the integration by parts formula, we have

∫_{a}^{x} (t − a) u′ (t) dt = u (x) (x − a) − ∫_{a}^{x} u (t) dt

and

∫_{x}^{b} (t − b) u′ (t) dt = u (x) (b − x) − ∫_{x}^{b} u (t) dt.

If we add the above two equalities, we get

(2.2)  u (x) (b − a) − ∫_{a}^{b} u (t) dt = ∫_{a}^{x} (t − a) u′ (t) dt + ∫_{x}^{b} (t − b) u′ (t) dt

for any x ∈ [a, b].

Taking the modulus, we have

|u (x) (b − a) − ∫_{a}^{b} u (t) dt| ≤ ∫_{a}^{x} (t − a) |u′ (t)| dt + ∫_{x}^{b} (t − b) |u′ (t)| dt
= : M (x) .

Now, it is obvious that

M (x) ≤ ∥u′∥_{[a,x],∞} ∫_{a}^{x} (t − a) dt + ∥u′∥_{[x,b],∞} ∫_{x}^{b} (b − t) dt
= ∥u′∥_{[a,x],∞} · \frac{(x − a)^{2}}{2} + ∥u′∥_{[x,b],∞} · \frac{(b − x)^{2}}{2}

≤ ∥u′∥_{[a,b],∞} [\frac{(x − a)^{2} + (b − x)^{2}}{2} ]

= ∥u′∥_{[a,b],∞} [\frac{1}{4} (b − a)^{2} + (x − \frac{a + b}{2})^{2}] ,

proving the first part of (2.1).

Using Hölder’s integral inequality, we may write:

M (x) ≤ ∥u′∥_{[a,x],p} (∫_{a}^{x} (t − a)^{q} dt)^{\frac{1}{q}} + ∥u′∥_{[x,b],p} (∫_{x}^{b} (b − t)^{q} dt)^{\frac{1}{q}}

= ∥u′∥_{[a,x],p} · [\frac{(x − a)^{q+1}}{q + 1} ]^{\frac{1}{q}} + ∥u′∥_{[x,b],p} · [\frac{(b − x)^{q+1}}{q + 1} ]^{\frac{1}{q}}
≤ ∥u′∥_{[a,b],p} \frac{1}{(q + 1)^{\frac{1}{q}}} [(x − a)^{1+ \frac{1}{q}} + (b − x)^{1+ \frac{1}{q}} ] ,

proving the second part of (2.1).

Finally, we observe that

M (x) ≤ (x − a) ∥u′∥_{[a,x],1} + (b − x) ∥u′∥_{[x,b],1}
≤ max {x − a, b − x} [∥u′∥_{[a,x],1} + ∥u′∥_{[x,b],1}]
= [\frac{1}{2} (b − a) + |x − \frac{a + b}{2}|] ∥u′∥_{[a,b],1}

and the lemma is proved.

The best inequalities we can get from (2.1) are embodied in the following corollary.

Corollary 1. With the assumptions of Lemma 1, we have

|u (\frac{a + b}{2}) (b − a) − ∫_{a}^{b} u (t) dt| (2.3)

≤ \{ \begin{matrix}\frac{1}{4} (b − a)^{2} ∥u′∥_{[a,b],∞} & if u′ ∈ L_{∞} [a, b] ; \\ _{\frac{1}{2(q+1)^{\frac{1}{q}}}} (b − a)^{1+ \frac{1}{q}} ∥u′∥_{[a,b],p} & if u′ ∈ L_{p} [a, b] , \\ & p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ \frac{1}{2} (b − a) ∥u′∥_{[a,b],1} . & if u′ ∈ L [a, b] \end{matrix}

The following theorem providing an estimate for the finite Hilbert transform, holds.

