APPROXIMATION OF BOUNDED VARIATION FUNCTIONS
BY A BÉZIER VARIANT OF THE BLEIMANN, BUTZER,
AND HAHN OPERATORS
VIJAY GUPTA AND OGÜN DOĞRU
Received 21 March 2005; Revised 13 May 2006; Accepted 28 May 2006
We give a sharp estimate on the rate of convergence for the Bézier variant of Bleimann,
Butzer, and Hahn operators for functions of bounded variation. We consider the case
when α ≥ 1 and our result improves the recently established results of Srivastava and
Gupta (2005) and de la Cal and Gupta (2005).
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
Bleimann et al. [3] introduced an interesting sequence of positive linear operators defined
on the space of real functions on the infinite interval [0, ∞) by
Ln ( f ,x) =
n
bn,k (x) f
k =0
k
,
n−k+1
x ∈ [0, ∞), n ∈ N,
(1.1)
where
bn,k (x) =
n
xk
.
k (1 + x)n
(1.2)
The Bézier variant of these operators for α ≥ 1 is defined in [6] as
Ln,α ( f ,x) =
n
k =0
(α)
Qn,k
(x) f
k
,
n−k+1
x ∈ [0, ∞), n ∈ N,
(1.3)
(α)
α
α
where Qn,k
(x) = Jn,k
(x) − Jn,k+1
(x) and Jn,k (x) = nj=k bn, j (x).
As a special case α = 1, Ln,α ( f ,x) reduce to the operators Ln,1 ( f ,x) ≡ Ln ( f ,x), defined
by (1.1). Some approximation properties of the Bleimann, Butzer, and Hahn operators
were discussed in [1, 2], and so forth. Very recently, de la Cal and Gupta [4] and Srivastava
Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2006, Article ID 37253, Pages 1–5
DOI 10.1155/IJMMS/2006/37253
2
Approximation by a Bézier variant of the BBH operators
and Gupta [6] studied the rate of approximation for the Bleimann, Butzer, and Hahn
operators and its Bézier variant (α ≥ 1), respectively.
We recall the Lebesgue-Stieltjes integral representation
Ln,α ( f ,x) =
∞
0
f (t)dt Kn,α (x,t) ,
(1.4)
where
⎧
⎪
⎨
Kn,α (x,t) = ⎪k≤(n−k+1)t
⎩
0,
(α)
Qn,k
(x), 0 < t < ∞,
(1.5)
t = 0.
In this paper, we give a different and improved estimate on the rate of approximation
for functions of bounded variation on the Bézier variant of Bleimann, Butzer, and Hahn
operators.
2. Auxiliary results
In this section, we recall two lemmas, which are essential for our main theorem.
Lemma 2.1 [6, Lemma 3]. For all x ∈ (0, ∞), α ≥ 1, and k ∈ N, there holds
α(1 + x)
(α)
.
(x) ≤ αbn,k (x) < √
Qn,k
2enx
(2.1)
Lemma 2.2 [5, Lemma 3]. For x ∈ (0, ∞),
−3/2 |1 − x|
1
≤ +
O
bn,k (x) − n
.
2 6 2π(n + 1)x
k/(n−k+1)>x
(2.2)
3. Rate of convergence
Our main result is stated as follows.
Theorem 3.1. Let f be a function of bounded variation on every finite subinterval of [0, ∞).
Let f (t) = O(t r ) for some r ∈ N as t → ∞. Then for x ∈ (0, ∞), α ≥ 1, and for n → ∞,
Ln,α ( f ,x) − 2−α f (x+) − 1 − 2−α f (x−)
α|1 − x| 9α(1 + x)2 x+x/√√k f (x+) − f (x−)
V
fx + (n + 2)x k=1 x−x/ k
6 2π(n + 1)x
n
≤
α(1 + x)
εn (x) f (x) − f (x−) + O n−1 ,
+ √
2enx
(3.1)
V. Gupta and O. Doğru
3
where
⎧
⎪
⎪
⎨1,
x(n + 1)
∈ N,
1+x
if
εn (x) = ⎪
⎪
⎩0, otherwise,
⎧
⎪
f (t) − f (x−),
⎪
⎪
⎪
⎨
fx (t) = ⎪0,
⎪
⎪
⎪
⎩
f (t) − f (x+),
(3.2)
if 0 ≤ t < x,
if t = x,
if x < t < ∞,
and Vab ( fx ) is the total variation of fx on [a,b].
Proof. We have
f (t) − 2−α f (x+) − 1 − 2−α f (x−)
= fx (t) + 2−α f (x+) − f (x−) sign(α) (t − x)
+ f (x) − 2−α f (x+) − 1 − 2
−α
(3.3)
f (x−) δx (t),
where
⎧
α
⎪
⎪
⎪2 − 1,
⎪
⎨
if t > x,
sign(α) (t − x) := ⎪0,
if t = x,
⎪
⎪
⎪
⎩
−1,
if t < x,
⎧
⎪
⎨1,
if x = t,
δx (t) = ⎪
⎩0, if x = t.
