Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 852897, 10 pages
doi:10.1155/2009/852897
Research Article
Rate of Convergence of a New Type Kantorovich
Variant of Bleimann-Butzer-Hahn Operators
Lingju Chen1 and Xiao-Ming Zeng2
1
2
Department of Mathematics, Minjiang University, Fuzhou 350108, China
Department of Mathematics, Xiamen University, Xiamen 361005, China
Correspondence should be addressed to Lingju Chen, lingjuchen@126.com
and Xiao-Ming Zeng, xmzeng@xmu.edu.cn
Received 28 September 2009; Accepted 16 November 2009
Recommended by Vijay Gupta
A new type Kantorovich variant of Bleimann-Butzer-Hahn operator Jn is introduced. Furthermore,
the approximation properties of the operators Jn are studied. An estimate on the rate of
convergence of the operators Jn for functions of bounded variation is obtained.
Copyright q 2009 L. Chen and X.-M. Zeng. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
1. Introduction
In 1980, Bleimann et al. 1 introduced a sequence of positive linear Bernstein-type operators
Ln abbreviated in the following by BBH operators defined on the infinite interval I 0, ∞
by
n
n
k
1
k
x
Ln f, x f
,
n1−k
1 xn k0 k
x ∈ I, n ∈ N,
1.1
where N denotes the set of natural numbers.
Bleimann et al. 1 proved that Ln f, x → fx as n → ∞ for f ∈ Cb I the space
of all bounded continuous functions on I and give an estimate on the rate of convergence of
Ln f, x → fx measured with the second modulus of continuity of f.
In the present paper, we introduce a new type of Kantorovich variant of BBH operator
Jn , also defined on I by
Jn f, x n
n
k0
k
k
px 1 − px
n−k
ftdt
,
dt
Ik
Ik
1.2
2
Journal of Inequalities and Applications
where px x/1 x x ≥ 0, Ik k/n 2 − k, k 1/n 1 − k, and dt is Lebesgue
measure.
The operator 1.2 is different from another type of Kantorovich variant of BBH
operator Kn :
k1/n1−k
n
n
ft
n2 xk
Kn f, x dt,
n
1 x k0 k
1 t2
k/n2−k
1.3
which was first considered by Abel and Ivan in 2. The integrand function ft/1 t2 in
the operator 1.3 has been replaced with new integrand function ft in the operator 1.2.
In this paper we will study the approximation properties of Jn for the functions of bounded
variation. The rate of convergence for functions of bounded variation was investigated by
many authors such as Bojanić and Vuilleumier 3, Chêng 4, Guo and Khan 5, Zeng and
Piriou 6, Gupta et al. 7, involving several different operators.
Throughout this paper the class of function Φ is defined as follows:
Φ f | f is of bounded variation on every finite subinterval of I 0, ∞ .
1.4
Our main result can be stated as follows.
Theorem 1.1. Letf ∈ Φ and let Vab f be the total variation of f on interval a, b. Then, for n
sufficiently large, one has
2
n
√
1
xx/ k Jn f, x − fx − fx− ≤ 51√ x fx − fx− 91 x
√
V
gx O
,
2
nx k1 x−x/ k
n
2 nx
1.5
where
⎧
⎪
ft − fx,
⎪
⎪
⎨
gx t 0,
⎪
⎪
⎪
⎩
ft − fx−,
x < t < ∞,
x t,
1.6
0 ≤ t < x.
2. Some Lemmas
In order to prove Theorem 1.1, we need the following lemmas for preparation. Lemma 2.1
is the well-known Berry-Esséen bound for the classical central limit theorem of probability
theory. Its proof and further discussion can be founded in Feller 8, page 515.
Lemma 2.1. Let {ξ}∞
i1 be a sequence of independent and identically distributed random variables.
And 0 < Dξ1 < ∞, β3 E|ξ1 − Eξ1 |3 < ∞, then, there holds
y
n
c β3
1 1
−t2 /2 maxP
e
dt < √
,
2.1
ξk − a1 ≤ y − √
√
y∈R n b1 3
b1 n k1
2π −∞
√
where a1 Eξ1 , b1 2 Dξ1 Eξ1 − Eξ1 2 , 1/ 2π ≤ c ≤ 0.82.
