Journal of Inequalities in Pure and
Applied Mathematics
RATE OF CONVERGENCE OF CHLODOWSKY TYPE
DURRMEYER OPERATORS
volume 6, issue 4, article 106,
2005.
ERTAN IBIKLI AND HARUN KARSLI
Ankara University, Faculty of Sciences,
Department of Mathematics,
06100 Tandogan - Ankara/Turkey.
Received 16 September, 2005;
accepted 23 September, 2005.
Communicated by: A. Lupaş
EMail: ibikli@science.ankara.edu.tr
EMail: karsli@science.ankara.edu.tr
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c 2000 Victoria University
ISSN (electronic): 1443-5756
276-05
Quit
Abstract
In the present paper, we estimate the rate of pointwise convergence of the
Chlodowsky type Durrmeyer Operators Dn (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.
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
3
Proof Of The Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
References
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Ertan Ibikli and Harun Karsli
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
1.
Introduction
Very recently, some authors studied some linear positive operators and obtained
the rate of convergence for functions of bounded variation. For example, Bojanic 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 operators Mn introduced by Durrmeyer [1]. Also let us note that
these operators were introduced by Lupaş [2]. The polynomial Mn f defined by
Z 1
n
X
Mn (f ; x) = (n + 1)
Pn,k (x)
f (t)Pn,k (t) dt,
0 ≤ x ≤ 1,
k=0
where
0
n
Pn,k (x) =
(x)k (1 − x)n−k .
k
These operators are the integral modification of Bernstein polynomials so as
to approximate Lebesgue integrable functions on the interval [0, 1]. The operators Mn were studied by several authors. Also, Guo S. [5] investigated Durrmeyer operators Mn and estimated the rate of convergence of operators Mn for
functions of bounded variation on the interval [0, 1].
Chlodowsky polynomials are given [6] by
k n−k
n
X
k
n
x
x
Cn (f ; x) =
f
bn
1−
, 0 ≤ x ≤ bn ,
n
k
bn
bn
k=0
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Ertan Ibikli and Harun Karsli
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
where (bn ) is a positive increasing sequence with the properties lim bn = ∞
n→∞
bn
n→∞ n
and lim
= 0.
Works on Chlodowsky operators are fewer, since they are defined on an unbounded interval [0, ∞).
This paper generalizes Chlodowsky polynomials by incorporating Durrmeyer
operators, hence the name Chlodowsky-Durrmeyer operators: Dn :BV [0,∞)→P,
n
(n + 1) X
Dn (f ; x) =
Pn,k
bn k=0
x
bn
Z
bn
f (t)Pn,k
0
t
bn
dt,
0 ≤ x ≤ bn
where P := {P : [0, ∞) → R}, is a polynomial functions set, (bn ) is a positive
increasing sequence with the properties,
lim bn = ∞
n→∞
and
and
bn
=0
n→∞ n
lim
n
Pn,k (x) =
(x)k (1 − x)n−k
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 Dn , 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, bn ], (n → ∞) extending infinity,
for functions of bounded variation on the interval [0, ∞).
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Ertan Ibikli and Harun Karsli
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
For the sake of brevity, let the auxiliary function gx be defined by


f (t) − f (x+), x < t ≤ bn ;



0,
t = x;
gx (t) =



 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 subinterval of [0, ∞). Then for every x ∈ (0, ∞) , and n sufficiently large, we have,
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Ertan Ibikli and Harun Karsli
(1.1)
1
Dn (f ; x) − (f (x+) + f (x−))
2


x+ bn√−x


n
k


X _
3An (x)b2n
≤ 2
(gx )