(3.4)
Therefore, we can write
Ln,α ( f ,x) − 2−α f (x+) − 1 − 2−α f (x−)
≤ Ln,α fx ,x + 2−α f (x+) − f (x−) Ln,α sign(α) (t − x),x
(3.5)
+ f (x) − 2−α f (x+) − 1 − 2−α f (x−) Ln,α δx ,x ,
and our first estimates are
Ln,α sign(α) (t − x),x = 2α
k>(n−k+1)x
=2
(α)
(α)
Qn,k
(x) − 1 + εn (x)Qn,k
(x)
α
α
bn,k (x)
k>(n−k+1)x
(α)
Ln,α δx ,x = εn (x)Qn,k
(x).
(α)
− 1 + εn (x)Qn,k
(x),
(3.6)
4
Approximation by a Bézier variant of the BBH operators
Then we have
G := 2−α f (x+) − f (x−) Ln,α sign(α) (t − x),x
+ f (x) − 2−α f (x+) − 1 − 2−α f (x−) Ln,α δx ,x α
−α f (x+) − f (x−) 2
= 2
k>(n−k+1)x
(α)
Qn,k
(x) − 1
+ f (x) − f (x−)
(α)
εn (x)Qn,k
(x).
(3.7)
Using the mean value theorem, we get
k>(n−k+1)x
α−1 (α)
Qn,k
(x) − 2−α = α ξn,k (x)
k>(n−k+1)x
bn,k (x) − 2 ,
−1 (3.8)
where ξn,k (x) lies between 2−1 and k>(n−k+1)x bn,k (x). Because of Lemma 2.2, it is easily
seen that the intermediate point ξn,k (x) is close to 2−1 for sufficiently large n. Then we can
write ξn,k (x) = (2 + ε)−1 for each ε > 0. Thus, we have
ξn,k (x)
α−1
= (2 + ε)1−α ≤ 1
(3.9)
for each α ≥ 1. By using (3.9) and Lemma 2.2 in (3.8), we obtain
k>(n−k+1)x
(α)
≤
Qn,k
(x) − 2−α α|1 − x|
+ O n−3/2 .
6 2π(n + 1)x
(3.10)
Hence, by using (3.10) in (3.7) and Lemma 2.1, we obtain
+ x)
α|1 − x| f (x+) − f (x−) + α(1
√
εn (x) f (x) − f (x−) + O n−3/2 .
G≤ 2enx
6 2π(n + 1)x
(3.11)
On the other hand, to estimate Ln,α ( fx ,x), we break the Lebesgue-Stieltjes integral into
four parts as follows:
Ln,α fx ,x =
x−x/√n x+x/√n 2x
+
0
√
x−x/ n
+
∞
√
x+x/ n
+
2x
fx (t)dt Kn,α (x,t)
(3.12)
then, by proceeding along the lines of [6], we get
n
2 √ −1 Ln,α fx ,x ≤ 9α(1 + x)
√k f
Vxx+x/
.
x +O n
−x/ k
(n + 2)x
(3.13)
k =1
Using (3.11) and (3.13) in (3.5), we get the desired result. This completes the proof of
Theorem 3.1.
Notice that for the case 0 < α < 1, these results can be found in [5].
V. Gupta and O. Doğru
5
Acknowledgment
The authors are thankful to the referees for making valuable suggestions leading to the
overall improvement of this paper.
References
[1] U. Abel and M. Ivan, Some identities for the operator of Bleimann, Butzer and Hahn involving
divided differences, Calcolo 36 (1999), no. 3, 143–160.
, Best constant for a Bleimann-Butzer-Hahn moment estimation, East Journal on Approx[2]
imations 6 (2000), no. 3, 349–355.
[3] G. Bleimann, P. L. Butzer, and L. Hahn, A Bernšteı̆n-type operator approximating continuous
functions on the semi-axis, Indagationes Mathematicae 42 (1980), no. 3, 255–262.
[4] J. de la Cal and V. Gupta, On the approximation of locally bounded functions by operators of
Bleimann, Butzer and Hahn, Journal of Inequalities in Pure and Applied Mathematics 6 (2005),
no. 1, 1–10, article 4.
[5] V. Gupta and A. Lupas, Rate of approximation for Bézier variant of Bleiman, Butzer and Hahn
operators, General Mathematics 13 (2005), no. 1, 41–54.
[6] H. M. Srivastava and V. Gupta, Rate of convergence for the Bézier variant of the Bleimann-ButzerHahn operators, Applied Mathematics Letters 18 (2005), no. 8, 849–857.
Vijay Gupta: School of Applied Sciences, Netaji Subhas Institute of Technology, Sector-3,
Dwarka, New Delhi 110045, India
E-mail address: vijay@nsit.ac.in
Ogün Doğru: Department of Mathematics, Faculty of Science, Ankara University,
06100 Tandogan, Ankara, Turkey
E-mail address: dogru@science.ankara.edu.tr