Journal of Inequalities and Applications
3
In addition, let {ξ}ni1 be the random variables with two-point distribution
Pξ i ⎧
⎨x,
ξi 1
2.2
⎩1 − x, ξ 0,
i
where i 1, 2, . . . , n. Then we can easily obtain that
Let ηn β3 E|ξ1 − Eξ1 |3 ≤ x1 − x 2x2 − 2x 1 .
b1 2 Dξ1 x1 − x,
a1 Eξ1 x,
n
i1 ξi ,
2.3
then we also have
P ηn k n
k
xk 1 − xn−k ,
k 0, 1, . . . , n.
2.4
On the other hand, Jn f, x can be written by following integral form:
n
x I ftdt ∞
k
pn,k
ftHn x, tdt,
Jn f, x 1x
dt
0
k0
Ik
2.5
where
n
x 1
,
χk t pn,k
Hn x, t 1
x
dt
k0
I
χk t k
⎧
⎨1,
t ∈ Ik ,
⎩0,
t/
∈ Ik ,
Ik k/n2−k, k1/n1−k, k 0, 1, 2, . . . , n. It is easy to verify that
2.6
∞
0
Hn x, udu 1.
Lemma 2.2. If x ∈ 0, ∞ is fixed and n is sufficiently large, then
a for 0 ≤ y < x, there holds
y
0
2x1 x2
2 n 1 ,
x−y
2.7
2x1 x2
.
z − x2 n 1
2.8
Hn x, tdt ≤ 1
b for x < z < ∞, there holds
∞
Hn x, tdt ≤
z
1
Proof. We first prove a. Since 0 ≤ y < x, t ∈ 0, y, then x − t/x − y ≥ 1. Hence, we have
y
0
Hn x, tdt ≤
y
1
x − t2
2
2 Hn x, tdt ≤ 2 Jn x − t , x .
0 x−y
x−y
2.9
4
Journal of Inequalities and Applications
Direct calculation gives
x1 x2
1 x4
1 x4 4x 1
o n−4 .
Jn x − t2 , x n1
3n 1n 2 3n 1n 2n 3
Hence
2.10
y
Hn x, tdt ≤ 1/x − y2 2x1 x2 /n 1, for n sufficiently large.
The proof of b is similar.
0
Lemma 2.3 see 9, Theorem 1 or, cf. 10. For every x ∈ 0, 1, there holds
pn,k x Cnk xk 1 − xn−k ≤ 1
2enx1 − x
.
2.11
3. Proof of Theorem 1.1
Let f ∈ Φ, and x ∈ I, Bojanic-Cheng decomposition yields
ft fx − fx−
fx − fx−
fx − fx−
gx t sgnt − x δx t fx −
,
2
2
2
3.1
where gx t is defined as in 1.6 and
δx t ⎧
⎨1,
t x,
⎩0,
t/
x.
3.2
Obviously, Jn δx t, x 0. Thus it follows from 3.1 that
Jn f, x − fx − fx− ≤ Jn gx , x Jn sgnt − x, x fx − fx− .
2
2
3.3
First of all, we estimate |Jn sgnt − x, x|
Jn sgnt − x, x n
n 1 − kn 2 − k
k0
n2
where Ik k/n 2 − k, k 1/n 1 − k.
x pn,k
sgnt − xdt,
1 x Ik
3.4
Journal of Inequalities and Applications
5
Assuming that x ∈ k /n 2 − k , k 1/n 1 − k , for some k 0 ≤ k ≤ n, then
we have
Jn sgnt − x, x pn,k
k/n2−k>x
x x −
pn,k
x1
x1
k1/n1−k<x
k 1/n1−k
x n 1 − k n 2 − k pn,k
n2
1x
1−2
k/n2−k≤x
2
pn,k
dt −
x
x
k /n2−k
dt
x x1
k 1/n1−k
x n 1 − k n 2 − k pn,k
n2
1x
dt.
x
3.5
Thus
x x 2pn,k
Jn sgnt − x, x ≤ 1 − 2
.
p
n,k
x 1 1x
k/n2−k≤x
3.6
By Lemma 2.3 combining some direct computations, we can easily obtain
2pn,k
x 2
1x
≤
≤ √ .