Theorem 1. Let f : [a, b] → ℝ be a function so that its derivative f′ : [a, b] → ℝ is absolutely continuous on [a, b]. Then we have the inequalities

(2.4) |(T f) (a, b; t) − \frac{f (t)}{π} ln (\frac{b − t}{t − a}) − \frac{b − a}{π} [f; λt + (1 − λ) b, λt + (1 − λ) a]|

≤ \{ \begin{matrix}\frac{1}{π} & [\frac{1}{4} + (λ − \frac{1}{2})^{2}] \\ × [\frac{1}{4} (b − a)^{2} + (t − \frac{a + b}{2})^{2}] ∥f′′∥_{[a,b],∞} & if f′′ ∈ L_{∞} [a, b] ; \\ \frac{1}{π} & \frac{q}{(q + 1)^{1+ \frac{1}{q}}} & [λ^{1+ \frac{1}{q}} + (1 − λ)^{1+ \frac{1}{q}} ] \\ × [(t − a)^{1+ \frac{1}{q}} + (b − t)^{1+ \frac{1}{q}} ] ∥f′′∥_{[a,b],p} & if f′′ ∈ L_{p} [a, b] , \\ & p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ \frac{1}{π} [\frac{1}{2} + |λ − \frac{1}{2}|] [\frac{1}{2} (b − a) + |t − \frac{a + b}{2}|] ∥f′′∥_{[a,b],1} , & if f′′ ∈ L [a, b] \end{matrix}

for any t ∈ (a, b) and λ ∈ [0, 1), where [f; α, β] is the divided difference, i.e.,

[f; α, β] = \frac{f (α) − f (β)}{α − β}.

Proof. Since f′ 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 f_{0} : (a, b) → ℝ, f_{0} (t) = 1, t ∈ (a, b) , we have

(T f_{0}) (a, b; t) = \frac{1}{π} ln (\frac{b − t}{t − a}) , t ∈ (a, b) ,

then, obviously,

(T f) (a, b; t) = \frac{1}{π} P V ∫_{a}^{b} \frac{f (τ) − f (t) + f (t)}{τ − t}dτ
= \frac{1}{π} P V ∫_{a}^{b} \frac{f (τ) − f (t)}{τ − t}dτ + \frac{f (t)}{π}P V ∫_{a}^{b} \frac{dτ}{τ − t},

from where we get the identity

(2.5)  (T f) (a, b; t) − \frac{f (t)}{π} ln (\frac{b − t}{t − a}) = \frac{1}{π} P V ∫_{a}^{b} \frac{f (τ) − f (t)}{τ − t}dτ.

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

|f (d) − f (c) − (d − c) f′ (λc + (1 − λ) d)|

≤ \{ \begin{matrix}(d − c)^{2} [\frac{1}{4} + (λ − \frac{1}{2})^{2}] ∥f′′∥_{[c,d],∞} & if f′′ ∈ L_{∞} [a, b] ; \\ \begin{matrix}|d − c| & 1+ \frac{1}{q} \\ (q + 1) & \frac{1}{q} \end{matrix} [λ^{1+ \frac{1}{q}} + (1 − λ)^{1+ \frac{1}{q}} ] |∥f′′∥_{[c,d],p}| & if f′′ ∈ L_{p} [a, b] , \\ & p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ |d − c| [\frac{1}{2} + |λ − \frac{1}{2}|] |∥f′′∥_{[c,d],1}| , \end{matrix}

FINITE HILBERT TRANSFORM VIA OSTROWSKI TYPE INEQUALITIES 5

which is equivalent to:

|\frac{f (d) − f (c)}{d − c} − f′ (λc + (1 − λ) d)| (2.6)

≤ \{ \begin{matrix}|d − c| [\frac{1}{4} + (λ − \frac{1}{2})^{2}] ∥f′′∥_{[c,d],∞} & if f′′ ∈ L_{∞} [a, b] ; \\ \begin{matrix}|d − c| & \frac{1}{q} \\ (q + 1) & \frac{1}{q} \end{matrix} [λ^{1+ \frac{1}{q}} + (1 − λ)^{1+ \frac{1}{q}} ] |∥f′′∥_{[c,d],p}| & if f′′ ∈ L_{p} [a, b] , \\ & p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ [\frac{1}{2} + |λ − \frac{1}{2}|] |∥f′′∥_{[c,d],1}| . \end{matrix}