x (bn − x)2 
 k=1 x− √x

k
+q
2
nx
(1
bn
−
x
)
bn
|{f (x+) − f (x−)}| ,
Title Page
Contents
JJ
J
II
I
Go Back
Close
where An (x) =
h
2 n x(bn −x)+2b2n
n2
i
and
b
W
(gx ) is the total variation of gx on [a, b].
Quit
a
Page 5 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
2.
Auxiliary Results
In this section we give certain results, which are necessary to prove our main
theorem.
Lemma 2.1. If s ∈ N and s ≤ n, then
s (n + 1)!bsn X s s!
n!
Dn (t ; x) =
·
(xbn )r .
(n + s + 1)! r=0 r r! (n − r)!
s
Proof.
n
x
bn
Z
bn
t
s
Pn,k
t dt
bn
0
#
"Z bn k n−k
n
x
n
t
t
n+1X
ts dt
=
Pn,k
1−
bn k=0
bn
k
b
b
n
n
0
Z 1
n
n+1X
x
t
s+1 n
=
Pn,k
bn
(u)k+s (1 − u)n−k du, set u =
bn k=0
bn
k
bn
0
n
x
(k + s)!
n!
n+1X
=
Pn,k
bs+1
·
.
n
bn k=0
bn
k!
(n + s + 1)!
n+1X
Dn (t ; x) =
Pn,k
bn k=0
s
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Ertan Ibikli and Harun Karsli
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Thus
n
(n + 1)!bsn X
Dn (t ; x) =
Pn,k
(n + s + 1)! k=0
s
x
bn
(k + s)!
.
k!
Page 6 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
For s ≤ n, we have
s n k n−k
n x
x+y
1 X n
x
y
(k + s)!
∂s
= s
s
∂x
bn
bn
bn k=0 k
bn
bn
k!
and from the Leibnitz formula
s n X
n−r
s 1
∂s
x
x+y
s s!
n!
x+y
r
=
·
(xbn )
s
∂x
bn
bn
r r! (n − r)!
bn
bsn
r=0
n−r
s 1 X s s!
n!
x+y
r
= s
·
(xbn )
bn r=0 r r! (n − r)!
bn
Let x + y = bn , we have
k n−k
n−r
s n X
x
y
(k + s)! X s s!
n
n!
x+y
r
=
·
(xbn )
k
b
b
k!
r
r!
(n
−
r)!
bn
n
n
r=0
k=0
Thus the proof is complete.
By the Lemma 2.1, we get
(2.1)
Dn (1; x) = 1
bn − 2x
n+2
[4nb
2b2n
n − 6(n + 1)x]
Dn (t2 ; x) = x2 +
x+
.
(n + 2)(n + 3)
(n + 2)(n + 3)
Dn (t; x) = x +
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Ertan Ibikli and Harun Karsli
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 7 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
By direct computation, we get
2b2n
2(n − 3)(bn − x)x
+
(n + 2)(n + 3)
(n + 2)(n + 3)
Dn ((t − x)2 ; x) =
and hence,
Dn ((t − x)2 ; x) ≤
(2.2)
2nx(bn − x) + 2b2n
.
n2
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Lemma 2.2. For all x ∈ (0, ∞) , we have
Z t
x t
x u
λn
,
=
Kn
,
du
bn bn
bn bn
0
2nx(bn − x) + 2b2n
1
(2.3)
·
,
≤
(x − t)2
n2
where
Kn
x u
,
bn bn
n
n+1X
Pn,k
=
bn k=0
x
bn
Pn,k
u
bn
Ertan Ibikli and Harun Karsli
Title Page
Contents
.
JJ
J
II
I
Go Back
Proof.
Close
λn
x t
,
bn bn
Z
=
t
Kn
x u
,
bn bn
du
2
Z t
x u
x−u
≤
Kn
,
du
bn bn
x−t
0
0
Quit
Page 8 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
Z t
1
x u
=
Kn
,
(x − u)2 du
(x − t)2 0
bn bn
1
=
Dn ((u − x)2 ; x)
(x − t)2
By the (2.2), we have,
x t
1
λn
,
≤
bn bn
(x − t)2
1
≤
(x − t)2
2(n − 3)(bn − x)x
2b2n
+
(n + 2)(n + 3)
(n + 2)(n + 3)
2
2nx(bn − x) + 2bn
