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volume 6, issue 4, article 106, 2005.

Received 16 September, 2005;

accepted 23 September, 2005.

Communicated by:A. Lupa¸s

Abstract Contents

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Journal of Inequalities in Pure and Applied Mathematics

RATE OF CONVERGENCE OF CHLODOWSKY TYPE DURRMEYER OPERATORS

ERTAN IBIKLI AND HARUN KARSLI

Ankara University, Faculty of Sciences, Department of Mathematics,

06100 Tandogan - Ankara/Turkey.

EMail:ibikli@science.ankara.edu.tr EMail:karsli@science.ankara.edu.tr

c

2000Victoria University ISSN (electronic): 1443-5756 276-05

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Rate of Convergence of Chlodowsky Type Durrmeyer

Operators Ertan Ibikli and Harun Karsli

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J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005

http://jipam.vu.edu.au

Abstract

In the present paper, we estimate the rate of pointwise convergence of the Chlodowsky type Durrmeyer OperatorsD_n(f, x)for functions, defined on the interval[0, bn],(bn→ ∞), extending infinity, of bounded variation. To prove our main result, we have used some methods and techniques of probability theory.

2000 Mathematics Subject Classification:41A25, 41A35, 41A36.

Key words: Approximation, Bounded variation, Chlodowsky polynomials, Durrmeyer Operators, Chanturiya’s modulus of variation, Rate of convergence.

1. Introduction

Very recently, some authors studied some linear positive operators and obtained the rate of convergence for functions of bounded variation. For example, Bo- janic R. and Vuilleumier M. [3] estimated the rate of convergence of Fourier Legendre series of functions of bounded variation on the interval [0,1], Cheng F. [4] estimated the rate of convergence of Bernstein polynomials of functions bounded variation on the interval[0,1], Zeng and Chen [9] estimated the rate of convergence of Durrmeyer type operators for functions of bounded variation on the interval[0,1].

Durrmeyer operatorsM_nintroduced by Durrmeyer [1]. Also let us note that these operators were introduced by Lupa¸s [2]. The polynomialM_nfdefined by

M_n(f;x) = (n+ 1)

n

X

k=0

P_n,k(x) Z 1

0

f(t)P_n,k(t)dt, 0≤x≤1, where

P_n,k(x) = n

k

(x)^k(1−x)^n−k.

These operators are the integral modification of Bernstein polynomials so as to approximate Lebesgue integrable functions on the interval [0,1].The operators M_n were studied by several authors. Also, Guo S. [5] investigated Dur- rmeyer operatorsMnand estimated the rate of convergence of operatorsMnfor functions of bounded variation on the interval[0,1].

Chlodowsky polynomials are given [6] by C_n(f;x) =

n

X

k=0

f k

nb_n n k

x b_n

k 1− x

b_n ^n−k

, 0≤x≤b_n,

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where (b_n) is a positive increasing sequence with the properties lim

n→∞b_n = ∞ and lim

n→∞

bn

n = 0.

Works on Chlodowsky operators are fewer, since they are defined on an un- bounded interval[0,∞).

This paper generalizes Chlodowsky polynomials by incorporating Durrmeyer operators, hence the name Chlodowsky-Durrmeyer operators:Dn:BV[0,∞)→P,

D_n(f;x) = (n+ 1) b_n

n

X

k=0

P_n,k x

b_n Z bn

0

f(t)P_n,k t

b_n

dt, 0≤x≤b_n whereP :={P : [0,∞)→R}, is a polynomial functions set,(bn)is a positive increasing sequence with the properties,

n→∞lim b_n =∞ and lim

n→∞

b_n n = 0 and

P_n,k(x) = n

k

(x)^k(1−x)^n−k is the Bernstein basis.

In this paper, by means of the techniques of probability theory, we shall estimate the rate of convergence of operators D_n, for functions of bounded variation in terms of the Chanturiya’s modulus of variation. At the points which one sided limit exist, we shall prove that operators Dn converge to the limit ¹₂[f(x+) +f(x−)] on the interval [0, b_n], (n → ∞) extending infinity, for functions of bounded variation on the interval[0,∞).