1x
nx
2enx/1 x · 1/1 x
Set y x/1 x < 1, then by 2.4 and using Lemma 2.1, we have
x 1 − 2
pn,k
x 1 k/n2−k≤x
1
1 − 2
pn,k y 2 − P ηn ≤ n 2y 2
k≤n2y
⎛
⎞
1
2y
⎜ ηn − ny
⎟
2 − P ⎝ ≤
⎠
2
ny 1 − y
ny 1 − y
⎛
⎞
2y−1/√ny1−y
2y
1
2
⎟
⎜ ηn − ny
2P ⎝ e−t /2 dt
≤ ⎠ − √2π
−∞
ny 1 − y
ny 1 − y
1
√
2π
2y−1/√ny1−y
0
2
e−t /2 dt
3.7
6
Journal of Inequalities and Applications
⎛
⎞
2y−1/√ny1−y
η
−
ny
2y
1
2
n
⎟
⎜
−t /2 ≤
e
dt
≤ 2P ⎝ −
√
⎠
2π −∞
ny 1 − y
ny 1 − y
2
√
2π
2y/√ny1−y
e−t /2 dt
2
0
2y/√ny1−y
2cβ3
2
2
e−t /2 dt
≤√ 3√
2π 0
nb1
2 × 0.82 × y 1 − y 2y2 − 2y 1
2y
2
≤
√ 2π ny 1 − y
y 1 − y ny 1 − y
≤
4
41 x
√nx .
ny 1 − y
3.8
Thus, by 3.7, 3.8 we have
x 1 x 51 x
Jn sgnt − x, x ≤ 41
√
√
.
√
nx
nx
nx
3.9
Finally, we estimate Jn gx , x.
First, interval I 0, ∞ can be decomposed into four parts as
x
D1 0, x − √ ,
n
x
x
D2 x − √ , x √ ,
n
n
x
D3 x √ , 2x ,
n
D4 2x, ∞.
3.10
So Jn gx , x can be divided into four parts
J n gx , x ∞
gx tHn x, tdt Δ1,n gx Δ2,n gx Δ3,n gx Δ4,n gx ,
3.11
0
where Δj,n gx Dj gx tHn x, tdt.
Noticing gx x 0 and for t ∈ D2 , we have gx t gx t − gx x.
Thus
Δ2,n gx ≤
xx/√n
n
√ √ gx t − gx xHn x, tdt ≤ V xx/√ n gx ≤ 1 V xx/√ k gx .
x−x/ n
√
n k1 x−x/ k
x−x/ n
t
√
Next, let y x − x/ n, λn x, t 0 Hn x, udu.
3.12
Journal of Inequalities and Applications
7
Now, we recall the Lebesgue-Stieltjes integral representation, and by using partial
Lebesgue-Stieltjes integration, we get
y
Δ1,n gx gx tdt λn x, t
0
y
gx y λn x, y − λn x, tdt gx t
0
y
gx y − gx x λn x, y − λn x, tdt gx t − gx x ≤ Vyx gx λn x, y y
0
3.13
0
λn x, tdt −Vtx gx .
An application of a in Lemma 2.2 yields
2
2x1 x2
Δ1,n gx ≤ Vyx gx 2x1 x
2
n 1
x − y n 1
y
0
1
2
x − t
dt −Vtx gx .
