A New Estimate on the Rate of Convergence of Durrmeyer-B ´ezier Operators

Volume 2009, Article ID 702680,7pages doi:10.1155/2009/702680

Research Article

A New Estimate on the Rate of Convergence of Durrmeyer-B ´ezier Operators

Pinghua Wang

and Yali Zhou

1Department of Mathematics, Quanzhou Normal University, Fujian 362000, China

2Liming University, Quanzhou, Fujian 362000, China

Correspondence should be addressed to Pinghua Wang,[email protected] 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.

Copyrightq2009 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 D_n,α which are defined as follows:

Dn,α

f, x

n1ⁿ

k0

Q^α_nkx ₁

0

ftpnktdt, 1.1

where f is defined on 0,1, α ≥ 1, Q^α_nkx J_nk^α x− J_n,k1^α x, J_nkx _n

jkp_njx, k 0,1,2, . . . , n are B´ezier basis functions, and pnkx n!/k!n− k!x^k 1−x^n−k, k0,1,2, . . . , nare Bernstein basis functions.

Whenα1,D_n,1fis just the well-known Durrmeyer operator

Dn,1

f, x

n1ⁿ

k0

pnkx ₁

0

ftpnktdt. 1.2

Concerning the approximation properties of operators Dn,1f and some results on approximation of functions of bounded variation by positive linear operators, one can refer

to2–7. Authors of1studied the rate of convergence of the operatorsDn,αffor functions of bounded variation and presented the following important result.

Theorem A. Letfbe a function of bounded variation on0,1, (f∈BV0,1),α≥1,then for every x∈0,1andn≥1/x1−xone has

D_n,α f, x

− 1

α1fx α

α1fx− ≤ 8α nx1−x

n k1

x1−x/√

k x−x/√

k

g_x

2α

nx1−xfx−fx−,

1.3

where_b

agxis the total variation ofgxona, band

g_xt

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

ft−fx, x < t≤1,

0, tx,

ft−fx−, 0≤t < x.

1.4

Since the Durrmeyer-B´ezier operatorsD_n,αare an important approximation operator of new type, the purpose of this paper is to continue studying the approximation properties of the operatorsDn,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. Letf be a function of bounded variation on0,1, (f ∈ BV0,1),α ≥ 1,then for everyx∈0,1andn >1 one has

Dn,α

f, x

− 1

α1fx α

α1fx− ≤ 4α1 nx1−x

n k1

x1−x/√

k x−x/√

k

gx

α

n1x1−xfx−fx−,

1.5

wheregxtis defined in1.4.

It is obvious that the estimate1.5is better than the estimate1.3. More important, the estimate1.5is true for alln >1. This is an important improvement comparing with the fact that estimate1.3holds only forn≥1/x1−x.

2. Some Lemmas

In order to proveTheorem 1.1, we need the following preliminary results.

Lemma 2.1. Let{ξk}^∞_k1 be a sequence of independent and identically distributed random variables, ξ₁is a random variable with two-point distributionPξ1i xⁱ1−x¹⁻ⁱ(i0,1,andx∈0,1is

a parameter). Setηn_n

k1ξk,with the mathematical expectationEηn μn∈−∞,∞,and with the varianceDηn σ_n² >0.Then fork1,2, . . . , n1,one has

P

η_n≤k−1

−P

η_n1≤k≤ σ_n1

μ_n1, 2.1

P η_n≤k

−P

η_n1≤k≤ σ_n1

n1−μ_n1. 2.2

Proof. Sinceηn _n

k1ξk, from the distribution series ofξk, by convolution computation we get

P η_nj

n!

j!

n−j

!x^j1−x^n−j, 0≤j≤n. 2.3 Furthermore by direct computations we have

μ_n1 n1x, P

η_n j−1 j

n1xP

η_n1j

, 1≤j≤n1. 2.4

Thus we deduce that P

η_n≤k−1

−P

η_n1≤k

k j1

P

η_nj−1

−^k

j1

P

η_n1j

−P

η_n10

k j0

j n1x−1

P

η_n1j

≤ 1 n1x

k j0

j−n1xP

η_n1j

≤ 1 n1x

n1 j0

j−n1xP

η_n1j

≤ 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 inequality2.1is proved.

Similarly, by using the identities

n1−μ_n1 n11−x, P

ηn j

n1−j n11−xP

η_n1j

, 1≤j≤n1,

2.7

we get the inequality2.2.Lemma 2.1is proved.

Lemma 2.2. Letα≥1, k0,1,2, . . . , n,pnkx n!/k!n−k!x^k 1−x^n−k be Bernstein basis functions, and letJ_nkx _n

jkp_njxbe B´ezier basis functions, then one has J_nk^α x−J_n1,k1^α x≤ α

n1x1−x, J_nk^αx−J_n1,k^α x≤ α

n1x1−x.

2.8

Proof. Note that 0≤J_nkx, Jn1,k1x≤1, μ_n1 n1x, σ²_n1 n1x1−x, andα≥1.