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For the sake of brevity, let the auxiliary functiong_xbe defined by

g_x(t) =











f(t)−f(x+), x < t≤b_n;

0, t=x;

f(t)−f(x+), 0≤t < x.

The main theorem of this paper is as follows.

Theorem 1.1. Let f be a function of bounded variation on every finite subin- terval of[0,∞).Then for everyx∈(0,∞),andnsufficiently large, we have, (1.1)

D_n(f;x)−1

2(f(x+) +f(x−))

≤ 3A_n(x)b²_n x²(b_n−x)²







n

X

k=1 x+^bn^√^−x

k

_

x−^√^x

k

(g_x)







+ 2

qnx

bn(1−_b^x

n)

|{f(x+)−f(x−)}|,

whereA_n(x) = h

2n x(bn−x)+2b²_n n²

i and

b

W

a

(g_x)is the total variation ofg_xon[a, b].

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2. Auxiliary Results

In this section we give certain results, which are necessary to prove our main theorem.

Lemma 2.1. Ifs∈Nands≤n,then D_n(t^s;x) = (n+ 1)!b^s_n

(n+s+ 1)!

s

X

r=0

s r

s!

r! · n!

(n−r)!(xb_n)^r. Proof.

D_n(t^s;x) = n+ 1 b_n

n

X

k=0

P_n,k x

b_n

Z bn

0

P_n,k t

b_n

t^sdt

= n+ 1 b_n

n

X

k=0

P_n,k x

b_n "

Z bn

0

n k

t b_n

k 1− t

b_n n−k

t^sdt

#

= n+ 1 b_n

n

X

k=0

P_n,k x

b_n

b^s+1_n n

k Z 1

0

(u)^k+s(1−u)^n−kdu, set u= t b_n

= n+ 1 b_n

n

X

k=0

P_n,k x

b_n

b^s+1_n (k+s)!

k! · n!

(n+s+ 1)!. Thus

Dn(t^s;x) = (n+ 1)!b^s_n (n+s+ 1)!

n

X

k=0

Pn,k

x b_n

(k+s)!

k! .

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Fors≤n,we have

∂^s

∂x^s x

b_n s

x+y b_n

n

= 1 b^s_n

n

X

k=0

n k

x b_n

k y b_n

n−k

(k+s)!

k!

and from the Leibnitz formula

∂^s

∂x^s x

b_n s

x+y b_n

n

=

s

X

r=0

s r

s!

r! · n!

(n−r)!(xbn)^r

x+y b_n

n−r

1 b^s_n

= 1 b^s_n

s

X

r=0

s r

s!

r! · n!

(n−r)!(xb_n)^r

x+y b_n

n−r

Letx+y =b_n,we have

n

X

k=0

n k

x b_n

k y b_n

n−k

(k+s)!

k! =

s

X

r=0

s r

s!

r!· n!

(n−r)!(xb_n)^r

x+y b_n

n−r

Thus the proof is complete.

By the Lemma2.1, we get D_n(1;x) = 1 (2.1)

D_n(t;x) = x+bn−2x n+ 2

D_n(t²;x) = x²+[4nbn−6(n+ 1)x]

(n+ 2)(n+ 3) x+ 2b²_n (n+ 2)(n+ 3).

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By direct computation, we get

Dn((t−x)²;x) = 2(n−3)(b_n−x)x

(n+ 2)(n+ 3) + 2b²_n (n+ 2)(n+ 3) and hence,

(2.2) D_n((t−x)²;x)≤ 2nx(b_n−x) + 2b²_n

n² .

Lemma 2.2. For allx∈(0,∞),we have λ_n

x b_n, t

b_n

= Z t

0

K_n x

b_n, u b_n

du

≤ 1

(x−t)² · 2nx(bn−x) + 2b²_n

n² ,

(2.3) where

K_n x

b_n, u b_n

= n+ 1 b_n

n

X

k=0

P_n,k x

b_n

P_n,k u

b_n

. Proof.

