Hindawi Publishing Corporation
International Journal of Mathematics and Mathematical Sciences
Volume 2009, Article ID 205649, 8 pages
doi:10.1155/2009/205649
Research Article
On Rate of Approximation by Modified
Beta Operators
Prerna Maheshwari (Sharma)1 and Vijay Gupta2
1
2
Department of Mathematics, SRM University, NCR, Campus Modinagar 201 204, India
School of Applied Sciences, Netaji Subhas Institute of Technology, Sector 3 Dwarka,
New Delhi 110 078, India
Correspondence should be addressed to
Prerna Maheshwari Sharma, mprerna anand@yahoo.com
Received 9 July 2009; Accepted 20 November 2009
Recommended by A. Zayed
We establish the rate of convergence for the modified Beta operators Bn f, x, for functions having
derivatives of bounded variation.
Copyright q 2009 P. M. Sharma and 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 on R as
∞
n − 1
Bn f, x bn,k x
n k0
∞
x ∈ R ,
pn,k tftdt,
0
1.1
where
xk
1
bn,k x ,
Bk 1, n 1 xnk1
pn,k t nk−1
k
tk 1 t−n−k .
1.2
These operators are linear positive operators and reproduce only the constantfunctions.
2
International Journal of Mathematics and Mathematical Sciences
We set
λn x, t n−1
n
t ∞
bn,k xpn,k sds.
0 k0
1.3
In particular, it can be observed by the definition of Beta function that
λn x, ∞ 1.
1.4
Gupta and Ahmad 1 estimated asymptotic formula and error estimate for these
operators in simultaneous approximation. Later Gupta et al. 2 studied 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.
By DBr 0, ∞, r ≥ 0 we mean the class of absolutely continuous functions f defined
on 0, ∞, which satisfy the following:
i ft Otr , t → ∞,
ii this class has a derivative f on the interval 0, ∞ coinciding a.e. with a function
which is of bounded variation on every finite subinterval of 0, ∞. It can be
observed that all functions f ∈ BDr 0, ∞ possess for each c > 0 the representation
fx fc x
ψtdt,
x ≥ c.
1.5
c
Very recently Gupta and Agrawal 3 and Gupta et al. 4 have obtained an interesting
result on the rate of convergence of certain Durrmeyer type operators in ordinary and
simultaneous approximation. İspir 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 2.1 1. Let the function Tn,m x, m ∈ ℵ0 , be defined as
∞
n − 1
Tn,m x bn,k x
n k0
∞
0
pn,k tt − xm dt.
2.1
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3
Then
Tn,0 x 1,
Tn,2 x Tn,1 x 3x 1
,
n−2
2n 7x2 2n 5x 2
.
n − 2n − 3
2.2
Also for each x ∈ 0, ∞, one has
Tn,m x O n−m1/2 .
2.3
Remark 2.2. From Lemma 2.1 and using Cauchy-Schwarz inequality, for n sufficiently large,
it follows that
Bn |t − x|, x ≤ Tn,2 x1/2 ≤
Cxx 1
,
n
2.4
where C > 2.
Lemma 2.3. Let x ∈ 0, ∞, then for C > 2 and n sufficiently large, one has
Cxx 1
λn x, y ≤ 2 ,
n x−y
1 − λn x, z ≤
Cxx 1
nz − x2
0 ≤ y < x,
2.5
x < z < ∞.
,
Proof. By using Lemma 2.1, for n sufficiently large, we have
n−1
λn x, y n
y ∞
≤ x−y
bn,k xpn,k tdt ≤
0 k0
−2
n−1
n
y
∞
x − t2 2 bn,k xpn,k tdt
0 x−y
k0
Cxx 1
Tn,2 x ≤ 2 .
n x−y
The proof of the second inequality is similar, and we skip the details.
3. Rate of Approximation
Our main result is stated as follows.
2.6
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Theorem 3.1. Let f ∈ DBr 0, ∞, r ∈ N, and x ∈ 0, ∞. Then for n sufficiently large, one has
⎞
⎛ √
√
n
xx/
n Cx 1 xx/k
x
Bn f, x − fx ≤
⎝
f x √
f x ⎠
n
n x−x/√n
k1 x−x/k
Cx 1 f2x − fx − xf x fx
nx
Cxx 1 −r/2 O n
f x n
1 3x 1
Cxx 1 f x − f x− f x f x− ,
4n
2
n−2
3.1
where the auxiliary function fx is given by
⎧
⎪
ft − fx− , 0 ≤ t < x,
⎪
⎪
⎨
fx t 0,
t x,
⎪
⎪
⎪
⎩
ft − fx , x < t < ∞.
b
a fx
3.2
denotes the total variation of fx on a, b.
Proof. By mean value theorem, we have
n−1
Bn f, x − fx n
∞ ∞
bn,k xpn,k t ft − fx dt
0 k0
∞ t
0
∞
n − 1
bn,k xpn,k t f udu dt.
n
x
k0
3.3
Applying the identity
f u 1 1
f x f x− f x u f x − f x− sgnu − x
2
2
1
f x − f x f x− χx u,
2
3.4
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5
Equation 3.3 becomes keeping in mind that the last term of the above identity vanishes
t
∞
∞
n − 1
Bn f, x − fx ≤ bn,k xpn,k tdt
f x udu
x
n k0
x
−
x t
0
x
f x udu
∞
n − 1
bn,k xpn,k tdt
n k0
3.5
1
f x − f x− Bn |t − x|, x
2
1
f x f x− Bn t − x, x.