·
.
n2
·
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Ertan Ibikli and Harun Karsli
Set
(2.4)
Title Page
α
Jn,j
x
bn
=
n
X
k=j
Pn,k
x
bn
!α
,
α
Jn,n+1
x
bn
where α ≥ 1.
Lemma 2.3. For all x ∈ (0, 1) and j = 0, 1, 2, . . . , n, we have
=0 ,
Contents
JJ
J
II
I
Go Back
Close
2α
α
α
Jn,j
(x) − Jn+1,j+1
(x) ≤ p
nx(1 − x)
Quit
Page 9 of 26
and
α
α
Jn,j
(x) − Jn+1,j
(x) ≤ p
2α
nx(1 − x)
.
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
Proof. The proof of this lemma is given in [9].
For α = 1, replacing the variable x with
ing lemma:
x
bn
in Lemma 2.3 we get the follow-
Lemma 2.4. For all x ∈ (0, bn ) and j = 0, 1, 2, . . . , n, we have
2
x
x
≤r
Jn,j
− Jn+1,j+1
bn
bn
x
x
n bn 1 − bn
and
(2.5)
Jn,j
x
bn
− Jn+1,j
x
bn
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Ertan Ibikli and Harun Karsli
≤r
n bxn
2
1−
x
bn
.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 10 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
3.
Proof Of The Main Result
Now, we can prove the Theorem 1.1.
Proof. For any f ∈ BV [0, ∞), we can decompose f into four parts on [0, bn ]
for sufficiently large n,
(3.1) f (t) =
1
(f (x+) + f (x−))
2
f (x+) − f (x−)
+ gx (t) +
sgn(t − x)
2
1
+ δx (t) f (x) − (f (x+) + f (x−))
2
where
Ertan Ibikli and Harun Karsli
Title Page
(
(3.2)
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
δx (t) =
1, x = t
0, x 6= t.
If we applying the operator Dn the both side of equality (3.1), we have
Contents
JJ
J
II
I
Go Back
Dn (f ; x) =
1
(f (x+) + f (x−)) Dn (1; x) + Dn (gx ; x)
2
f (x+) − f (x−)
+
Dn (sgn(t − x); x)
2 1
+ f (x) − (f (x+) + f (x−)) Dn (δx ; x).
2
Close
Quit
Page 11 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
Hence, since (2.1) Dn (1; x) = 1, we get,
1
(f (x+) + f (x−))
2
f (x+) − f (x−)
≤ |Dn (gx ; x)| +
|Dn (sgn(t − x); x)|
2
1
+ f (x) − (f (x+) + f (x−)) |Dn (δx ; x)| .
2
Dn (f ; x) −
For operators Dn , using (3.2) we can see that Dn (δx ; x) = 0.
Hence we have
Dn (f ; x) −
1
(f (x+) + f (x−))
2
≤ |Dn (gx ; x)| +
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Ertan Ibikli and Harun Karsli
Title Page
f (x+) − f (x−)
|Dn (sgn(t − x); x)|
2
In order to prove above inequality, we need the estimates for Dn (gx ; x) and
Dn (sgn(t − x); x).
We first estimate |Dn (gx ; x)| as follows:
Contents
JJ
J
II
I
Go Back
Close
|Dn (gx ; x)| =
n+1
bn
n
X
k=0
Pn,k
x
bn
Z
bn
Pn,k
0
t
bn
gx (t)dt
Quit
Page 12 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
n
n+1X
=
Pn,k
bn k=0
n
n+1X
≤
Pn,k
bn k=0
x
bn
" Z
x
bn
Z
n
n+1X
Pn,k
+
bn k=0
n
n+1X
Pn,k
+
bn k=0
x− √xn
Z
√−x
x+ bn
n
+
x− √xn
Pn,k
0
x
bn
Z
x
bn
Z
bn
+
√−x
x+ bn
n
x− √xn
0
Z
t
bn
!
Pn,k
t
bn
#
gx (t)dt
√−x
x+ bn