λ_n x

bn

, t bn

= Z t

0

K_n x

bn

, u bn

du

≤ Z t

0

K_n x

bn

, u bn

x−u x−t

2

du

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= 1

(x−t)² Z t

0

K_n x

b_n, u b_n

(x−u)²du

= 1

(x−t)²D_n((u−x)²;x) By the (2.2), we have,

λ_n x

b_n, t b_n

≤ 1

(x−t)² ·2(n−3)(b_n−x)x

(n+ 2)(n+ 3) + 2b²_n (n+ 2)(n+ 3)

≤ 1

(x−t)² ·2nx(b_n−x) + 2b²_n

n² .

Set

(2.4) J_n,j^α x

b_n

=

n

X

k=j

Pn,k

x b_n

!α

,

J_n,n+1^α x

b_n

= 0

,

whereα≥1.

Lemma 2.3. For allx∈(0,1)andj = 0,1,2, . . . , n,we have J_n,j^α (x)−J_n+1,j+1^α (x)

≤ 2α pnx(1−x) and

J_n,j^α (x)−J_n+1,j^α (x)

≤ 2α pnx(1−x).

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Proof. The proof of this lemma is given in [9].

Forα= 1,replacing the variablexwith _b^x

n in Lemma2.3we get the following lemma:

Lemma 2.4. For allx∈(0, b_n)andj = 0,1,2, . . . , n,we have

J_n,j x

bn

−J_n+1,j+1 x

bn

≤ 2

r n_b^x

n

1−_b^x

n

and (2.5)

J_n,j x

bn

−J_n+1,j x

bn

≤ 2

r n_b^x

n

1− _b^x

n

.

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3. Proof Of The Main Result

Now, we can prove the Theorem1.1.

Proof. For any f ∈ BV[0,∞),we can decompose f into four parts on[0, b_n] for sufficiently largen,

(3.1) f(t) = 1

2(f(x+) +f(x−))

+gx(t) + f(x+)−f(x−)

2 sgn(t−x) +δ_x(t)

f(x)− 1

2(f(x+) +f(x−))

where

(3.2) δ_x(t) =

( 1, x=t 0, x6=t.

If we applying the operatorD_nthe both side of equality (3.1), we have D_n(f;x) = 1

2(f(x+) +f(x−))D_n(1;x) +D_n(g_x;x) + f(x+)−f(x−)

2 D_n(sgn(t−x);x) +

f(x)− 1

2(f(x+) +f(x−))

Dn(δx;x).

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Hence, since (2.1)D_n(1;x) = 1,we get,

D_n(f;x)− 1

2(f(x+) +f(x−))

≤ |D_n(g_x;x)|+

f(x+)−f(x−) 2

|D_n(sgn(t−x);x)|

+

f(x)− 1

2(f(x+) +f(x−))

|D_n(δ_x;x)|. For operatorsD_n, using (3.2) we can see thatD_n(δ_x;x) = 0.

Hence we have

D_n(f;x)− 1

2(f(x+) +f(x−))

≤ |D_n(g_x;x)|+

f(x+)−f(x−) 2

|D_n(sgn(t−x);x)|

In order to prove above inequality, we need the estimates forD_n(g_x;x)and D_n(sgn(t−x);x).

We first estimate|D_n(g_x;x)|as follows:

|Dn(gx;x)|=

n+ 1 b_n

n

X

k=0

Pn,k

x b_n

Z bn

0

Pn,k

t b_n

gx(t)dt

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=

n+ 1 bn

n

X

k=0

P_n,k x

bn

"