Using (2.6), we may write

|\frac{1}{π} P V ∫_{a}^{b} \frac{f (τ) − f (t)}{τ − t}dτ − \frac{1}{π} P V ∫_{a}^{b} f′ (λt + (1 − λ) τ) dτ| (2.7)

≤ \{ \begin{matrix}\frac{1}{π} [\frac{1}{4} + (λ − \frac{1}{2})^{2}] P V ∫_{a}^{b} |t − τ| |∥f′′∥_{[t,τ],∞}| dτ \\ \frac{1}{π} · \frac{1}{(q + 1)^{\frac{1}{q}}} [λ^{1+ \frac{1}{q}} + (1 − λ)^{1+ \frac{1}{q}} ] P V ∫_{a}^{b} |t − τ|^{\frac{1}{q}} |∥f′′∥_{[t,τ],p}| dτ \\ \frac{1}{π} [\frac{1}{2} + |λ − \frac{1}{2}|] P V ∫_{a}^{b} |∥f′′∥_{[t,τ],1}| dτ \end{matrix}

≤ \{ \begin{matrix}\frac{1}{π} [\frac{1}{4} + (λ − \frac{1}{2})^{2}] ∥f′′∥_{[a,b],∞} P V ∫_{a}^{b} |t − τ| dτ \\ \frac{1}{π} · \frac{1}{(q + 1)^{\frac{1}{q}}} [λ^{1+ \frac{1}{q}} + (1 − λ)^{1+ \frac{1}{q}} ] ∥f′′∥_{[a,b],p} P V ∫_{a}^{b} |t − τ|^{\frac{1}{q}} dτ \\ \frac{1}{π} [\frac{1}{2} + |λ − \frac{1}{2}|] P V [∫_{a}^{t} ∥f′′∥_{[τ,t],1} dτ + ∫_{t}^{b} ∥f′′∥_{[t,τ],1} dτ] \end{matrix}

≤ \{ \begin{matrix}\frac{1}{π} & [\frac{1}{4} + (λ − \frac{1}{2})^{2}] & ∥f′′∥ & [a,b],∞ & [\frac{1}{4} (b − a)^{2} + (t − \frac{a + b}{2})^{2}] \\ \frac{1}{π} · \frac{1}{(q + 1)^{\frac{1}{q}}} [λ^{1+ \frac{1}{q}} + (1 − λ)^{1+ \frac{1}{q}} ] ∥f′′∥_{[a,b],p} \\ ×\frac{q}{(q + 1)} [(t − a)^{1+ \frac{1}{q}} + (b − t)^{1+ \frac{1}{q}} ] \\ \frac{1}{π} [\frac{1}{2} + |λ − \frac{1}{2}|] [\frac{1}{2} (b − a) + |t − \frac{a + b}{2}|] ∥f′′∥_{[a,b],1} . \end{matrix}

Since (note that λ ≠ 1)

\frac{1}{π} P V ∫_{a}^{b} f′ (λt + (1 − λ) τ) dτ

= \frac{1}{π} lim_{ε→0+} [∫_{a}^{t−ε} + ∫_{t+ε}^{b}] (f′ (λt + (1 − λ) τ) dτ)

= \frac{1}{π} lim_{ε→0+} [ \frac{1}{1 − λ}f (λt + (1 − λ) τ)|_{a}^{t−ε} + \frac{1}{1 − λ}f (λt + (1 − λ) τ)|_{t+ε}^{b}]

= \frac{1}{π} \frac{f (t) − f (λt + (1 − λ) a) + f (λt + (1 − λ) a) − f (t)}{1 − λ}
= \frac{b − a}{π} [f; λt + (1 − λ) b, λt + (1 − λ) a] ,

then by (2.5) and (2.7) we deduce the desired inequality (2.4).

