c
Applied Mathematics E-Notes, 8(2008), 203-213 Available free at mirror sites of http://www.math.nthu.edu.tw/∼amen/
ISSN 1607-2510
Rate Of Convergence Of Chlodowsky Operators For
Functions With Derivatives Of Bounded Variation∗
Harun Karsli†
Received 28 March 2007
Abstract
In the present paper, we estimate the rate of pointwise convergence of the
Chlodowsky operators Cn for functions, defined on the interval [0, bn ] with derivatives of bounded variation, where limn→∞ bn = ∞.
1
Introduction
For a function defined on the interval [0, 1], the classical Bernstein operators are
n
X
k
Bn (f; x) =
f
pk,n(x), 0 ≤ x ≤ 1, n ≥ 1,
n
(1)
k=0
n
xk (1 − x)n−k is the Bernstein basis. Bernstein [1] used these
k
operators (1) to give the first constructive proof of the Weierstrass theorem. It is
well known that if we make the substitution x = bxn and replacing the discrete values
f( kn ) by f( kn bn ), in the polynomial of Bernstein Bn (f; x) corresponding to a function
f defined on [0, 1], then one can obtain the following polynomials
where pk,n(x) =
k n−k
n
X
x
x
k
n
Cn (f; x) =
1−
f
, 0 ≤ x ≤ bn
bn
k
n
bn
bn
(2)
k=0
where (bn ) is a positive increasing sequence with the properties
lim bn = ∞,
n→∞
lim
n→∞
bn
= 0.
n
These operators are called Bernstein-Chlodowsky operators or Chlodowsky operators.
Operators of type (2) were introduced by Chlodowsky [2] and further modified and
studied by many authors [3-4]. Since the behaviour of Chlodowsky operators are very
∗ Mathematics
† Department
Subject Classifications: 41A25, 41A36.
of Mathematics, Faculty of Science and Arts, Abant Izzet Baysal University, 14280,
Bolu/Turkey
203
204
Rate of Convergence of Chlodowsky Operators
similar to the Bernstein operators, these operators allow us to investigate approximation properties of functions defined on the infinite interval 0 ≤ x < ∞ by using
the similar techniques and methods on clasical and modified Bernstein operators. For
example, in [4] Ibikli and Karsli approximated integrable functions on the interval
[0, bn] by what they called “Chlodowsky Type Durrmeyer operators” defined as follows: Dn : BV [0, ∞) → P,
n
Dn (f; x) =
(n + 1) X
pk,n
bn
k=0
x
bn
Zbn
f(t)pk,n
t
bn
dt, 0 ≤ x ≤ bn ,
0
where pk,n (x) is the Bernstein basis. They estimated the convergence rate of these
Dn operators for functions in BV [0, ∞). After this study Karsli [5] obtained their rate
of convergence for functions whose derivatives had bounded variation in [0, ∞). It is
useful to mention very recent papers by Karsli and Ibikli [6-7], which deal with the
rate of pointwise convergence of the operators (2) and its Bézier variant in the space
BV [0, bn] respectively.
The concern of this paper is to study the rate of convergence of operators Cn to the
limit f of functions with derivatives of bounded variation on the interval [0, bn], (n → ∞)
extending infinity. At the point x, which is a discontinuity of the first kind of the derivative, we shall prove that Cn (f; x) converge to the limit f(x).
Some authors studied some linear positive operators and obtained the rate of convergence for functions in DBV (I). For example, Bojanic and Cheng investigated the rate
of convergence of Hermite-Fejer polynomials for functions with derivatives of bounded
variation [8] and they also investigated in the paper [9] the asymptotic behavior of
Bernstein polynomials for functions in DBV [0, 1] of all functions f that can be written
as
Zx
f(x) = f(0) + Ψ(t)dt, x ∈ [0, 1],
0
where Ψ ∈ BV [0, 1]. We also mention some recent studies in this area by Gupta et
al. [10], in which they estimated the rate of convergence of summation-integral-type
operators for functions in γ-weighted space DBVγ (0, ∞), (γ ≥ 0), and by Gupta et al.
[11] and very recent papers by Karsli [12].
Let DBV (I) denote the class of differentiable functions defined on a set I ⊂ R,
whose derivatives are bounded variation on I,
DBV (I) = {f : f 0 ∈ BV (I)} .
It is clear that the class of functions DBV (I) considered here is much more general
than the class of functions with continuous derivative on I.
For the sake of brevity, let the auxiliary function fx be defined by