Z x−^√^x

n

0

+

Z x+^bn^√^−x_n x−^√^x

n

+ Z bn

x+^bn−x^√_n

! P_n,k

t bn

g_x(t)dt

#

≤

n+ 1 b_n

n

X

k=0

P_n,k x

b_n

Z x−^√^x

n

0

P_n,k t

b_n

g_x(t)dt +

n+ 1 b_n

n

X

k=0

P_n,k x

b_n

Z x+^bn^√^−x_n x−^√^x

n

P_n,k t

b_n

g_x(t)dt +

n+ 1 bn

n

X

k=0

P_n,k x

bn

Z bn

x+^bn−x^√_n

P_n,k t

bn

g_x(t)dt

=|I₁(n, x)|+|I₂(n, x)|+|I₃(n, x)|

We shall evaluateI1(n, x), I2(n, x)andI3(n, x).To do this we first observe thatI₁(n, x), I₂(n, x)andI₃(n, x)can be written as Lebesque-Stieltjes integral,

|I1(n, x)|=

Z x−^√^x

n

0

gx(t)dt

λn

x b_n, t

b_n

|I₂(n, x)|=

Z x+^bn^√^−x_n x−^√^x

n

g_x(t)d_t

λ_n x

b_n, t b_n

|I₃(n, x)|=

Z bn

x+^bn−x^√_n

g_x(t)d_t

λ_n x

bn

, t bn

, where

λ_n x

b_n, t b_n

= Z t

0

K_n x

b_n, u b_n

du

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and

K_n x

b_n, t b_n

= n+ 1 b_n

n

X

k=0

P_n,k x

b_n

P_n,k t

b_n

.

First we estimateI₂(n, x).Fort∈h

x− ^√^x_n, x+ ^bⁿ^√^−x_n i

,we have

|I2(n, x)|=

Z x+^bn^√^−x_n x−^√^x

n

(gx(t)−gx(x))dt

λn

x b_n, t

b_n

≤

Z x+^bn^√^−x_n x−^√^x

n

|g_x(t)−g_x(x)|

d_t

λ_n x

bn

, t bn

≤

x+^bn^√^−x_n

_

x−^√^x

n

(g_x)≤ 1 n−1

n

X

k=2 x+^bn^√^−x_n

_

x−^√^x

n

(g_x).

(3.3)

Next, we estimateI₁(n, x).Using partial Lebesque-Stieltjes integration, we obtain

I₁(n, x) =

Z x−^√^x

n

0

g_x(t)d_t

λ_n x

b_n, t b_n

=g_x

x− x

√n

λ_n x

b_n,x−^√^x_n b_n

−gx(0)λn

x b_n,0

−

Z x−^√^x

n

0

λn

x b_n, t

b_n

dt(gx(t)).

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Since

gx

x− x

√n

=

gx

x− x

√n

−gx(x)

≤

x

_

x−^√^x

n

(gx),

it follows that

|I₁(n, x)| ≤

x

_

x−^√^x

n

(g_x)

λ_n x

bn

,x−^√^x_n bn

+ Z x−^√^x

n

0

λ_n x

bn

, t bn

d_t −

x

_

t

(g_x)

! .

From (2.3), it is clear that λ_n

x bn

,x− ^√^x_n bn

≤ 1 √x

n

2

2n x(b_n−x) + 2b²_n n²

.

It follows that

|I₁(n, x)| ≤

x

_

x−^√^x

n

(g_x) 1 √x

n

2

2nx(b_n−x) + 2b²_n n²

+

Z x−^√^x

n

0

1 (x−t)²

2nx(b_n−x) + 2b²_n n²

d_t −

x

_

t

(g_x)

!

=

x

_

x−^√^x

n

(g_x)A_n(x) √x

n

2 +A_n(x)

Z x−^√^x

n

0

1

(x−t)²d_t −

x

_

t

(g_x)

! .

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Furthermore, since Z x−^√^x

n

0

1

(x−t)²dt −

x

_

t

(g_x)

!

=− 1

(x−t)²

x

_

t

(g_x)

x−^√^x

n

0

+

Z x−^√^x

n

0

2 (x−t)³

x

_

t

(g_x)dt

=− 1 (^√^x_n)²

x

_

x−^√^x

n

(gx) + 1 x²

x

_

0

(gx) +

Z x−^√^x

n

0

2 (x−t)³

x

_

t

(gx)dt.

