Volume 2009, Article ID 702680, 7 pagesdoi:10.1155/2009/702680 Research Article A New Estimate on the Rate of Convergence of Durrmeyer-B ´ezier Operators 1 Department of Mathematics, Qua
Trang 1Volume 2009, Article ID 702680, 7 pages
doi:10.1155/2009/702680
Research Article
A New Estimate on the Rate of Convergence of
Durrmeyer-B ´ezier Operators
1 Department of Mathematics, Quanzhou Normal University, Fujian 362000, China
2 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-B´ezier operaters for functions
of bounded variation by means of some probabilistic methods and inequality techniques Our estimate improves the result of Zeng and Chen2000
Copyrightq 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-B´ezier operators Dn,α which are defined as follows:
D n,α
f, x
n 1n
k0
Q α nk x
1
0
f tp nk tdt, 1.1
where f is defined on 0, 1, α ≥ 1, Q α nk x J α
nk x − J α
n,k1x, J nk x n
j k p nj x,
k 0, 1, 2, , n are B´ezier basis functions, and p nk x n!/k!n − k!x k 1 − x n −k,
k 0, 1, 2, , n are Bernstein basis functions.
When α 1, D n,1 f is just the well-known Durrmeyer operator
D n,1
f, x
n 1n
k0
p nk x
1
0
f tp nk tdt. 1.2
Concerning the approximation properties of operators D n,1 f and some results on
approximation of functions of bounded variation by positive linear operators, one can refer
Trang 2to2 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
D n,α
f, x
−
1
α 1f x
α
α 1f x−
≤ nx 1 − x 8α n
k1
x 1−x/√k
x −x/√k
g x
2α
nx 1 − xf x − fx−, 1.3
where b
a g x is the total variation of g x on a, b and
g x t
⎧
⎪
⎨
⎪
⎩
f t − fx, x < t ≤ 1,
0, t x,
f t − fx−, 0 ≤ t < x.
1.4
Since the Durrmeyer-B´ezier operators D n,αare an important approximation operator
of new type, the purpose of this paper is to continue studying the approximation properties
of the operators D n,αfor 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
D n,α
f, x
−
1
α 1f x
α
α 1f x−
≤ nx 4α 1 − x 1 n
k1
x 1−x/√k
x −x/√k
g x
α
n 1x1 − xf x − fx−, 1.5
where g x t is defined in 1.4.
It is obvious that the estimate1.5 is better than the estimate 1.3 More important, the estimate1.5 is true for all n > 1 This is an important improvement comparing with the fact that estimate1.3 holds only for n ≥ 1/x1 − x
2 Some Lemmas
In order to proveTheorem 1.1, we need the following preliminary results
Lemma 2.1 Let {ξ k}∞k1be a sequence of independent and identically distributed random variables,
ξ1is a random variable with two-point distribution P ξ1 i x i 1−x1−i(i 0, 1, and x ∈ 0, 1 is
Trang 3a parameter) Set η nn
k1ξ k , with the mathematical expectation E η n μ n ∈ −∞, ∞, and with
the variance D η n σ2
n > 0 Then for k 1, 2, , n 1,one has
P
η n ≤ k − 1− Pη n1≤ k ≤ σ n1
μ n1, 2.1
P
η n ≤ k− Pη n1≤ k ≤ σ n1
n 1 − μ n1. 2.2
Proof Since η n n
k1ξ k , from the distribution series of ξ k, by convolution computation we get
P
η n j n!
j!
n − j!x
j 1 − x n −j , 0≤ j ≤ n. 2.3
Furthermore by direct computations we have
μ n1 n 1x,
P
η n j − 1 n 1x j P
η n1 j, 1≤ j ≤ n 1. 2.4
Thus we deduce that
P
η n ≤ k − 1− Pη n1≤ k
k
j1
P
η n j − 1−k
j1
P
η n1 j− Pη n1 0
k
j0
j
n 1x− 1
P
η n1 j
≤ 1
n 1x
k
j0
j − n 1xP
η n1 j
≤ n 1x1 n1
j0
j − n 1xP
ηn1 j
≤ 1
μ n1Eη n1− μ n1.