3.14
Furthermore, since
y
0
1
x − t
d −Vtx
2 t
gx
y x −Vyx gx
V0x gx
Vt gx
2
dt,
2 2
3
x
0 x − t
x−y
3.15
we have
x−x/√n x Vt gx
2x1 x2 V0x gx
Δ1,n gx ≤
2
dt .
n1
x2
x − t3
0
3.16
√
Putting t x − x/ u in the last integral, we have
x−x/√n x n
Vt gx
1 n x
1
x
√ gx .
dt
Vx−x/√u gx du ≤ 2 Vx−x/
2
3
2
k
x 1
x k1
x − t
0
3.17
It follows from 3.16 and 3.17 that
2
Δ1,n gx ≤ 2x1 x
n 1x2
V0x
gx n
k1
x
√
Vx−x/
k
gx
≤
n
41 x2 x
√ gx .
Vx−x/
k
n 1x k1
3.18
8
Journal of Inequalities and Applications
By a similar method and using Lemma 2.2b, we obtain
2
n
√ Δ3,n gx ≤ 81 x
Vxxx/ k gx .
n 1x k1
3.19
Now, the remainder of our work is to estimate Δ4,n gx .
For fx satisfying the growth condition ft 0tr for some positive integer r as
t → ∞, we obviously have
Δ4,n gx ≤
x pn,k
1x
k/n2−k>2x
dt
g
Ik x t
.
dt
Ik
3.20
Thus, for n sufficiently large, there exists a M > 0, such that the following inequalities
hold:
x I tr dt
k
pn,k
1
x
dt
k/n2−k>2x
Ik
r
Ik t dt
M
pn,k y dt
k/n2−k>2y/1−y
Ik
Δ4,n gx ≤ M
≤M
pn,k y
k/n2−k>2y/1−y
k1
n1−k
3.21
r
,
where y x/1 x. By the definition of the Stirling numbers Sr, s of the second kind, we
readily have
ar r
Sr, saa − 1 · · · a − s 1,
r ∈ N,
3.22
s1
where the Stirling numbers Sr, s satisfy
Sn, 0 ⎧
⎨1
n 0,
⎩0
n ∈ N.
3.23
Thus we can write
k/n2−k>2y/1−y
k1
n1−k
r
r
pn,k y Sr, sAs ,
s1
3.24
Journal of Inequalities and Applications
9
where
As k 1k · · · k − s 2
pn,k y
r
n 1 − k
k/n2−k>2y/1−y
n−k
1
n!k 1
.
yk 1 − y
r ·
−
s
1!n
−
k!
k
1
−
k
n
k/n2−k>2y/1−y
3.25
From k/n 2 − k > 2x, x/1 x y, we can easily find k > 2n 4y/1 y. For a fixed
x > 0, when n > 2r r/x, we have k 1/k 1 − s < 2. Thus there holds
n−k
1
n!
yk 1 − y
.
r ·
n 1 − k k − s!n − k!
k>2n4y/1y
As ≤ 2
3.26
Now using the similar method as that in the proof of Lemma 4 of 11, we deduce that
r−s2
24r!n!ys−1 1 y
As ≤
,
n r − s!n r − s 2
for n > 2r r
.
x
3.27
From 3.21, 3.24, and 3.27, we obtain
Δ4,n gx ≤ M
pn,k y
k/n2−k>2y/1−y
k1
n1−k
r
1
.
M Sr, sAs O
n
s1
r
3.28
Finally, by combining 3.12, 3.18, 3.19, and 3.28, we deduce that
Jn gx t, x ≤ Δ1,n gx Δ2,n gx Δ3,n gx Δ4,n gx n
n
√ 1 xx/√k 81 x2 41 x2 1
x
xx/ k
√
√
≤
V
V
gx Vx−x/ k gx gx O
n
n
n 1x k1 x−x/ k
n 1x k1 x
≤
n
√
1 xx/√k 81 x2 1
xx/ k √
Vx−x/√k gx Vx−x/
g
O
x
k
n
n
n 1x k1
≤
n
√
1
91 x2 xx/ k √
.
Vx−x/
g
O
x
k
nx k1
n
Theorem 1.1 now follows from 3.3, 3.9, and 3.29.
3.29
10
Journal of Inequalities and Applications
Acknowledgments
This work is supported by the National Natural Science Foundation of China and the Fujian
Provincial Science Foundation of China. The authors thank the associate editor and the
referees for their several important comments and suggestions which improve the quality
of the paper.
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