Puttingt=x− ^√^x_u in the last integral, we get Z x−^√^x

n

0

2 (x−t)³

x

_

t

(g_x)dt= 1 x²

Z n 1

x

_

x−^√^x

u

(g_x)du= 1 x²

n

X

k=1 x

_

x−^√^x

k

(g_x).

Consequently,

|I1(n, x)| ≤

x

_

x−^√^x

n

(gx)A_n(x) (^√^x_n)²

+A_n(x)







− 1 (^√^x_n)²

x

_

x−^√^x

n

(g_x) + 1 x²

x

_

0

(g_x) + 1 x²

n

X

k=1 x

_

x−^√^x

k

(g_x)







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=A_n(x)





 1 x²

x

_

0

(g_x) + 1 x²

n

X

k=1 x

_

x−^√^x

k

(g_x)







= A_n(x) x²







bn

_

0

(g_x) +

n

X

k=1 x

_

x−^√^x

k

(g_x)





 . (3.4)

Using the similar method for estimating|I₃(n, x)|,we get

|I₃(n, x)| ≤ A_n(x) (b_n−x)²







bn

_

x

(g_x) +

n

X

k=1

x+^bn−x^√

k

_

x

(g_x)







≤ An(x) (b_n−x)²







bn

_

0

(g_x) +

n

X

k=1

x+^bn^√^−x

k

_

x−^√^x

k

(g_x)





 . (3.5)

Hence from (3.3), (3.4) and (3.5), it follows that

|D_n(g_x;x)| ≤ |I₁(n, x)|+|I₂(n, x)|+|I₃(n, x)|

≤ A_n(x) x²







bn

_

0

(g_x) +

n

X

k=1 x

_

x−^√^x

k

(g_x)







+ A_n(x) (b_n−x)²







bn

_

0

(g_x) +

n

X

k=1 x+^bn^√^−x

k

_

x−^√^x

k

(g_x)







+ 1

n−1

n

X

k=2

x+^bn^√^−x_n

_

x−^√^x

k

(g_x).

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Obviously,

1

x² + 1

(b_n−x)² = b²_n x²(b_n−x)², for _b^x

n ∈[0,1]and

x

_

x−^√^x

k

(gx)≤

x+^bn−x^√

k

_

x−^√^x

k

(gx).

Hence,

|D_n(g_x;x)| ≤

A_n(x)

x² + A_n(x) (bn−x)²







bn

_

0

(g_x) +

n

X

k=1 x+^bn−x^√

k

_

x−^√^x

k

(g_x)







+ 1

n−1

n

X

k=2 x+^bn^√^−x

k

_

x−^√^x

k

(g_x)

= A_n(x)b²_n x²(b_n−x)²







bn

_

0

(gx) +

n

X

k=1 x+^bn−x^√

k

_

x−^√^x

k

(gx)







+ 1

n−1

n

X

k=2

x+^bn−x^√

k

_

x−^√^x

k

(gx)

= A_n(x)b²_n x²(b_n−x)²







bn

_

0

(g_x) +

n

X

k=1 x+^bn^√^−x

k

_

x−^√^x

k

(g_x)







+ 1

n−1

n

X

k=2

x+^bn^√^−x

k

_

x−^√^x

k

(g_x).

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On the other hand, note that

bn

_

0

(g_x)≤

n

X

k=1

x+^bn−x^√

k

_

x−^√^x

k

(g_x).

By(2.3), we have

|D_n(g_x;x)| ≤ 2A_n(x)b²_n x²(b_n−x)²







n

X

k=1 x+^bn^√^−x

k

_

x−^√^x

k

(g_x)







+ 1

n−1

n

X

k=2 x+^bn^√^−x

k

_

x−^√^x

k

(g_x).

Note that _n−1¹ ≤ _x^A2(bⁿ^(x)n−x)^b²ⁿ², forn >1, _b^x

n ∈[0,1].Consequently (3.6) |D_n(g_x;x)| ≤ 3A_n(x)b²_n

x²(b_n−x)²







n

X

k=1

x+^bn^√^−x

k

_

x−^√^x

k

(g_x)





 .