The best inequality one may obtain from (2.4) is embodied in the following corollary.

Corollary 2. With the assumptions of Theorem 1, one has the inequality

|(T f) (a, b; t) − \frac{f (t)}{π} ln (\frac{b − t}{t − a}) − \frac{b − a}{π} [f; \frac{t + b}{2}, \frac{a + t}{2}]| (2.8)

≤ \{ \begin{matrix}\frac{1}{4π} [\frac{1}{4} (b − a)^{2} + (t − \frac{a + b}{2})^{2}] ∥f′′∥_{[a,b],∞} & if f′′ ∈ L_{∞} [a, b] ; \\ \frac{1}{π} \frac{q}{2^{\frac{p}{q}} (q + 1)^{1+ \frac{1}{q}}} [(t − a)^{1+ \frac{1}{q}} + (b − t)^{1+ \frac{1}{q}} ] ∥f′′∥_{[a,b],p} & if f′′ ∈ L_{p} [a, b] , \\ & p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ \frac{1}{2π} [\frac{1}{2} (b − a) + |t − \frac{a + b}{2}|] ∥f′′∥_{[a,b],1} , & if f′′ ∈ L [a, b] ; \end{matrix}

for any t ∈ (a, b) .

3. A Quadrature Formula

The following lemma is of interest in itself.

Lemma 2. Let u : [a, b] → ℝ be an absolutely continuous function on [a, b]. Then for all n ≥ 1, λ_{i} ∈ [0, 1) (i = 0, …, n − 1) and t, τ ∈ [a, b] with t ≠ τ, we have the inequality

|\frac{1}{τ − t} ∫_{t}^{τ} u (s) ds − \frac{1}{n} ∑_{i=0}^{n−1} u [t + (i + 1 − λ_{i}) \frac{τ − t}{n}]| (3.1)

≤ \{ \begin{matrix}\frac{|t − τ|}{n} [\frac{1}{4} + \frac{1}{n} ∑_{i=0}^{n−1} (λ_{i} − \frac{1}{2})^{2}] ∥u′∥_{[t,τ],∞}; \\ \frac{|t − τ|^{\frac{1}{q}}}{(q + 1)^{\frac{1}{q}} n} & [\frac{1}{n} ∑_{i=0}^{n−1} (λ_{i}^{1+ \frac{1}{q}} + (1 − λ_{i})^{1+ \frac{1}{q}} )^{q}] & \frac{1}{q} & ∥u′∥ & [t,τ],p \\ p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ \frac{1}{n} [\frac{1}{2} + max |λ_{i} − \frac{1}{2}|] ∥u′∥_{[t,τ],1} ; \end{matrix}

≤ \{ \begin{matrix}\frac{|t − τ|}{2n} ∥u′∥_{[t,τ],∞}; \\ \frac{|t − τ|^{\frac{1}{q}}}{(q + 1)^{\frac{1}{q}} n} ∥u′∥_{[t,τ],p} & p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ \frac{1}{n} ∥u′∥_{[t,τ],1} , \end{matrix}

where

∥u′∥_{[t,τ],p} := |∫_{t}^{τ} |u′ (s)|^{p} ds|^{\frac{1}{p}} , p ≥ 1

and

∥u′∥_{[t,τ],∞} := ess sup_{\begin{matrix}s∈[t,τ] \\ (s∈[τ,t]) \end{matrix}} |u′ (s)| .

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

(3.2)  E_{n} : x_{i} = t + i · \frac{τ − t}{n}, i = \overline{0, n}.

Then the points ξ_{i} := λ_{i} [t + i · \frac{τ−t}{n}]+(1 − λ_{i}) [t + (i + 1) · \frac{τ−t}{n}] (λ_{i} ∈ [0, 1), i = \overline{0, n − 1}) are between x_{i} and x_{i+1}. We observe that we may write for simplicity ξ_{i} = t + (i + 1 − λ_{i}) \frac{τ−t}{n} (i = \overline{0, n − 1}) . We also have

ξ_{i} − \frac{x_{i} + x_{i+1}}{2} = \frac{τ − t}{n} (\frac{1}{2} − λ_{i});
ξ_{i} − x_{i} = (1 − λ_{i}) \frac{τ − t}{n}

and

x_{i+1} − ξ_{i} = λ_{i} · \frac{τ − t}{n}

for any i = \overline{0, n − 1}.