n
gx (t)dt
t
Pn,k
gx (t)dt
bn
t
Pn,k
gx (t)dt
bn
x− √xn
bn
√−x
x+ bn
n
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Ertan Ibikli and Harun Karsli
= |I1 (n, x)| + |I2 (n, x)| + |I3 (n, x)|
We shall evaluate I1 (n, x), I2 (n, x) and I3 (n, x). To do this we first observe
that I1 (n, x), I2 (n, x) and I3 (n, x) can be written as Lebesque-Stieltjes integral,
Z x− √x
n
x t
|I1 (n, x)| =
gx (t)dt λn
,
bn bn
0
Z x+ bn√−x
n
x t
|I2 (n, x)| =
gx (t)dt λn
,
bn bn
x− √xn
Z bn
x t
|I3 (n, x)| =
gx (t)dt λn
,
,
bn bn
√−x
x+ bn
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
n
Page 13 of 26
where
λn
x t
,
bn bn
Z
=
t
Kn
0
x u
,
bn bn
du
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
and
n
x
t
n+1X
Kn
=
Pn,k
Pn,k
.
bn k=0
bn
bn
i
h
bn
−x
x
√
√
First we estimate I2 (n, x). For t ∈ x − n , x + n , we have
x t
,
bn bn
Z
|I2 (n, x)| =
√−x
x+ bn
n
x− √xn
Z
≤
√−x
x+ bn
n
x− √xn
√−x
x+ bn
n
(3.3)
≤
_
x− √xn
x t
(gx (t) − gx (x)) dt λn
,
bn bn
x t
|gx (t) − gx (x)| dt λn
,
bn bn
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Ertan Ibikli and Harun Karsli
√−x
x+ bn
n
n
1 X _
(gx ).
(gx ) ≤
n − 1 k=2
√x
x−
n
Next, we estimate I1 (n, x). Using partial Lebesque-Stieltjes integration, we
obtain
Z x− √x
n
x t
I1 (n, x) =
gx (t)dt λn
,
bn bn
0
x x
x x − √n
= gx x − √
λn
,
bn
bn
n
Z x− √x
n
x
x t
− gx (0)λn
,0 −
λn
,
dt (gx (t)) .
bn
bn bn
0
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
Since
gx
x
x− √
n
= gx
x
x− √
n
− gx (x) ≤
x
_
(gx ),
x− √xn
it follows that
|I1 (n, x)| ≤
x
_
(gx ) λn
x− √xn
x
x x − √n
,
bn
bn
Z
+
x− √xn
λn
0
x t
,
bn bn
dt −
x
_
!
(gx ) .
t
From (2.3), it is clear that
x x x − √n
1
2 n x(bn − x) + 2b2n
≤ 2
.
λn
,
bn
n2
bn
√x
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Ertan Ibikli and Harun Karsli
Title Page
n
Contents
It follows that
|I1 (n, x)| ≤
x
_
x− √xn
(gx ) 1
√x
n
x− √xn
2
2nx(bn − x) +
n2
2b2n
JJ
J
Go Back
!
x
_
2nx(bn − x) + 2bn
+
dt − (gx )
n2
0
t
!
x
Z
x
x
√
x−
_
_
n
An (x)
1
=
(gx ) 2 + An (x)
dt − (gx ) .
(x − t)2
0
√x
√x
t
Z
x−
n
1
(x − t)2
n
II
I
2
Close
Quit
Page 15 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
Furthermore, since
x− √xn
Z
0
!
x
_
1
dt − (gx )
(x − t)2
t
=−
1
( √xn )2
x−
Putting t = x −
Z
x− √xn
0
√x
u
x− √xn
x− √xn
x
_
2
(gx )
+
(gx )dt
(x − t)3 t
0
t
0
Z x− √x
x
x
x
_
_
n
2
1 _
(gx )dt.
(gx ) + 2 (gx ) +
3
(x
−
t)
x
0
x
√
t
0
1
=−
(x − t)2
x
_
Z
n
Ertan Ibikli and Harun Karsli
in the last integral, we get
Z
n
x
x
x
_
1 n _
1 X _
2
(gx ).
(gx )dt = 2
(gx )du = 2
x k=1 √x
(x − t)3 t
x 1
√x
x−
u
x−
|I1 (n, x)| ≤
Title Page
Contents
k
JJ
J
Consequently,
x
_
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
An (x)
( √xn )2
x− √xn