Now secondly, we can estimateD_n(sgn(t−x);x).If we apply operatorD_n to the signum function, we get

D_n(sgn(t−x);x)

= n+ 1 bn

n

X

k=0

P_n,k x

bn

Z bn

x

P_n,k t

bn

dt−

Z x 0

P_n,k t

bn

dt

= n+ 1 b_n

n

X

k=0

Pn,k

x b_n

Z bn

0

Pn,k

t b_n

dt−2

Z x 0

Pn,k

t b_n

dt

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using (2.1), we have

(3.7) D_n(sgn(t−x);x) = 1−2n+ 1 bn

n

X

k=0

P_n,k x

bn

Z x 0

P_n,k t

bn

dt.

Now, we differentiate both side of the following equality J_n+1,k+1

x bn

=

n+1

X

j=k+1

P_n+1,j x

bn

.

Fork = 0,1,2, . . . , nwe get, d

dxJ_n+1,k+1 x

b_n

= d dx

n+1

X

j=k+1

P_n+1,j x

b_n

= d

dxP_n+1,k+1 x

bn

+ d

dxP_n+1,k+2 x

bn

+· · ·+ d

dxP_n+1,n+1 x

bn

d

dxJ_n+1,k+1 x

b_n

= (n+ 1) b_n

P_n,k

x b_n

−P_n,k+1 x

b_n

+

P_n,k+1 x

b_n

−P_n,k+2 x

b_n

+· · ·+

Pn,n−1

x b_n

−P_n,n x

b_n

+

P_n,n x

b_n

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= (n+ 1) b_n

n+1

X

j=k+1

Pn,j−1

x b_n

−P_n,j x

b_n

= (n+ 1) b_n P_n,k

x b_n

andJn+1,k+1(0) = 0.Taking the integral from zero tox, we have (n+ 1)

b_n Z x

0

Pn,k

t b_n

dt=Jn+1,k+1

x b_n

and therefore from (2.4) J_n+1,k+1

x b_n

=

n+1

X

j=k+1

P_n+1,j x

b_n

=

n+1

X

j=0

P_n+1,j x

b_n

−

k

X

j=0

P_n+1,j x

b_n

= 1−

k

X

j=0

P_n+1,j x

b_n

.

Hence

(n+ 1) b_n

Z x 0

P_n,k t

b_n

dt= 1−

k

X

j=0

P_n+1,j x

b_n

.

From (3.7), we get D_n(sgn(t−x);x) = 1−2

n

X

k=0

P_n,k x

b_n "

1−

k

X

j=0

P_n+1,j x

b_n #

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= 1−2

n

X

k=0

P_n,k x

b_n

+ 2

n

X

k=0

P_n,k x

b_n ^k

X

j=0

P_n+1,j x

b_n

=−1 + 2

n

X

k=0

P_n,k x

b_n k

X

j=0

P_n+1,j x

b_n

.

Set

Q⁽²⁾_n+1,j x

b_n

=J_n+1,j² x

b_n

−J_n+1,j+1² x

b_n

. Also note that

n

X

k=0 k

X

j=0

∗=

n

X

j=0 n

X

k=j

∗,

n+1

X

k=j

Q⁽²⁾_n+1,k x

b_n

=J_n+1,j² x

b_n

and J_n,n+1 x

b_n

= 0, we have

D_n(sgn(t−x);x) =−1 + 2

n

X

j=0

P_n+1,j x

b_n n

X

k=j

P_n,k x

b_n

=−1 + 2

n

X

j=0

P_n+1,j x

b_n

J_n,j x

b_n

=−1 + 2

n+1

X

j=0

P_n+1,j x

b_n

J_n,j x

b_n

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= 2

n+1

X

j=0

P_n+1,j x

b_n

J_n,j x

b_n

−1.