 f(t) − f(x+), x < t ≤ bn
fx (t) =
0,
t=x
.

f(t) − f(x−), 0 ≤ t < x
H. Karsli
205
The main theorem of this paper is as follows.
THEOREM. Let f be a function with derivatives of bounded variation on every
finite subinterval of [0, ∞) and limx→∞ f 0 (x) exists. Then for every x ∈ (0, ∞) , we
have
√
0
r
[ n] x+ bn −x
f (x+) − f 0 (x−) x(bn − x) 2bn X _k
|Cn(f; x) − f(x)| ≤ (fx0 ). (3)
+
2
n
n
x
k=1
where
b
W
a
2
x− k
(fx0 ) is the total variation of fx0 on [a, b].
Auxiliary Results
In this section we give certain results, which are necessary to prove our main theorems.
As before we let
Zt
x t
x u
∂
λn
:=
du
,
,
Kn
bn bn
∂u
bn bn
0
where 0 ≤ t ≤ bn , and
(
x u
Kn
=
,
bn bn
P
kbn ≤nu
pk,n( bxn ),
0 < u ≤ bn
.
0,
(4)
u=0
Then λn ( bxn , btn ) ≤ 1. Since the operators (1) and (2) are special cases of Stieltjes
integrals, alternatively we can rewrite the operators (2) in the form of a Stieltjes integral
as follows:
Zbn
x t
∂
Cn(f; x) = f(t) Kn
dt.
,
∂t
bn bn
0
LEMMA 1 ([6]). For Cn (ts; x), s = 0, 1, 2, one has
Cn (1; x) =
1,
Cn(t; x) =
x,
Cn (t2; x) =
x2 +
x(bn − x)
.
n
By direct calculation, we find the following equalities:
x(bn − x)
(5)
, Cn((t − x); x) = 0 .
n
LEMMA 2. For all x ∈ (0, ∞) , let Kn bxn , bun be defined by (4), we have for t < x,
Cn ((t − x)2; x) =
λn
x t
,
bn bn
=
Zt
0
∂
Kn
∂u
x u
,
bn bn
du ≤
1
x(bn − x)
.
(x − t)2
n
(6)
206
Rate of Convergence of Chlodowsky Operators
PROOF.
λn
x t
,
bn bn
=
Zt
∂
Kn
∂u
x u
,
bn bn
du ≤
Zt 0
=
x−u
x−t
2
∂
Kn
∂u
x u
,
bn bn
du
0
1
(x − t)2
Zt
(x − u)2
∂
Kn
∂u
x u
,
bn bn
du ≤
1
Cn ((u − x)2; x).
(x − t)2
0
From (5), it is easy to see that
x(bn − x)
x t
1
λn
≤
,
.
2
bn bn
(x − t)
n
REMARK. From Cauchy-Schwarz-Bunyakowsky inequality, one has
r
12
x(bn − x)
2
Cn(|t − x| ; x) ≤ Cn((t − x) ; x) =
.
n
3
(7)
Main Result
Now we can prove the main theorem. From (6), we can write the difference between
Cn(f; x) and f(x) as a Lebesgue-Stieltjes integral,
Zbn
x t
∂
Cn(f; x) − f(x) = [f(t) − f(x)] Kn
dt.
,
∂t
bn bn
(8)
0
Since f(t) ∈ DBV [0, bn], we may rewrite (8) as follows:
=
=
Cn(f; x) − f(x)
Zx
Zbn
x t
x t
∂
∂
dt + [f(t) − f(x)] Kn
dt
[f(t) − f(x)] Kn
,
,
∂t
bn bn
∂t
bn bn
x
0