x
x
n
x


_
_
X
_
1
1
1
+ An (x) − x 2
(gx ) + 2 (gx ) + 2
(gx )
 ( √n )

x 0
x k=1 √x
√x
(gx )
x−
n
x−
II
I
Go Back
Close
Quit
Page 16 of 26
k
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
(3.4)


x
n
x
1 _

_
X
1
(gx ) + 2
= An (x)
(gx )

 x2 0
x k=1 √x
x−
k


bn
n
x


_
X
_
An (x)
=
(gx ) .
(gx ) +

x2 
x
√
0
k=1 x−
k
Using the similar method for estimating |I3 (n, x)| ,we get


x+ bn√−x


b
n
k
n

X _
An (x) _
(g
)
+
(g
)
|I3 (n, x)| ≤
x
x

(bn − x)2 
x

x
k=1


x+ bn√−x


b
n
k
n


_
X _
An (x)
(3.5)
≤
(gx ) +
(gx ) .

(bn − x)2 
0

k=1 x− √x
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Ertan Ibikli and Harun Karsli
Title Page
Contents
k
Hence from (3.3), (3.4) and (3.5), it follows that
|Dn (gx ; x)| ≤ |I1 (n, x)| + |I2 (n, x)| + |I3 (n, x)|


b
n
x
n


X _
An (x) _
≤
(gx ) +
(gx )

x2  0
k=1 x− √x
k


bn
√−x
x+ bn√−x


b
n
n x+_n
k
n


_
X
_
X
An (x)
1
+
(gx ) +
(gx ) +
(gx ).
2

(bn − x) 
0
 n − 1 k=2 x− √x
k=1 x− √x
k
k
JJ
J
II
I
Go Back
Close
Quit
Page 17 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
Obviously,
1
1
b2n
+
=
,
x2 (bn − x)2
x2 (bn − x)2
for
x
bn
∈ [0, 1] and
x+ bn√−x
x
_
(gx ) ≤
x− √x
_k
(gx ).
x− √x
k
k
Hence,
|Dn (gx ; x)| ≤
An (x)
An (x)
+
2
x
(bn − x)2
+
1
n−1
n
X


bn
_


0
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators


n

X
_k
(gx ) +
(gx )


k=1 x− √x
x+ bn√−x
Ertan Ibikli and Harun Karsli
k
Title Page
x+ bn√−x
k
_
Contents
(gx )
k=2 x− √x
k


bn
_

bn
√−x

n
n x+_k

2
X
_
X
An (x) bn
1
= 2
(gx ) +
(gx ) +
(gx )
2

x (bn − x) 
0
 n − 1 k=2 x− √x
k=1 x− √x
k
k


bn
−x
b
−x
n
√
√
 bn

n x+_k
n x+ k

X
An (x) b2n _
1 X _
= 2
(gx ) +
(gx ) +
(gx ).

x (bn − x)2 
n
−
1
x
x
0

k=1 x− √
k=2 x− √
x+ bn√−x
k
k
JJ
J
II
I
Go Back
Close
Quit
Page 18 of 26
k
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
On the other hand, note that
bn
_
x+ bn√−x
(gx ) ≤
n
X
_k
(gx ).
k=1 x− √x
0
k
By (2.3), we have
|Dn (gx ; x)| ≤
Note that
(3.6)
1
n−1