2.5
By Schwarz’s inequality, it follows that
1
μ n1Eη n1− μ n1 ≤E
η n1− μ n12
μ n1 σ n1
μ n1. 2.6 The inequality2.1 is proved
Trang 4Similarly, by using the identities
n 1 − μ n1 n 11 − x,
P
η n j n 11 − x n 1 − j P
η n1 j, 1≤ j ≤ n 1, 2.7
we get the inequality2.2.Lemma 2.1is proved
Lemma 2.2 Let α ≥ 1, k 0, 1, 2, , n, p nk x n!/k!n − k!x k 1 − x n −k be Bernstein basis functions, and let J nk x n
j k p nj x be B´ezier basis functions, then one has
J α
nk x − J α
n 1,k1 x ≤ α
n 1x1 − x ,
J α
nk x − J α
n 1,k x ≤ α
n 1x1 − x .
2.8
Proof Note that 0 ≤ J nk x, J n 1,k1 x ≤ 1, μ n1 n 1x, σ2
n1 n 1x1 − x, and α ≥ 1.
Thus
J α
nk x − J α
n 1,k1 x ≤ α|J
nk x − J n 1,k1 x|
α
n
j k
p nj− n1
j k1
p n 1,j
α
⎛
⎝1 −n
j k
p nj
⎞
⎠ −
⎛
⎝1 − n1
j k1
p n 1,j
⎞
⎠
αP
η n ≤ k − 1− Pη n1≤ k.
2.9
Now by inequality2.1 ofLemma 2.1we obtain
J α
nk x − J α
n 1,k1 x ≤ α 1− x
n 1x1 − x ≤
α
n 1x1 − x . 2.10
Similarly, by using inequality2.2, we obtain
J α
nk x − J α
n 1,k x ≤ α x
n 1x1 − x ≤
α
n 1x1 − x . 2.11
ThusLemma 2.2is proved
Trang 53 Proof of Theorem 1.1
Let f satisfy the conditions ofTheorem 1.1, then f can be decomposed as
f t 1
α 1f x
α
α 1f x− g x t
f x − fx−
2
sgnt − x α− 1
α 1
δ x t
f x −1
2f x −1
2f x−
,
3.1
where
sgnt
⎧
⎪
⎨
⎪
⎩
1, t > 0
0, t 0,
−1, t < 0,
δ x t
⎧
⎨
⎩
0, t / x,
1, t x. 3.2
Obviously D n,α δ x , x 0, thus from 3.1 we get
D n,α
f, x
−
1
α 1f x
α
α 1f x−
≤D n,α
g x, , x f x − fx−
2
D n,α
sgnt − x, xα− 1
α 1
.
3.3
We first estimate|D n,α sgnt − x, x α− 1/α 1|, from 1, page 11 we have the following
equation:
D n,α
sgnt − x, xα− 1
α 1 2
n1
k0
p n 1,k xJ α
nk x − 2n1
k0
p n 1,k xγ α
nk x, 3.4
where J n α 1,k1 x < γ α
nk x < J α
n 1,k x.
Thus by Lemma 2.2, we get |J α
nk x − γ α
nk x| ≤ α/ n 1x1 − x Note that
n1
k0p n 1,k x 1, we have
D n,α
sgnt − x, xα− 1
α 1
2
n1
k0
p n 1,k xJ nk α x − γ α
nk x
≤
2α
n 1x1 − x . 3.5
Next we estimate|D n,α g x , x| From 15 of 1, it follows the inequality
D n,α
g x , x ≤ 4αnx1 − x 1
n2x21 − x2
n
k1
x 1−x/√k
x −x/√k
g x
Trang 6That is,
n2x21 − x2D n,α
g x , x ≤ 4αnx1 − x 1n
k1
x 1−x/√k
x −x/√k
g x
. 3.7
On the other hand, note that g x x 0, we have
D n,α
g x , x ≤ D n,αg x t − g x x, x
≤1
0
g x
D n,α 1, x
1
0
g x
≤n
k1
x 1−x/√k
x −x/√k
g x
.
3.8
From3.7 and 3.8 we obtain
D n,α
g x , x ≤ 4αnx1 − x 4α 4α
n2x21 − x2 4α
n
k1
x 1−x/√k
x −x/√k
g x
. 3.9
Using inequality
n2x21 − x2 16α2 4α > 8αnx1 − x, 3.10
we get
4αnx1 − x 4α 4α
n2x21 − x2 4α <
4α 1
nx 1 − x , ∀n > 1. 3.11
Thus from3.9 we obtain
D n,α
g x , x ≤ 4α 1
nx 1 − x
n
k1
x 1−x/√k
x −x/√k
g x
Theorem 1.1now follows by collecting the estimations3.3, 3.5, and 3.12
Acknowledgment
The present work is supported by Project 2007J0188 of Fujian Provincial Science Foundation
of China
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