Hindawi Publishing Corporation
Journal of Inequalities and Applications
Volume 2009, Article ID 702680, 7 pages
doi:10.1155/2009/702680
Research Article
A New Estimate on the Rate of Convergence of
Durrmeyer-Bézier Operators
Pinghua Wang1 and Yali Zhou2
1
2
Department of Mathematics, Quanzhou Normal University, Fujian 362000, China
Liming University, Quanzhou, Fujian 362000, China
Correspondence should be addressed to Pinghua Wang, xxc570@163.com
Received 20 February 2009; Accepted 13 April 2009
Recommended by Vijay Gupta
We obtain an estimate on the rate of convergence of Durrmeyer-Bézier operaters for functions
of bounded variation by means of some probabilistic methods and inequality techniques. Our
estimate improves the result of Zeng and Chen 2000.
Copyright q 2009 P. Wang and Y. Zhou. 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. Introdution
In 2000, Zeng and Chen 1 introduced the Durrmeyer-Bézier operators Dn,α which are
defined as follows:
n
α
Dn,α f, x n 1 Qnk x
k0
1
1.1
ftpnk tdt,
0
n
α
α
α
x − Jn,k1
x, Jnk x where f is defined on 0, 1, α ≥ 1, Qnk x Jnk
jk
pnj x,
k 0, 1, 2, . . . , n are Bézier basis functions, and pnk x n!/k!n − k!x 1 − xn−k ,
k 0, 1, 2, . . . , n are Bernstein basis functions.
When α 1, Dn,1 f is just the well-known Durrmeyer operator
k
Dn,1 f, x n 1
n
k0
pnk x
1
ftpnk tdt.
1.2
0
Concerning the approximation properties of operators Dn,1 f and some results on
approximation of functions of bounded variation by positive linear operators, one can refer
2
Journal of Inequalities and Applications
to 2–7. Authors of 1 studied the rate of convergence of the operators Dn,α f for functions
of bounded variation and presented the following important result.
Theorem A. Let f be a function of bounded variation on 0, 1, (f ∈ BV0, 1), α ≥ 1, then for every
x ∈ 0, 1 and n ≥ 1/x1 − xone has
√
n x1−x/
k 8α
α
1
Dn,α f, x −
fx fx− ≤
gx
√
α1
α1
nx1 − x k1
x−x/ k
where
b
a gx 2α
nx1 − x
fx − fx−,
1.3
is the total variation of gx on a, b and
⎧
⎪
ft − fx,
⎪
⎪
⎨
gx t 0,
⎪
⎪
⎪
⎩
ft − fx−,
x < t ≤ 1,
1.4
t x,
0 ≤ t < x.
Since the Durrmeyer-Bézier operators Dn,α are an important approximation operator
of new type, the purpose of this paper is to continue studying the approximation properties
of the operators Dn,α for functions of bounded variation, and give a better estimate than that
of Theorem A by means of some probabilistic methods and inequality techniques. The result
of this paper is as follows.
Theorem 1.1. Let f be a function of bounded variation on 0, 1, (f ∈ BV0, 1), α ≥ 1, then for
every x ∈ 0, 1 and n > 1 one has
√
n x1−x/
k 4α
1
α
1
Dn,α f, x −
fx fx− ≤
gx
√
α1
α1
nx1 − x k1
x−x/ k
α
n 1x1 − x
fx − fx−,
1.5
where gx t is defined in 1.4.
It is obvious that the estimate 1.5 is better than the estimate 1.3. More important,
the estimate 1.5 is true for all n > 1. This is an important improvement comparing with the
fact that estimate 1.3 holds only for n ≥ 1/x1 − x.
2. Some Lemmas
In order to prove Theorem 1.1, we need the following preliminary results.
Lemma 2.1. Let {ξk }∞
k1 be a sequence of independent and identically distributed random variables,
ξ1 is a random variable with two-point distribution P ξ1 i xi 1 − x1−i (i 0, 1, and x ∈ 0, 1 is
Journal of Inequalities and Applications
3
a parameter). Set ηn nk1 ξk , with the mathematical expectation Eηn μn ∈ −∞, ∞, and with
the variance Dηn σn2 > 0. Then for k 1, 2, . . . , n 1,one has
P ηn ≤ k − 1 − P ηn1 ≤ k ≤ σn1 ,
μn1
P ηn ≤ k − P ηn1 ≤ k ≤ σn1
.
n 1 − μn1
Proof. Since ηn get
n
k1 ξk ,
2.1
2.2
from the distribution series of ξk , by convolution computation we
P ηn j n!
xj 1 − xn−j ,
j! n − j !
0 ≤ j ≤ n.
2.3
Furthermore by direct computations we have
μn1 n 1x,
P ηn j − 1 j
P ηn1 j ,
n 1x
1 ≤ j ≤ n 1.
2.4
Thus we deduce that
k
k
P ηn ≤ k − 1 − P ηn1 ≤ k P ηn j − 1 − P ηn1 j − P ηn1 0 j1
j1
k
j
− 1 P ηn1 j j0 n 1x
≤
k 1
j − n 1xP ηn1 j
n 1x j0
≤
n1 1
j − n 1xP ηn1 j
n 1x j0
≤
1 E ηn1 − μn1 .
μn1
2.5
By Schwarz’s inequality, it follows that
1 E ηn1 − μn1 ≤
μn1
The inequality 2.1 is proved.