bn
√−x

n x+_k
X
2 An (x) b2n
x2 (bn − x)2 
 k=1
≤
An (x) b2n
,
x2 (bn −x)2
x− √x
k

bn
√−x

n x+ k

1 X _
(gx ) +
(gx ).

 n − 1 k=2 x− √x
k
x
bn
∈ [0, 1]. Consequently


bn
√−x
x+


n
k

3 An (x) b2n X _
|Dn (gx ; x)| ≤ 2
(g
)
.
x

x (bn − x)2 
 k=1 x− √x

for n > 1,
k
Now secondly, we can estimate Dn (sgn(t − x); x). If we apply operator Dn
to the signum function, we get
Dn (sgn(t − x); x)
Z bn
Z x
n
n+1X
x
t
t
=
Pn,k
Pn,k
dt −
Pn,k
dt
bn k=0
bn
bn
bn
x
0
Z bn
Z x
n
n+1X
x
t
t
Pn,k
Pn,k
dt − 2
Pn,k
dt
=
bn k=0
bn
bn
bn
0
0
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Ertan Ibikli and Harun Karsli
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 19 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
using (2.1), we have
n
(3.7)
n+1X
Dn (sgn(t − x); x) = 1 − 2
Pn,k
bn k=0
x
bn
Z
x
Pn,k
0
t
bn
dt.
Now, we differentiate both side of the following equality
n+1
X
x
x
Jn+1,k+1
=
Pn+1,j
.
bn
b
n
j=k+1
For k = 0, 1, 2, . . . , n we get,
n+1
d
x
d X
x
Jn+1,k+1
=
Pn+1,j
dx
bn
dx j=k+1
bn
d
x
d
x
Pn+1,k+1
+ Pn+1,k+2
=
dx
bn
dx
b
n
d
x
+ · · · + Pn+1,n+1
dx
bn
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Ertan Ibikli and Harun Karsli
Title Page
Contents
JJ
J
II
I
Go Back
d
x
Jn+1,k+1
dx
b
n (n + 1)
x
x
x
x
=
Pn,k
− Pn,k+1
+ Pn,k+1
− Pn,k+2
bn
bn
b
bn
bn
n x
x
x
+ · · · + Pn,n−1
− Pn,n
+ Pn,n
bn
bn
bn
Close
Quit
Page 20 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
n+1 (n + 1) X
x
x
(n + 1)
x
=
Pn,j−1
− Pn,j
=
Pn,k
bn j=k+1
bn
bn
bn
bn
and Jn+1,k+1 (0) = 0. Taking the integral from zero to x, we have
Z
(n + 1) x
t
x
Pn,k
dt = Jn+1,k+1
bn
bn
bn
0
and therefore from (2.4)
n+1
X
x
x
Jn+1,k+1
=
Pn+1,j
bn
bn
j=k+1
X
n+1
k
X
x
x
=
Pn+1,j
−
Pn+1,j
bn
bn
j=0
j=0
k
X
x
=1−
Pn+1,j
.
b
n
j=0
Hence
(n + 1)
bn
Z
x
Pn,k
0
t
bn
dt = 1 −
k
X
Pn+1,j
j=0
x
bn
Ertan Ibikli and Harun Karsli
Title Page
Contents
JJ
J
II
I
Go Back
.
Close
Quit
From (3.7), we get
Dn (sgn(t − x); x) = 1 − 2
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
n
X
k=0
Pn,k
x
bn
"
1−
k
X
j=0
Pn+1,j
x
bn
#
Page 21 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
n
X
X
k
n
X
x
x
x
=1−2
Pn,k
Pn,k
Pn+1,j
+2
bn
bn j=0
bn
k=0
k=0
X
k
n
X
x
x
= −1 + 2
Pn,k
Pn+1,j
.
bn j=0
bn
k=0
Set
(2)
Qn+1,j
x
bn
=
2
Jn+1,j
x
bn
−
2
Jn+1,j+1
x
bn
.
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Also note that
n X
k
X
∗=
k=0 j=0
n+1
X
k=j
(2)
Qn+1,k
x
bn
n X
n
X
Ertan Ibikli and Harun Karsli
∗,
j=0 k=j
2
= Jn+1,j
x
bn
and Jn,n+1
x
bn
Title Page
= 0,
we have
Dn (sgn(t − x); x) = −1 + 2
= −1 + 2
n
X
j=0
n
X
Pn+1,j
Pn+1,j
j=0
= −1 + 2
n+1
X
j=0
x
bn
X
n
x
bn
x
bn
Pn,k
k=j
Jn,j
x
bn
x
bn
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 22 of 26
Pn+1,j
Jn,j
x
bn
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
=2
n+1
X
Pn+1,j
j=0
Since
n+1
P
j=0
(2)
Qn+1,j
x
bn
Dn (sgn(t − x); x) = 2