Zx Zx
Zbn Zt
x t
x t
∂
∂
−  f 0 (u) du Kn
dt +  f 0 (u) du Kn
dt
,
,
∂t
bn bn
∂t
bn bn
0
=
where
t
x


Zx Zx
x t
∂
0


I1 (x) =
dt
f (u) du
,
Kn
∂t
bn bn
0
and
x
−I1 (x) + I2 (x),


Zbn Zt
x t
∂
I2 (x) =  f 0 (u) du Kn
dt.
,
∂t
bn bn
x
(9)
t
x
(10)
H. Karsli
207
For any f(t) ∈ DBV [0, bn], we decompose f(t) into four parts as
f 0 (t)
1 0
f 0 (x+) − f 0 (x−)
(f (x+) + f 0 (x−)) + fx0 (t) +
sgn(t − x)
2
2
1
+δx (t) f 0 (x) − (f 0 (x+) + f 0 (x−)) ,
2
=
where
δx (t) =
1, x=t
.
0 , x 6= t
If we use this equality in (9) and (10), we have the following expressions
I1 (x)
=

Zx Zx
=

Zbn Zt
1 0
f 0 (x+) − f 0 (x−)
(f (x+) + f 0 (x−)) + fx0 (u) +
sgn(u − x)
 2
2
0
t
x t
1
∂
+δx (u) f 0 (x) − (f 0 (x+) + f 0 (x−)) du
dt
,
Kn
2
∂t
bn bn
and
I2 (x)
1 0
f 0 (x+) − f 0 (x−)
(f (x+) + f 0 (x−)) + fx0 (u) +
sgn(u − x)
 2
2
x
x
x t
1
∂
+δx (u) f 0 (x) − (f 0 (x+) + f 0 (x−)) du
dt.
,
Kn
2
∂t
bn bn
Firstly, we evaluate I1 (x):
I1 (x)
Zx
x t
f 0 (x+) + f 0 (x−)
∂
dt
=
(x − t) Kn
,
2
∂t
bn bn
0


Zx Zx
x t
∂
0


+
dt
fx (u) du
,
Kn
∂t
bn bn
0
t
x t
∂
dt
,
Kn
∂t
bn bn
0


Zx Zx
0
0
f (x+) + f (x−)
 δx (u) du ∂ Kn x , t dt.
+ f 0 (x) −
2
∂t
bn bn
−
f 0 (x+) − f 0 (x−)
2
Zx
(x − t)
0
t
208
Rate of Convergence of Chlodowsky Operators
It is obvious that
Rx
δx (u) du = 0. From this fact, we get
t
I1 (x) =
Zx
x t
f 0 (x+) + f 0 (x−)
∂
dt
(x − t) Kn
,
2
∂t
bn bn
0


Zx Zx
x t
∂
0


+
dt
fx (u) du
,
Kn
∂t
bn bn
0
t
f 0 (x+) − f 0 (x−)
−
2
Zx
∂
(x − t) Kn
∂t
x t
,
bn bn
dt.
(11)
0
Using a similar method, for evaluating I2 (x), we find that
I2 (x) =
Zbn
x t
f 0 (x+) + f 0 (x−)
∂
dt
(t − x) Kn
,
2
∂t
bn bn
x


