On Rate of Approximation by Modified Beta Operators

Volume 2009, Article ID 205649,8pages doi:10.1155/2009/205649

Research Article

On Rate of Approximation by Modified

Beta Operators

Prerna Maheshwari (Sharma)

1

_{and Vijay Gupta}

2

1_{Department of Mathematics, SRM University, NCR, Campus Modinagar 201 204, India} 2_{School of Applied Sciences, Netaji Subhas Institute of Technology, Sector 3 Dwarka,}

New Delhi 110 078, India

Correspondence should be addressed to

Prerna MaheshwariSharma,mprerna [email protected]

Received 9 July 2009; Accepted 20 November 2009

Recommended by A. Zayed

We establish the rate of convergence for the modified Beta operatorsBnf, x, for functions having

derivatives of bounded variation.

Copyrightq2009 P. M.Sharmaand V. Gupta. 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 the year 1995 Gupta and Ahmad 1 proposed modified Beta operators so as to approximate Lebesgue integrable functions onRas

Bn

f, x n−1 n

∞

k0

bn,kx

_∞

0

pn,ktftdt, x∈R, 1.1

where

bn,kx 1

Bk1, n xk

1xnk1, pn,kt

nk−1

k

tk1t−n−k. 1.2

We set

λnx, t n−1

n

t

0 ∞

k0

bn,kxpn,ksds. 1.3

In particular, it can be observed by the definition of Beta function that

λnx,∞ 1. 1.4

Gupta and Ahmad 1 estimated asymptotic formula and error estimate for these operators in simultaneous approximation. Later Gupta et al.2studied the approximation properties of these operators in Lp norm and they obtained some direct results for the linear combinations. Here we continue our studies on these operators. First we consider the following class of functions.

ByDBr0,∞, r ≥ 0 we mean the class of absolutely continuous functionsfdefined on0,∞, which satisfy the following:

ift Otr_{, t → ∞,}

iithis class has a derivativefon the interval0,∞coinciding a.e. with a function which is of bounded variation on every finite subinterval of 0,∞. It can be observed that all functionsf∈BDr0,∞possess for eachc >0 the representation

fx fc

x

c

ψtdt, x≥c. 1.5

Very recently Gupta and Agrawal3and Gupta et al.4have obtained an interesting result on the rate of convergence of certain Durrmeyer type operators in ordinary and simultaneous approximation. ˙Ispir et al. 5 estimated similar results for Kantorovich operators for functions with derivatives of bounded variation. We now extend the study for the modified Beta operators and estimate the rate of convergence of the operators 1 for functions having derivatives of bounded variation.

2. Auxiliary Results

We shall use the following lemmas to prove our main theorem.

Lemma 2.11. Let the functionTn,mx,m∈ ℵ0, be defined as

Tn,mx n− 1

n

∞

k0

bn,kx

_∞

0

Then

Tn,0x 1, Tn,1x 3x1

n−2 ,

Tn,2x

2n7x2_2n5x2 n−2n−3 .

2.2

Also for eachx∈0,∞, one has

Tn,mx O

n−m1/2 . 2.3

Remark 2.2. FromLemma 2.1and using Cauchy-Schwarz inequality, fornsufficiently large, it follows that

Bn|t−x|, x≤Tn,2x1/2≤

Cxx1

n , 2.4

whereC >2.

Lemma 2.3. Letx∈0,∞, then forC >2 andnsufficiently large, one has

λn

x, y≤ Cxx1

nx−y2, 0≤y < x,

1−λnx, z≤ Cxx 1

nz−x2, x < z <∞.

2.5

Proof. By usingLemma 2.1, fornsufficiently large, we have

λn

x, y n−1 n

_y

0 ∞

k0

bn,kxpn,ktdt≤ n− 1

n

_y

0

x−t2

x−y2

∞

k0

bn,kxpn,ktdt

≤x−y−2Tn,2x≤ Cxx1

nx−y2.

2.6

The proof of the second inequality is similar, and we skip the details.

3. Rate of Approximation

Theorem 3.1. Letf ∈DBr0,∞,r∈N, andx∈0,∞. Then fornsufficiently large, one has

_Bn

f, x−fx≤ Cx1 n

⎛

⎝

√_n

k1

xx/k

x−x/k

f_x√x n

xx/√n

x−x/√n

f_x

⎞ ⎠

Cx1

nx f2x−fx−xf

_xfx

Cxx1

n

On−r/2 fx

Cxx1 4n f

_x−fx−1

2f

_{x fx}−3x1

n−2 ,

3.1

where the auxiliary functionfxis given by

fxt

⎧ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

ft−fx−, 0≤t < x,

0, tx,

ft−fx, x < t <∞.

3.2

_b

afxdenotes the total variation offxona, b.

Proof. By mean value theorem, we have

Bn

f, x−fx n−1 n

_∞

0 ∞

k0

bn,kxpn,kt

ft−fxdt

_∞

0 t

x

n−1

n

∞

k0

bn,kxpn,kt

fududt.

3.3

Applying the identity

fu 1

2

fx fx−f_xu 1

2

fx−fx−sgnu−x

fx−1

2

fx fx−χxu,

Equation3.3becomeskeeping in mind that the last term of the above identity vanishes

_Bn

f, x−fx≤

_∞

x

t

x

f_xudu

n−1

n

∞

k0

bn,kxpn,ktdt

− _x

0

t

x

f_xudu

n−1

n

∞

k0

bn,kxpn,ktdt

1 2f

_x−fx−_B

n|t−x|, x

1 2f

_{x fx}−_B

nt−x, x.