x
bn
Jn,j
x
bn
− 1.
= 1, thus
n+1
X
j=0
Pn+1,j
x
bn
Jn,j
x
bn
−
n+1
X
(2)
Qn+1,j
j=0
x
bn
By the mean value theorem, we have
x
x
x
(2)
2
2
Qn+1,j
= Jn+1,j
− Jn+1,j+1
bn
b
b
n n
x
x
= 2Pn+1,j
γn,j
bn
bn
where
x
x
x
Jn+1,j+1
< γn,j
< Jn+1,j
.
bn
bn
bn
Hence it follows from (2.5) that
n+1
X
x
x
x
|Dn (sgn(t − x); x)| = 2
Jn,j
− γn,j
Pn+1,j
bn
bn
bn
j=0
n+1
X
x
x
x
Jn,j
− γn,j
≤2
Pn+1,j
bn
bn
bn
j=0
.
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Ertan Ibikli and Harun Karsli
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 23 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
Jn,j
since γn,j
x
bn
− Jn+1,j+1
x
= Jn,j
bn
x
bn
− Jn+1,j+1
Jn,j
x
bn
x
b
n
x
bn
x
x
x
,
− γn,j
+ γn,j
− Jn+1,j+1
bn
bn
bn
x
bn
− γn,j
x
bn
> 0,then we have
≤ Jn,j
x
bn
− Jn+1,j+1
x
bn
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
.
Ertan Ibikli and Harun Karsli
Hence
Title Page
|Dn (sgn(t − x); x)| ≤ 2
n+1
X
Pn+1,j
j=0
≤2
n+1
X
Pn+1,j
j=0
(3.8)
=q
x
bn
x
bn
4
n bxn (1
−
x
)
bn
Jn,j
x
bn
− Jn+1,j+1
x
bn
2
q
n bxn (1 −
x
)
bn
.
Combining (3.6) and (3.8) we get (1.1). Thus, the proof of the theorem is
completed.
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 24 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
References
[1] J.L. DURRMEYER, Une formule d’inversion de la ransformee de
Laplace: Applicationsa la theorie de moments , these de 3e cycle, Faculte des sciences de 1’universite de Paris,1971.
[2] A. LUPAŞ, Die Folge Der Betaoperatoren, Dissertation, Stuttgart Universität, 1972.
[3] R. BOJANIC AND M. VUILLEUMIER, On the rate of convergence of
Fourier Legendre series of functions of bounded variation, J. Approx. Theory, 31 (1981), 67–79.
[4] F. CHENG, On the rate of convergence of Bernstein polynomials of functions of bounded variation, J. Approx. Theory, 39 (1983), 259–274.
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Ertan Ibikli and Harun Karsli
Title Page
[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 dĕveloppment des fonctions dĕfines dans un
interval infinien sĕries de polynŏmes de S.N. Bernstein, Compositio Math.,
4 (1937), 380–392.
[7] XIAO-MING ZENG, Bounds for Bernstein basis functions and MeyerKönig-Zeller basis functions, J. Math. Anal. Appl., 219 (1998), 364–376.
[8] XIAO-MING ZENG AND A. PIRIOU, On the rate of convergence of two
Bernstein-Bezier type operators for bounded variation functions, J. Approx. Theory, 95 (1998), 369–387.
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 25 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au
[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.
Rate of Convergence of
Chlodowsky Type Durrmeyer
Operators
Ertan Ibikli and Harun Karsli
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 26 of 26
J. Ineq. Pure and Appl. Math. 6(4) Art. 106, 2005
http://jipam.vu.edu.au