2
E ηn1 − μn1
μn1
σn1
.
μn1
2.6
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Journal of Inequalities and Applications
Similarly, by using the identities
n 1 − μn1 n 11 − x,
P ηn j n 1 − j
P ηn1 j ,
n 11 − x
2.7
1 ≤ j ≤ n 1,
we get the inequality 2.2. Lemma 2.1 is proved.
Lemma 2.2. Let α ≥ 1, k 0, 1, 2, . . . , n, pnk x n!/k!n − k!xk 1 − xn−k be Bernstein basis
functions, and let Jnk x njk pnj x be Bézier basis functions, then one has
α
α
α
,
x ≤ Jnk x − Jn1,k1
n 1x1 − x
α
α
α
.
x ≤ Jnk x − Jn1,k
1x1
− x
n
2.8
Proof. Note that 0 ≤ Jnk x, Jn1,k1 x ≤ 1, μn1 n 1x, σ 2n1 n 1x1 − x, and α ≥ 1.
Thus
α
α
x ≤ α|Jnk x − Jn1,k1 x|
Jnk x − Jn1,k1
n1
n
α pnj −
pn1,j jk
jk1
⎛
⎞ ⎛
⎞
n
n1
α⎝1 − pnj ⎠ − ⎝1 −
pn1,j ⎠
jk
jk1
2.9
αP ηn ≤ k − 1 − P ηn1 ≤ k .
Now by inequality 2.1 of Lemma 2.1 we obtain
1−x
α
α
α
≤
.
x ≤ α Jnk x − Jn1,k1
n 1x1 − x
n 1x1 − x
2.10
Similarly, by using inequality 2.2, we obtain
α
α
x ≤ α Jnk x − Jn1,k
Thus Lemma 2.2 is proved.
x
n 1x1 − x
≤
α
n 1x1 − x
.
2.11
Journal of Inequalities and Applications
5
3. Proof of Theorem 1.1
Let f satisfy the conditions of Theorem 1.1, then f can be decomposed as
ft α
1
fx fx− gx t
α1
α1
fx − fx−
α−1
sgnt − x 2
α1
1
1
δx t fx − fx − fx− ,
2
2
3.1
where
⎧
⎪
1,
⎪
⎪
⎨
sgnt 0,
⎪
⎪
⎪
⎩
−1,
t>0
t 0,
δx t t < 0,
⎧
⎨0,
t/
x,
⎩1,
t x.
3.2
Obviously Dn,α δx , x 0, thus from 3.1 we get
α
1
Dn,α f, x −
fx
fx−
α1
α1
fx − fx−
α−1 .
≤ Dn,α gx, , x Dn,α sgnt − x, x 2
α1 3.3
We first estimate |Dn,α sgnt − x, x α − 1/α 1|, from 1, page 11 we have the following
equation:
n1
n1
α−1
α
α
2 pn1,k xJnk
Dn,α sgnt − x, x x − 2 pn1,k xγnk
x,
α1
k0
k0
3.4
α
α
α
where Jn1,k1
x < γnk
x < Jn1,k
x.
α
α
x − γnk
x| ≤ α/ n 1x1 − x. Note that
Thus by Lemma 2.2, we get |Jnk
n1
k0 pn1,k x 1, we have
n1
α
α − 1 2α
α
2 pn1,k x J x − γ x ≤ Dn,α sgnt − x, x .
nk
nk
α 1 k0
n 1x1 − x
3.5
Next we estimate |Dn,α gx , x|. From 15 of 1, it follows the inequality
n
Dn,α gx , x ≤ 4α nx1 − x 1
2
n2 x2 1 − x k1
√
x1−x/
k
√
x−x/ k
gx .
3.6
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Journal of Inequalities and Applications
That is,
2
n2 x2 1 − x Dn,α
n
gx , x ≤ 4αnx1 − x 1
√
x1−x/
k
√
x−x/ k
k1
gx .
3.7
On the other hand, note that gx x 0, we have
Dn,α gx , x ≤ Dn,α gx t − gx x, x
≤
1
gx Dn,α 1, x
3.8
0
1
gx ≤
0
√
n x1−x/
k
k1
√
x−x/ k
gx .
From 3.7 and 3.8 we obtain
n
Dn,α gx , x ≤ 4αnx1 − x 4α 4α
n2 x2 1 − x2 4α k1
√
x1−x/
k
√
x−x/ k
gx .
3.9
Using inequality
n2 x2 1 − x2 16α2 4α > 8αnx1 − x,
3.10
we get
4αnx1 − x 4α 4α
n2 x2 1
2
− x 4α
<
4α 1
,
nx1 − x
∀n > 1.
3.11
Thus from 3.9 we obtain
Dn,α gx , x ≤
n
4α 1 nx1 − x k1
√
x1−x/
k
√
x−x/ k
gx .
3.12
Theorem 1.1 now follows by collecting the estimations 3.3, 3.5, and 3.12.
Acknowledgment
The present work is supported by Project 2007J0188 of Fujian Provincial Science Foundation
of China.
Journal of Inequalities and Applications
7
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