Since

n+1

P

j=0

Q⁽²⁾_n+1,j

x bn

= 1, thus

D_n(sgn(t−x);x) = 2

n+1

X

j=0

P_n+1,j x

b_n

J_n,j x

b_n

−

n+1

X

j=0

Q⁽²⁾_n+1,j x

b_n

.

By the mean value theorem, we have Q⁽²⁾_n+1,j

x b_n

=J_n+1,j² x

b_n

−J_n+1,j+1² x

b_n

= 2P_n+1,j x

b_n

γ_n,j x

b_n

where

J_n+1,j+1 x

b_n

< γ_n,j x

b_n

< J_n+1,j x

b_n

.

Hence it follows from (2.5) that

|D_n(sgn(t−x);x)|= 2

n+1

X

j=0

P_n+1,j x

b_n J_n,j x

b_n

−γ_n,j x

b_n

≤2

n+1

X

j=0

P_n+1,j x

b_n

J_n,j x

b_n

−γ_n,j x

b_n

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J_n,j x

b_n

−J_n+1,j+1 x

b_n x b_n

=

J_n,j x

b_n

−γ_n,j x

b_n

+γ_n,j x

b_n

−J_n+1,j+1 x

b_n

,

sinceγ_n,j

x bn

−J_n+1,j+1

x bn

>0,then we have

J_n,j x

b_n

−γ_n,j x

b_n

≤

J_n,j x

b_n

−J_n+1,j+1 x

b_n

.

Hence

|D_n(sgn(t−x);x)| ≤2

n+1

X

j=0

P_n+1,j x

b_n

J_n,j x

b_n

−J_n+1,j+1 x

b_n

≤2

n+1

X

j=0

Pn+1,j

x b_n

2 qn_b^x

n(1− _b^x

n)

= 4

q n_b^x

n(1− _b^x

n) . (3.8)

Combining (3.6) and (3.8) we get (1.1). Thus, the proof of the theorem is completed.

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References

[1] J.L. DURRMEYER, Une formule d’inversion de la ransformee de Laplace: Applicationsa la theorie de moments , these de 3e cycle, Fac- ulte des sciences de 1’universite de Paris,1971.

[2] A. LUPA ¸S, Die Folge Der Betaoperatoren, Dissertation, Stuttgart Univer- sität, 1972.

[3] R. BOJANIC AND M. VUILLEUMIER, On the rate of convergence of Fourier Legendre series of functions of bounded variation, J. Approx. The- ory, 31 (1981), 67–79.

[4] F. CHENG, On the rate of convergence of Bernstein polynomials of func- tions of bounded variation, J. Approx. Theory, 39 (1983), 259–274.

[5] S. GUO, On the rate of convergence of Durrmeyer operator for functions of bounded variation, J. Approx. Theory, 51 (1987), 183–197.

[6] I. CHLODOWSKY, Sur le d˘eveloppment des fonctions d˘efines dans un interval infinien s˘eries de polyn˘omes de S.N. Bernstein, Compositio Math., 4 (1937), 380–392.

[7] XIAO-MING ZENG, Bounds for Bernstein basis functions and Meyer- König-Zeller basis functions, J. Math. Anal. Appl., 219 (1998), 364–376.

[8] XIAO-MING ZENGANDA. PIRIOU, On the rate of convergence of two Bernstein-Bezier type operators for bounded variation functions, J. Ap- prox. Theory, 95 (1998), 369–387.

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[9] XIAO-MING ZENG AND W. CHEN, On the rate of convergence of the generalized Durrmeyer type operators for functions of bounded variation, J. Approx. Theory, 102 (2000), 1–12.

[10] A.N. SHIRYAYEV, Probability, Springer-Verlag, New York, 1984.

[11] G.G. LORENTZ, Bernstein Polynomials, Univ. of Toronto Press, Toronto, 1953.

[12] Z.A. CHANTURIYA, Modulus of variation of function and its application in the theory of Fourier series, Dokl. Akad. Nauk. SSSR, 214 (1974), 63–

66.