Zbn Zt
x t
∂
+  fx0 (u) du Kn
dt
,
∂t
bn bn
x
−
x
f 0 (x+) − f 0 (x−)
2
Zbn
x t
∂
dt.
(t − x) Kn
,
∂t
bn bn
x
Combining (11) and (12), we get
−I1 (x) + I2 (x)
=
f 0 (x+) + f 0 (x−)
2
Zbn
x t
∂
dt
(t − x) Kn
,
∂t
bn bn
0
Zbn
f 0 (x+) − f 0 (x−)
x t
∂
+
dt
|t − x| Kn
,
2
∂t
bn bn
0


Zx Zx
x t
∂
−  fx0 (u) du Kn
dt
,
∂t
bn bn
0
t


Zbn Zt
x t
∂
+  fx0 (u) du Kn
dt.
,
∂t
bn bn
x
x
(12)
H. Karsli
209
From the last expression, we can rewrite (8) as follows:
Cn (f; x) − f(x)
f 0 (x+) + f 0 (x−)
2
=
Zbn
x t
∂
dt
(t − x) Kn
,
∂t
bn bn
0
Zbn
f 0 (x+) − f 0 (x−)
x t
∂
+
dt
|t − x| Kn
,
2
∂t
bn bn
0


Zx Zx
x t
∂
dt
−  fx0 (u) du Kn
,
∂t
bn bn
t
0


Zbn Zt
x t
∂
+  fx0 (u) du Kn
dt.
,
∂t
bn bn
x
(13)
x
On the other hand, since
Zbn
|t − x|
∂
Kn
∂t
x t
,
bn bn
dt = Cn (|t − x| ; x)
0
and
Zbn
x t
∂
dt = Cn (t − x; x),
(t − x) Kn
,
∂t
bn bn
0
using these equalities in (13) and taking absolute value, we can express (13) as follows;
|Cn(f; x) − f(x)|
≤
0
f (x+) + f 0 (x−) |Cn(t − x; x)|
2
0
0
f (x+) − f (x−) |Cn (|t − x| ; x)|
+
2
 x

x
Z Z
x t
∂
+ −  fx0 (u) du Kn
dt
,
∂t
bn bn
t
0
b  t

Z n Z
x t
∂
0


+
dt .
fx (u) du
,
Kn
∂t
bn bn
x
(14)
x
According to (4), we write




Zx Zx
Zx Zx
 fx0 (u) du ∂ Kn x , t dt =  fx0 (u) du ∂ λn x , t dt.
∂t
bn bn
∂t
bn bn
0
t
0
t
(15)
210
Rate of Convergence of Chlodowsky Operators
Using partial integration on the right hand side of (15), we obtain


Zx Zx
Zx
x
x t
∂
t
0
 fx (u) du λn
dt = fx0 (t) λn
dt.
,
,
∂t
bn bn
bn bn
t
0
0
Thus
x x

Z Z
Zx
x
x t
∂
t
0
−  f 0 (u) du Kn
dt ≤ |fx (t)| λn
dt
,
,
x
∂t
bn bn
bn bn
t
0
0
and
x x