8 S.S. DRAGOMIR

If we apply the inequality (2.1) on the interval [x_{i}, x_{i+1}] and the intermediate points ξ_{i} (i = \overline{0, n − 1}) , then we may write that

(3.3)  |\frac{τ − t}{n}u [t + (i + 1 − λ_{i}) \frac{τ − t}{n}] − ∫_{x_{i}}^{x_{i+1}} u (s) ds|

(3.4)

≤ \{ \begin{matrix}[\frac{1}{4} \frac{(t − τ)^{2}}{n^{2}} + \frac{(t − τ)^{2}}{4n^{2}} (1 − 2λ_{i})^{2}] ∥u′∥_{[x_{i},x_{i+1}],∞} & if u′ ∈ L_{∞} [a, b] , \\ \frac{1}{(q + 1)^{\frac{1}{q}}} [\frac{|t − τ|^{1+ \frac{1}{q}}}{n^{1+ \frac{1}{q}}} λ_{i}^{1+ \frac{1}{q}} + \frac{|t − τ|^{1+ \frac{1}{q}}}{n^{1+ \frac{1}{q}}} (1 − λ_{i})^{1+ \frac{1}{q}} ] ∥u′∥_{[x_{i},x_{i+1}],p} & if u′ ∈ L_{p} [a, b] , \\ & p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ [\frac{1}{2} \frac{|τ − t|}{n} + \frac{|τ − t|}{n} |\frac{1}{2} − λ_{i}|] ∥u′∥_{[t,τ],1} ; \end{matrix}

Summing (2.3), we get:

|∫_{t}^{τ} u (s) ds − \frac{τ − t}{n} ∑_{i=0}^{n−1} u [t + (i + 1 − λ_{i}) \frac{τ − t}{n}]| (3.5)

≤ \{ \begin{matrix}\frac{(t − τ)^{2}}{n^{2}} ∑_{i=0}^{n−1} [\frac{1}{4} + (λ_{i} − \frac{1}{2})^{2}] ∥u′∥_{[x_{i},x_{i+1}],∞}; \\ \frac{|t − τ|^{1+ \frac{1}{q}}}{(q + 1)^{\frac{1}{q}} n^{1+ \frac{1}{q}}} & ∑_{i=0}^{n−1} & [λ_{i}^{1+ \frac{1}{q}} + (1 − λ_{i})^{1+ \frac{1}{q}} ] & ∥u′∥ & [x_{i},x_{i+1}],p \\ p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ \frac{|t − τ|}{n} ∑_{i=0}^{n−1} [\frac{1}{2} + |λ_{i} − \frac{1}{2}|] ∥u′∥_{[x_{i},x_{i+1}],1} . \end{matrix}

However,

∑_{i=0}^{n−1} [\frac{1}{4} + (λ_{i} − \frac{1}{2})^{2}] ∥u′∥_{[x_{i},x_{i+1}],∞} (3.6)

= ∥u′∥_{[t,τ],∞} [\frac{1}{4}n + ∑_{i=0}^{n−1} (λ_{i} − \frac{1}{2})^{2}] ,

∑_{i=0}^{n−1} [λ_{i}^{1+ \frac{1}{q}} + (1 − λ_{i})^{1+ \frac{1}{q}} ] ∥u′∥_{[x_{i},x_{i+1}],p} (3.7)