Thus

J_nk^α x−J_n1,k1^α x≤α|Jnkx−J_n1,k1x|

α

n jk

p_nj− ⁿ¹

jk1

p_n1,j α

⎛

⎝1−ⁿ

jk

pnj

⎞

⎠−

⎛

⎝1− ⁿ¹

jk1

p_n1,j

⎞

⎠ αP

η_n≤k−1

−P

η_n1≤k.

2.9

Now by inequality2.1ofLemma 2.1we obtain J_nk^α x−J_n1,k1^α x≤α 1−x

n1x1−x ≤ α

n1x1−x. 2.10

Similarly, by using inequality2.2, we obtain J_nk^α x−J_n1,k^α x≤α x

n1x1−x ≤ α

n1x1−x. 2.11

ThusLemma 2.2is proved.

3. Proof of Theorem 1.1

Letfsatisfy the conditions ofTheorem 1.1, thenfcan be decomposed as ft 1

α1fx α

α1fx− gxt fx−fx−

2

sgnt−x α−1 α1

δ_xt

fx−1

2fx−1 2fx−

,

3.1

where

sgnt

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

1, t >0 0, t0,

−1, t <0,

δ_xt

⎧⎨

⎩

0, t /x,

1, tx. 3.2

ObviouslyD_n,αδx, x 0,thus from3.1we get D_n,α

f, x

− 1

α1fx α

α1fx−

≤Dn,α

gx,, x

fx−fx−

2

Dn,α

sgnt−x, x α−1

α1 .

3.3

We first estimate|Dn,αsgnt−x, x α−1/α1|, from1, page 11we have the following equation:

Dn,α

sgnt−x, x α−1

α1 2

n1

k0

p_n1,kxJ_nk^α x−2 n1 k0

p_n1,kxγ_nk^α x, 3.4

whereJ_n1,k1^α x< γ_nk^αx< J_n1,k^α x.

Thus by Lemma 2.2, we get |J_nk^α x − γ_nk^α x| ≤ α/

n1x1−x. Note that _n1

k0p_n1,kx 1, we have Dn,α

sgnt−x, x α−1

α1

2 n1 k0

p_n1,kx

J_nk^α x−γ_nk^α x

≤ 2α

n1x1−x. 3.5

Next we estimate|Dn,αgx, x|. From15of1, it follows the inequality D_n,α

g_x, x≤4αnx1−x 1 n²x²1−x²

n k1

x1−x/√

k x−x/√

k

g_x

. 3.6

That is,

n²x²1−x²D_n,α

g_x, x≤4αnx1−x 1ⁿ

k1

x1−x/√

k x−x/√

k

g_x

. 3.7

On the other hand, note thatg_xx 0, we have D_n,α

g_x, x≤D_n,αg_xt−g_xx, x

≤¹

0

gx

Dn,α1, x

¹

0

gx

≤ⁿ

k1

x1−x/√

k x−x/√

k

gx

.

3.8

From3.7and3.8we obtain D_n,α

g_x, x≤ 4αnx1−x 4α4α n²x²1−x²4α

n k1

x1−x/√

k x−x/√

k

g_x

. 3.9

Using inequality

n²x²1−x²16α²4α >8αnx1−x, 3.10

we get

4αnx1−x 4α4α

n²x²1−x²4α < 4α1

nx1−x, ∀n >1. 3.11

Thus from3.9we obtain Dn,α

gx, x≤ 4α1 nx1−x

n k1

x1−x/√

k x−x/√

k

gx

. 3.12

Theorem 1.1now follows by collecting the estimations3.3,3.5, and3.12.

Acknowledgment

The present work is supported by Project 2007J0188 of Fujian Provincial Science Foundation of China.

References

1 X.-M. Zeng and W. Chen, “On the rate of convergence of the generalized Durrmeyer type operators for functions of bounded variation,” Journal of Approximation Theory, vol. 102, no. 1, pp. 1–12, 2000.

2 R. Bojani´c and F. H. Chˆeng, “Rate of convergence of Bernstein polynomials for functions with derivatives of bounded variation,” Journal of Mathematical Analysis and Applications, vol. 141, no. 1, pp. 136–151, 1989.

3 M. M. Derriennic, “Sur l’approximation de fonctions int´egrables sur 0, 1 par des polyn ˆomes de Bernstein modifies,” Journal of Approximation Theory, vol. 31, no. 4, pp. 325–343, 1981.

4 S. S. Guo, “On the rate of convergence of the Durrmeyer operator for functions of bounded variation,”

Journal of Approximation Theory, vol. 51, no. 2, pp. 183–192, 1987.

5 V. Gupta and R. P. Pant, “Rate of convergence for the modified Sz´asz-Mirakyan operators on functions of bounded variation,” Journal of Mathematical Analysis and Applications, vol. 233, no. 2, pp. 476–483, 1999.

6 A. N. Shiryayev, Probability, vol. 95 of Graduate Texts in Mathematics, Springer, New York, NY, USA, 1984.

7 X.-M. Zeng and V. Gupta, “Rate of convergence of Baskakov-B´ezier type operators for locally bounded functions,” Computers & Mathematics with Applications, vol. 44, no. 10-11, pp. 1445–1453, 2002.