2
As
n−1
n
∞ t
0
n−1
n
∞
− 1 bn,k xpn,k tdt
sgnu − xdu
f x − f x
x2
k0
1 f x − f x− Bn |t − x|, x,
2
∞ t
0
∞
1 f x f x− du
2
x
3.6
bn,k xpn,k tdt
k0
1 f x f x− Bn t − x, x.
2
Thus in view of the above values, Lemma 2.1, and Remark 2.2, 3.5 reduces to
Bn f, x − fx ≤ En f, x Fn f, x 1 f x − f x− 2
3x 1
1
.
f x f x− 2
n−2
Cxx 1
n
3.7
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In order to complete the proof of the theorem, it is sufficient to estimate the terms En f, x
√
and Fn f, x. Applying integration by parts, using Lemma 2.3 and taking y x − x/ n, we
have
t
x
Fn f, x f x udu dt λn x, t
0
x
x
0
λn x, t f x tdt ≤
y
x
0
y
f t|λn x, t|dt
x
x
x
x
1
Cxx 1 y dt
f x
f x dt
≤
2
n
x − t
0 t
y t
Cxx 1
≤
n
y
x
f x
0 t
3.8
x
x dt
f x .
√
2
√
n x−x/ n
x − t
1
Let u x/x − t. Then we have
Cxx 1
n
y
x
f x
0 t
Cxx 1
dt 2
n
x − t
1
√n x
f x du
1 x−x/u
√
n x Cxx 1 f x .
≤
n
k1 x−x/k
3.9
Thus,
√
n x
x
x
Fn f, x ≤ Cxx 1
f x √
f x .
n
n x−x/√n
k1 x−x/u
On the other hand, we have
t
∞
n − 1 ∞
En f, x bn,k xpn,k tdt
f x udu
n
x
x
k0
∞
n − 1∞ t bn,k xpn,k tdt
f x udu
n
2x
x
k0
f x udu dt 1 − λn x, t
x
2x t
x
3.10
International Journal of Mathematics and Mathematical Sciences
∞
n − 1∞ ≤
bn,k xpn,k tdt
ft − fx
n
2x
k0
7
∞
n − 1 ∞
f x t − x bn,k xpn,k tdt
n
2x
k0
2x
2x f t|1 − λn x, t|dt
f x udu|1 − λn x, 2x| x
x
x
≤
∞
C1 n − 1 ∞ bn,k xpn,k ttr |t − x|dt
x n
2x k0
∞
fx n − 1 ∞ bn,k xpn,k tt − x2 dt
n
x2
2x k0
n − 1
f x n
∞ ∞
Cx 1 f2x − fx − xf x bn,k xpn,k t|t − x|dt nx
2x k0
√
√
n n Cx 1 xx/k
x xx/
f x √
f x .
n
n
x
x
k1
3.11
For the estimation of the first two terms in the right-hand side of 3.11. we proceed as follows.
Applying Holder’s inequality, Remark 2.2, and Lemma 2.1, we have
C1 n − 1
x n
∞ ∞
∞
fx n − 1 ∞ r
bn,k xpn,k tt |t − x|dt bn,k xpn,k tt − x2 dt
n
x2
2x k0
2x k0
1/2 ∞
1/2
∞
∞
n − 1 ∞
2
2r
bn,k xpn,k tt dt
bn,k xpn,k tt − x dt
n
2x k0
0 k0
∞
fx n − 1 ∞ 2
bn,k xpn,k tt − x dt
n
x2
2x k0
Cxx 1 Cx 1
−r/2
.
fx
≤ C1 O n
√
nx
n
C1
≤
x
3.12
Also, by Remark 2.2, the third term of the right side of 3.11 is given by
n − 1
f x n
∞ ∞
∞
n − 1 ∞
bn,k xpn,k t|t − x|dt ≤ f x bn,k xpn,k t|t − x|dt
n
2x k0
0 k0
3.13
Cxx 1
≤ f x .
√
n
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Thus
−r/2
En f, x ≤ C1 O n
Cxx 1 Cx 1 Cxx 1
f x fx
√
√
nx
n
n
Cx 1 f2x − fx − xf x nx
3.14
√
√
n n Cx 1 xx/k
x xx/
f x √
f x .
n
n
x
x
k1
Combining the estimates 3.7, 3.10, and 3.14 we get the desired result. This completes the
proof of the theorem.
References
1 V. Gupta and A. Ahmad, “Simultaneous approximation by modified beta operators,” İstanbul
Üniversitesi Fen Fakültesi 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.
5 N. İspir, A. Aral, and O. Doğru, “On Kantorovich process of a sequence of the generalized linear
positive operators,” Numerical Functional Analysis and Optimization, vol. 29, no. 5-6, pp. 574–589, 2008.