≤ (∑_{i=0}^{n−1} [λ_{i}^{1+ \frac{1}{q}} + (1 − λ_{i})^{1+ \frac{1}{q}} ]^{q})^{\frac{1}{q}} (∑_{i=0}^{n−1} ∥u′∥_{[x_{i},x_{i+1}],p}^{p})^{\frac{1}{p}}
= ∥u′∥_{[t,τ],p} [∑_{i=0}^{n−1} (λ_{i}^{1+ \frac{1}{q}} + (1 − λ_{i})^{1+ \frac{1}{q}} )^{q}]^{\frac{1}{q}}

and

(3.8)  ∑_{i=0}^{n−1} [\frac{1}{2} + |λ_{i} − \frac{1}{2}|] ∥u′∥_{[x_{i},x_{i+1}],1}

≤ [\frac{1}{2} + max |λ_{i} − \frac{1}{2}|] ∑_{i=0}^{n−1} ∥u′∥_{[x_{i},x_{i+1}],1} = [\frac{1}{2} + max |λ_{i} − \frac{1}{2}|] ∥u′∥_{[t,τ],1} .

Now, using (3.5) – (3.8), we deduce the first part of (3.1).

The second part is obvious.

We may now state the following theorem in approximating the finite Hilbert transform of a differentiable function whose derivative is absolutely continuous. Theorem 2. Let f : [a, b] → ℝ be a differentiable function so that its derivative f′ is absolutely continuous on [a, b]. If λ = (λ_{i})_{i=\overline{0,n−1}} , λ_{i} ∈ [0, 1) (i = \overline{0, n − 1}) and

(3.9) S_{n} (f; λ, t) := \frac{b − a}{nπ} ∑_{i=0}^{n−1} [f; t + (i + 1 − λ_{i}) \frac{b − t}{n}, t − (i + 1 − λ_{i}) \frac{t − a}{n}]

then we have

(3.10)  (T f) (a, b; t) = \frac{f (t)}{π} ln (\frac{b − t}{t − a}) + S_{n} (f; λ, t) + R_{n} (f; λ, t)

and the remainder R_{n} (f; λ, t) satisfies the estimate:

(3.11)  |R_{n} (f; λ, t)|

≤ \frac{1}{π} ×\{ \begin{matrix}\frac{1}{n} & [\frac{1}{4} + \frac{1}{n} ∑_{i=0}^{n−1} (λ_{i} − \frac{1}{2})^{2}] \\ × [\frac{1}{4} (b − a)^{2} + (t − \frac{a + b}{2})^{2}] ∥f′′∥_{[a,b],∞} & if f′′ ∈ L_{∞} [a, b] ; \\ \frac{1}{n} · \frac{q}{(q + 1)^{\frac{1}{q} +1}} [\frac{1}{n} ∑_{i=0}^{n−1} (λ_{i}^{1+ \frac{1}{q}} + (1 − λ_{i})^{1+ \frac{1}{q}} )^{q}]^{\frac{1}{q}} \\ × [(t − a)^{1+ \frac{1}{q}} + (b − t)^{1+ \frac{1}{q}} ] ∥f′′∥_{[a,b],p} & if f′′ ∈ L_{p} [a, b] , \\ & p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ \frac{1}{n} [\frac{1}{2} + max |λ_{i} − \frac{1}{2}|] [\frac{1}{2} (b − a) + |t − \frac{a + b}{2}|] ∥f′′∥_{[a,b],1} , \end{matrix}

(3.12)

≤ \frac{1}{π} ×\{ \begin{matrix}\frac{1}{4n} ∥f′′∥_{[a,b],∞} (b − a)^{2} & if f′′ ∈ L_{∞} [a, b] ; \\ \frac{q}{(q + 1)^{\frac{1}{q} +1} n} (b − a)^{1+ \frac{1}{q}} ∥f′′∥_{[a,b],p} & if f′′ ∈ L_{p} [a, b] , p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ \frac{1}{n} (b − a) ∥f′′∥_{[a,b],1} , \end{matrix}

10 S.S. DRAGOMIR

Proof. Applying Lemma 2 for the function f′, we may write that

|\frac{f (t) − f (τ)}{τ − t} − \frac{1}{n} ∑_{i=0}^{n−1} f′ [t + (i + 1 − λ_{i}) \frac{τ − t}{n}]| (3.13)