3.5

As

n−1

n

_∞

0

t

x 1 2

fx−fx−sgnu−xdu

_∞

k0

bn,kxpn,ktdt

1 2

fx−fx−Bn|t−x|, x,

n−1

n

_∞

0

t

x 1 2

fx fx−du

_∞

k0

bn,kxpn,ktdt

1 2

fx fx−Bnt−x, x.

3.6

Thus in view of the above values,Lemma 2.1, andRemark 2.2,3.5reduces to

_Bn

f, x−fx≤En

f, xFn

f, x 1

2f

_x−fx−

Cxx1

n

1 2f

_{x fx}−3x1

n−2 .

In order to complete the proof of the theorem, it is sufficient to estimate the termsEnf, x andFnf, x.Applying integration by parts, usingLemma 2.3and takingy x−x/√n, we have _Fn f, x x 0 t x

f_xudu

dtλnx, t

x

0

λnx, t

f_xtdt≤

y 0 x y _f

xt|λnx, t|dt

≤ Cxx1

n y 0 x t

f_x 1

x−t2dt

x y x t

f_xdt

≤ Cxx1

n _y 0 x t

f_x 1

x−t2dt x

√

n

x

x−x/√n

f_x.

3.8

Letux/x−t.Then we have

Cxx1

n y 0 x t

f_x 1

x−t2dt

Cxx1

n

√_n

1

x

x−x/u

f_xdu

≤ Cxx1

n

√n

k1

x

x−x/k

f_x.

3.9

Thus,

_Fn

f, x≤ Cxx1 n

√n

k1

x

x−x/u

f_x√x

n

x

x−x/√n

f_x. 3.10

On the other hand, we have

_En

f, xn−1 n _∞ x t x

f_xudu

_∞

k0

bn,kxpn,ktdt

n−1

n _∞ 2x t x

f_xudu

_∞

k0

bn,kxpn,ktdt

2x x t x

f_xudu

dt1−λnx, t

≤n−1

n

_∞

2x

ft−fx

∞

k0

bn,kxpn,ktdt

fxn−1 n

_∞

2x t−x

∞

k0

bn,kxpn,ktdt

2x x

f_xudu|1−λnx,2x|

2x

x

_f

xt|1−λnx, t|dt

≤ C1

x n−1

n

_∞

2x

∞

k0

bn,kxpn,kttr|t−x|dt

fx

x2

n−1

n

_∞

2x

∞

k0

bn,kxpn,ktt−x2dt

fxn−1 n

_∞

2x

∞

k0

bn,kxpn,kt|t−x|dtCx 1

nx f2x−fx−xf

_x

Cx1

n

√n

k1

xx/k

x

f_x√x

n

xx/√n

x

f_x.

3.11

For the estimation of the first two terms in the right-hand side of3.11. we proceed as follows. Applying Holder’s inequality,Remark 2.2, andLemma 2.1, we have

C1

x n−1

n

_∞

2x

∞

k0

bn,kxpn,kttr|t−x|dt

_fx

x2

n−1

n

_∞

2x

∞

k0

bn,kxpn,ktt−x2dt

≤ C1

x

n−1

n

_∞

2x

∞

k0

bn,kxpn,ktt2rdt

1/2_∞

0 ∞

k0

bn,kxpn,ktt−x2dt

1/2

fx

x2

n−1

n

_∞

2x

∞

k0

bn,kxpn,ktt−x2dt

≤C1O

n−r/2 Cx_√x1

n fx

Cx1

nx .

3.12

Also, byRemark 2.2, the third term of the right side of3.11is given by

_fxn−1

n

_∞

2x

∞

k0

bn,kxpn,kt|t−x|dt≤fxn−1

n

_∞

0 ∞

k0

bn,kxpn,kt|t−x|dt

≤fx

Cxx1

√

n .

Thus

En

f, x≤C1O

n−r/2 Cx√x1

n fx

Cx1

nx f

_x

Cxx1

√

n

Cx1

nx f2x−fx−xf

_x

Cx1

n

√n

k1

xx/k

x

f_x√x n

xx/√n

x

f_x.

3.14

Combining the estimates3.7,3.10, and3.14we get the desired result. This completes the proof of the theorem.

References

1 V. Gupta and A. Ahmad, “Simultaneous approximation by modified beta operators,” ˙Istanbul

¨

Universitesi Fen Fak ¨ultesi Matematik Dergisi, vol. 54, pp. 11–22, 1995.

2 V. Gupta, P. Maheshwari, and V. K. Jain, “Lp-inverse theorem for modified beta operators,” International

Journal of Mathematics and Mathematical Sciences, vol. 2003, no. 20, pp. 1295–1303, 2003.

3 V. Gupta and P. N. Agrawal, “Rate of convergence for certain Baskakov Durrmeyer type operators,”

Annals of Oradea University, Fascicola Matematica, vol. 14, pp. 33–39, 2007.

4 V. Gupta, P. N. Agrawal, and A. R. Gairola, “On the integrated Baskakov type operators,” Applied

Mathematics and Computation, vol. 213, no. 2, pp. 419–425, 2009.