Z Z
x
∂
t
−  fx0 (u) du Kn
≤
,
∂t
b
b
n
n 0
x− √xn
Z
t
|fx0 (t)|
λn
x t
,
bn bn
dt
0
Zx
+
|fx0 (t)| λn
x− √xn
x t
,
bn bn
dt.
Since fx0 (x) = 0 and λn ( bxn , btn ) ≤ 1, one has
Zx
|fx0 (t)| λn
x− √xn
x t
,
bn bn
Zx
dt =
|fx0 (t) − fx0 (x)| λn
x− √xn
x t
,
bn bn
Zx _
x
dt ≤
x− √xn
(fx0 ) dt.
t
Make the change of variables t = x − xu , then
Zx _
x
x− √xn
(fx0 ) dt
≤
x
_
x− √xn
t
Zx
(fx0 )
dt.
x− √xn
From (6), we can write
x− √xn
Z
|fx0 (t)| λn
x t
,
bn bn
x− √xn
dt ≤
x(bn − x)
n
0
Z
|fx0 (t)|
dt
(x − t)2
0
x− √xn
=
x(bn − x)
n
Z
|fx0 (t) − fx0 (x)|
dt
(x − t)2
0
x− √xn
≤
x(bn − x)
n
Z
0
x
_
t
(fx0 )
dt
.
(x − t)2
H. Karsli
211
Make the change of variables t = x −
x− √xn
Z
x(bn − x)
n
x
u
again, we have
√
x
_
0
(fx0 )
t
dt
(x − t)2
x(bn − x)
n
=
Zn_
x
(fx0 )
x
1 x− u
√
[ n] x
( ux2 ) du
(− xu )2
(bn − x) X _ 0
(fx )
n
x
=
k=1 x− k
and hence, we obtain
√
n] x
Zx Zx
[X
x
_
_
−
x)
∂
x
t
x
(b
n
0
0
− [ f (u) du] Kn ( , ) ≤ √
(f
)
+
(fx0 ).
x
x
∂t
b
b
n
n
n
n x− √x
k=1 x− x
k
0
t
n
Since
√
x
√
n
x
_
(fx0 ) ≤
x− √xn
[ n] x
2x X _
n
(fx0 ),
k=1 x− x
k
it follows that
√
[ n] x
x
x _
(bn − x) X _ 0
0
√
(f ) +
(fx ) ≤
n √x x
n
x
x−
k=1 x− k
n
√
√
k=1 x− k
k=1 x− k
[ n] x
[ n] x
2x X _ 0
2(bn − x) X _ 0
(fx ) +
(fx )
n
n
x
x
√
≤
[ n] x
2bn X _ 0
(fx ).
n
x
k=1 x− k
Therefore
x x
√

n] x
Z Z
[X
_
x
2b
∂
t
n
−  fx0 (u) du Kn
dt
≤
,
(fx0 ).
∂t
bn bn
n
x
k=1 x−
0
t
(16)
k
Using a similar method for estimating, we have
b 

√
−x
√
x+ bn
n] x+ bnk−x
Z n Z t
[X
n
_
_
x
b
−
x
∂
t
x
n
0
0
 f (u) du Kn
dt ≤ √
,
(fx ) +
(fx0 ).
x
∂t
b
b
n
n
n
n
x
x
k=1
x
x
Furthermore, since
bn − x
√
n
−x
√
x+ bn
n
_
x
√
[ n] x+ bn −x
2(bn − x) X _k
0
(fx ) ≤
(fx0 ),
n
x
k=1
212
Rate of Convergence of Chlodowsky Operators
we can write the following inequality
bn − x
√
n
√
−x
√
x+ bn
n
_
x
(fx0 ) +
[ n] x+ bn −x
x X _k
n
k=1
√
(fx0 )
≤
x
[ n] x+ bn −x
2(bn − x) X _k
(fx0 )
n
x
k=1
√
+
n] x+ bnk−x
[
2x X
n
k=1
√
≤
_
(fx0 )
x
[ n] x+ bn −x
2bn X _k
(fx0 ).
n
x
k=1
Thus we get
b  t

√
n] x+ bnk−x
Z n Z
[X
_
x
2b
∂
t
n
 f 0 (u) du Kn
dt ≤
,
(fx0 ).
x
∂t
bn bn
n
x
k=1
x
(17)
x
Putting (5), (7), (16) and (17) in (14), we get (3), i.e.,
√
0
r
[ n] x+ bn −x
f (x+) − f 0 (x−) x(bn − x) 2bn X _k
|Cn(f; x) − f(x)| ≤ (fx0 ).
+
2
n
n
x
k=1
x− k
This completes the proof of the theorem.
Acknowledgments. The author is thankful to the referees for their valuable
comments and suggestions leading to a better presentation of this paper. The author
also wishes to dedicate this paper to Prof. Akif D. Gadjiev on his 70th birthday
occasion.
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H. Karsli
213
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