≤ \{ \begin{matrix}\frac{|t − τ|}{n} [\frac{1}{4} + \frac{1}{n} ∑_{i=0}^{n−1} (λ_{i} − \frac{1}{2})^{2}] ∥f′′∥_{[t,τ],∞}; \\ \frac{|t − τ|^{\frac{1}{q}}}{(q + 1)^{\frac{1}{q}} n} & [\frac{1}{n} ∑_{i=0}^{n−1} (λ_{i}^{1+ \frac{1}{q}} + (1 − λ_{i})^{1+ \frac{1}{q}} )^{q}] & \frac{1}{q} & ∥f′′∥ & [t,τ],p \\ \frac{1}{n} & [\frac{1}{2} + max |λ_{i} − \frac{1}{2}|] & ∥f′′∥ & [t,τ],1 \end{matrix}

for any t, τ ∈ [a, b] , t ≠ τ.

Taking the PV, we may write

(3.14) |\frac{1}{π} P V ∫_{a}^{b} \frac{f (t) − f (τ)}{τ − t}dτ − \frac{1}{nπ} ∑_{i=0}^{n−1} P V ∫_{a}^{b} f′ [t + (i + 1 − λ_{i}) \frac{τ − t}{n}] dτ|

(3.15)

≤ \frac{1}{π} ×\{ \begin{matrix}\frac{1}{n} [\frac{1}{4} + \frac{1}{n} ∑_{i=0}^{n−1} (λ_{i} − \frac{1}{2})^{2}] P V ∫_{a}^{b} |t − τ| ∥f′′∥_{[t,τ],∞} dτ; \\ \frac{1}{n (q + 1)^{\frac{1}{q}}} [\frac{1}{n} ∑_{i=0}^{n−1} (λ_{i}^{1+ \frac{1}{q}} + (1 − λ_{i})^{1+ \frac{1}{q}} )^{q}]^{\frac{1}{q}} P V ∫_{a}^{b} |t − τ|^{\frac{1}{q}} ∥f′′∥_{[t,τ],p} dτ \\ \frac{1}{n} [\frac{1}{2} + max |λ_{i} − \frac{1}{2}|] P V ∫_{a}^{b} ∥f′′∥_{[t,τ],1} dτ. \end{matrix}

However,

P V ∫_{a}^{b} |t − τ| ∥f′′∥_{[t,τ],∞} dτ ≤ ∥f′′∥_{[a,b],∞} P V ∫_{a}^{b} |t − τ| dτ
= ∥f′′∥_{[a,b],∞} [\frac{(t − a)^{2} + (b − t)^{2}}{2} ] ,
P V ∫_{a}^{b} |t − τ|^{\frac{1}{q}} ∥f′′∥_{[t,τ],p} dτ ≤ ∥f′′∥_{[a,b],p} P V ∫_{a}^{b} |t − τ|^{\frac{1}{q}} dτ
= ∥f′′∥_{[a,b],p} [\frac{(t − a)^{\frac{1}{q} +1} + (b − t)^{\frac{1}{q} +1}}{\frac{1}{q} + 1} ]
= \frac{q}{(q + 1)} [(t − a)^{\frac{1}{q} +1} + (b − t)^{\frac{1}{q} +1}] ∥f′′∥_{[a,b],p} ,

P V ∫_{a}^{b} ∥f′′∥_{[t,τ],1} dτ = P V [∫_{a}^{t} ∥f′′∥_{[τ,t],1} dτ + ∫_{t}^{b} ∥f′′∥_{[t,τ],1} dτ]
≤ max {t − a, b − t} ∥f′′∥_{[a,b],1}
= [\frac{1}{2} (b − a) + |t − \frac{a + b}{2}|] ∥f′′∥_{[a,b],1}

and using the inequality (3.14) we obtain the desired estimate (3.11).

