The research paper published by IJSER journal is about Some approximation results on modified positive linear operators 1
ISSN 2229-5518
Some approximation results on modified positive linear operators
Dr. R .P. Pathak and Shiv Kumar Sahoo
Recently Deo N.et.al. [1] Introduced new Bernstein type special operators defined as,
Again Deo N.et.al. [1] gave the integral modification of the operators (1.1) which are defined as ,
and prove some approximation results on the operators
(1.2).
Singh S.P. [4] studied some approximation results on a
sequence of Szȧ sz type operators defined as,
which map the space of bounded continuous funtions into itself following [3].
Kasana H.S. et. el. [2] obtained a sequence of modified
Szậsz operators for integrable function on defined as,
Motivated by Deo N.et.al.[1] we introduce a sequence of positive linear operators which are defined as,
.
we shall study some approximation results on the operators
(1.5).
Again following Kasana H.S. et. el. [2] we introduce a sequence of positive linear operators which are defined as,
and shall study some approximation results on the operators (1.6).
In order to prove our main result, the following basic results are needed.
1.
2.
........(2.2)
IJSER © 2012
The research paper published by IJSER journal is about Some approximation results on modified positive linear operators 2
ISSN 2229-5518
3. 5.
4. 6.
)3 3+7( + )2 2+( + ) ….(2.4) 7.
We know that
Differentiating with respect to , we get
Multiplying both sides, we get
This completes the proof of (2.1).
Again differentiating (2.6) with respect to , we get
Multiplying both sides, we get
This completes the proof of (2.2).
In the same way after differentiations and calculations, we get required result s (2.3) and (2.4).
1.
2.
3.
4.
8.
9.
By putting in equation (1.5), we get
This completes the proof of (2.8).
By putting in equation (1.5), we get
IJSER © 2012
The research paper published by IJSER journal is about Some approximation results on modified positive linear operators 3
ISSN 2229-5518
This completes the proof of (2.9).
By putting in equation (1.5), we get
This completes the proof of (2.10).
In the same way by taking respectively in (1.5) and after little calculations we get required results (2.11) to (2.16).
This completes the proof.
In this section we shall give our main result.
bounded in the interval and let if exists at a point in , then one gets that
where are defined in (1.5).
interval and let if exists at a point in
, then one gets that
where are defined in (1.6).
Proof : Since exists at a point in , then
by using Taylor’s expansion, we write
where
Now for each , there corresponds such that
Again for then there exist a positive number such that
Thus for all and , we get
Applying on (3.6), we get
IJSER © 2012
The research paper published by IJSER journal is about Some approximation results on modified positive linear operators 4
ISSN 2229-5518
Since is arbitrary and small, we get
Thus
Multiplying both sides, we get
17−12 2−6 +6 2 +12 +2 +3 + ,
Here we write,
Using (3.2) in equation (3.4), we get
This completes the proof.
The authors are thankful to Director of National Institute of
Technology, Raipur (C.G.) for encouragement.
By choosing we get that
Dr. R .P. Pathak Associate Professor Department of Mathematics, National Institute of Technology, Raipur
Shiv Kumar Sahoo Department of Mathematics, National Institute of Technology, Raipur
G.E. Road, Raipur -492010
IJSER © 2012
The research paper published by IJSER journal is about Some approximation results on modified positive linear operators 5
ISSN 2229-5518
G.E. Road, Raipur -492010 (C.G.) India.
e-mail id:
r.p.pathak8@gmail.com.
(C.G.) India. e-mail id:
maths10sks@gmail.com.
IJSER © 2012