The following particular case which may be easily numerically implemented holds. Corollary 3. let f be as in Theorem 2. Define

S_{M,n} (f; t) := \frac{b − a}{nπ} ∑_{i=0}^{n−1} [f; t + (i + \frac{1}{2}) \frac{b − t}{n}, t − (i + \frac{1}{2}) \frac{t − a}{n}]

and the remanider R_{M,n} (f; t) satisfies the estimate

(T f) (a, b; t) = \frac{f (t)}{π} ln (\frac{b − t}{t − a}) + S_{M,n} (f; t) + R_{M,n} (f; t)

Then we have the representation:

|R_{M,n} (f; t)|

≤ \frac{1}{π} ×\{ \begin{matrix}\frac{1}{4nπ} [\frac{1}{4} (b − a)^{2} + (t − \frac{a + b}{2})^{2}] ∥f′′∥_{[a,b],∞} & if f′′ ∈ L_{∞} [a, b] ; \\ \frac{1}{n} · \frac{1}{2^{\frac{1}{q}} πn (q + 1)^{\frac{1}{q} +1}} [(t − a)^{1+ \frac{1}{q}} + (b − t)^{1+ \frac{1}{q}} ] ∥f′′∥_{[a,b],p} & if f′′ ∈ L_{p} [a, b] , \\ & p > 1, \frac{1}{p} + \frac{1}{q} = 1; \\ \frac{1}{2πn} [\frac{1}{2} (b − a) + |t − \frac{a + b}{2}|] ∥f′′∥_{[a,b],1} , \end{matrix}

for any t ∈ (a, b) .

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.

[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 quasiinterpolating 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.

[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 and S. WANG, Applications of Ostrowski’s inequality to the estimation of error bounds for some special means and to some numerical quadrature rules, Comp. Math. Appl., 33 (1997), 15-20.

[14] S.S. DRAGOMIR and S. WANG, A new inequality of Ostrowski type in L_{1} norm and applications to some special means and some numerical quadrature rules, Tamkang J. of Math., 28 (1997), 239-244.

[15] S.S. DRAGOMIR and S. WANG, A new inequality of Ostrowski’s type in L_{p} norm and applications to some numerical quadrature rules, Indian J. of Math., 40 (3) (1998), 245-304.

[16] D. ELLIOTT, On the convergence of Hunter’s quadrature rule for Cauchy principal value integrals, BIT, 19 (1979), 457-462.

[17] 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.

[18] 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.

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

[20] T. HASEGAWA and T. TORII, An automatic quadrature for Cauchy principal value integrals, Math. Comp., 56 (1991), 741-754.

[21] T. HASEGAWA and T. TORII, Hilbert and Hadamard transforms by generalised Chebychev expansion, J. Comput. Appl. Math., 51 (1994), 71-83.

[22] D.B. HUNTER, Some Gauss type formulae for the evaluation of Cauchy principal value integrals, Numer. Math., 19 (1972), 419-424.

[23] N.I. IOAKIMIDIS, Further convergence results for two quadrature rules for Cauchy type principal value integrals, Apl. Mat., 27 (1982), 457-466.

[24] 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.

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

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

[27] 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.

[28] D.F. PAGET and D. ELLIOTT, An algorithm for the numerical evaluation of certain Cauchy principal value integrals, Numer. Math., 19 (1972), 373-385.

[29] P. RABINOWITZ, On an interpolatory product rule for evaluating Cauchy principal value integrals, BIT, 29 (1989), 347-355.

[30] P. RABINOWITZ, Numerical integration based on approximating splines, J. Comput. Appl. Math., 33 (1990), 73-83.

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

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

[33] P. RABINOWITZ and E. SANTI, On the uniform convergence of Cauchy principle value of quasi-interpolating splines, Bit, 35 (1995), 277-290.

[34] M.S. SHESHKO, On the convergence of quadrature processes for a singular integral, Izv, Vyssh. Uohebn. Zaved Mat., 20 (1976), 108-118.

[35] 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.

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