IMPACT: International Journal of Research in Applied, Natural and Social Sciences (IMPACT: IJRANSS) ISSN(E): 2321-8851; ISSN(P): 2347-4580 Vol. 2, Issue 7, Jul 2014, 95-106 © Impact Journals
ON CERTAIN GENERALIZED MODIFIED BETA OPERATORS PRERNA MAHESHWARI SHARMA1 & SANGEETA GARG2 1
Department of Mathematics, SRM University, NCR Campus, Ghaziabad, Uttar Pradesh, India 2
Research Scholar, Mewar University, Chittorgarh, Rajasthan, India
ABSTRACT In the present paper we study about the Stancu variants of modified Beta operators. We obtain some direct results in simultaneous approximation and asymptotic formula for these operators. We also modify these operators so as to preserve the linear moments, by applying the King’s approach.
KEYWORDS: Simultaneous Approximation, Modified Beta Operators, King’s Approach, Stancu Type Operators, Asymptotic Formula, Peetre’s K-Functional
AMS: Primary 41a25, 41a35 1. INTRODUCTION Stancu type generalization of Bernstein polynomials was introduced by Stancu [10] as n k S nα ( f , x ) = ∑ f p nk,α ( x ), 0 ≤ x ≤ 1 k =0 n k −1
n k
where p n ,α ( x ) = k
(1.1)
n − k −1
∏ (x + αs ) ∏ (1 − x + αs ) s =0
s =0
n −1
∏ (1 + αs ) s =0
As a special case if α = 1, we get the classical Bernstein polynomials. Starting with two parameters
satisfying 0 ≤
≤
and
in the year 1983, Stancu [11] gave the other generalization of the operators (1.1) as follows
n k +α , S nα , β ( f , x ) = ∑ pn ,k ( x ) f k =0 k+β n k n−k where p n ,k ( x ) = x (1 − x ) , k
(1.2)
(1.3)
is the Bernstein basis function. In the year 2010, Buyukyazici and collaborators [1] and [2] have proposed the Stancu variants of several well known operators and estimated some direct results. Recently Gupta-Yadav [4] established some interesting results for BBS operators. Motivating with their work, we propose Stancu type generalization of modified Beta operators (1.1).
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96
Prerna Maheshwari Sharma & Sangeeta Garg
For0 ≤
≤ , Stancu type generalization for modified Beta operators is as follows α ,β
Pn
∞ nt + α n − 1 ∞ dt, x ∈ [0, ∞) ( f , x) = ∑ bn,v ( x )∫ p n,v (t ) f n v =0 n+β 0
where
1 + 1,
=
,
+
=
,
The operators
,
,
1+
−1
(1.4)
1+
are called modified Beta-Stancu operators. For
modified Beta operators defined by Gupta-Ahmad [4], as
=
= 0, the operators (1.4) reduce to the
∞
n −1 ∞ Pn ( f , x ) = ∑ bn,v ( x )∫ p n,v (t ) f (t )dt, x ∈ [0, ∞) n v =0 0 ,
,
and
(1.5)
are defined as above. Some approximation properties of these operators were discussed by
Maheshwari [6] and [7], Maheshwari-Gupta [8], and Maheshwari-Sharma [9] etc.
Also
,
In the analysis, Hewitt-Stromberg [5] showed that the operators 1,
are linear positive operators.
= 1, howsoever smooth the function may be, it turns out the order of approximation for the operators
(1.4) are at best of
2. BASIC RESULTS
.
In this section, we give certain lemmas, which will be useful for the proof of main theorem. Lemma-1 [3]: If for $
,%
=
1
& ∞
∈ " and )
,
'(
+1
∈ #0, ∞ ,
− *
%
then there exists the following recurrence relation+1 $
,%
Consequently • •
$
$
,% ,%
=
1+
is a polynomial in = -
#%
//0
+$ ′ ,%
+
$
,%
of degree ≤ m .
1 where #20 denotes the integral part of 2.
Lemma-2 [3]: There exists the polynomial 34,5,6 6
1+
6
86 8 6 1+
,
= &
/4 596, 4,5:(
+ 1 4# −
independent of + 1 05 34,5,6
7 8 , such that 1+
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97
On Certain Generalized Modified Beta Operators
Lemma-3: Let ; be < times < = 1,2,3, … differentiable on @ , then ;,
C PB, ,
=
−<−1 ! +<−1 ! ! −2 !
6
+
& ∞
E
∞
6,
'(
6,
(
;
6
+ +
6
8
Proof: We have PB,C ,
;,
=
−1
&
E
6 ,
∞
'(
∞
(
Using Leibnitz theorem for C PB, ,
;,
=
−1
×E
∞
−1
=
−1
=
& ∞
'(
< &&) * F 6
∞
4'(
(
'4
;
,
+ +< ! . −1 ! ! 1+
+<−1 ! & n! ∞
+ +
8 .
+ +<−F ! −1 −1 ! −F ! 8
∞ 6
E & −1
6
( 4'(
∞ 6
E & −1
6,
'(
+ +
;
,
( 4'(
6 4
6 4
< ) * F
1+
< ) * F
6 4
,
4
,
+ +
;
4
+ +
;
4
6 4
8
8
Again using Leibnitz theorem, we get 6
6,
6
=
< −1 ! & −1 4 ) * −<−1 ! F 6
4'(
,
4
Hence we get P, C,
;,
=
−<−1 ! +<−1 ! & −2 ! ! ∞
E −1 ∞
6,
'(
(
6
6
6,
6
;
+ +
8
Solving integrals by parts ‘r’ times, we have the required Lemma 6 , ,
;,
=
−<−1 ! +<−1 ! ! −2 !
× E
∞
(
6,
;
6
6
+
+ +
6
8
& ∞
'(
6,
Lemma-4: Let Us Define I6,
,
,%
=
−<−1 & +< ∞
'(
6,
E
∞
(
6,
6
+ +
−
%
8
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98
Prerna Maheshwari Sharma & Sangeeta Garg
Then, we have I6,
,
I6,
I
,
L
=
,
, ,6,/
/
= 1,
,(
−<−2
−
=
/
−
+ { 2< + 3 −<−2
+
−<−2 +2 +
+ <+1 }
{ 2< + 3 − −<−2
/
2.1
+ <+1 }
+
/
+
2.2
×
2 − 1 + 2< + 3 2 − 1 + 2< + 3 < + 2 N+ L N+ <+1 <+2 2< + 5 + 2< + 5 < + 1 −<−2 −<−3
2.3
and the following recurrence relation is satisfied
=
+
1+
−<−
OI6,
+O
− 2 I6,
,
+<+1 +
B β B
,
Q+
,%
)
−
+
L
+
− * n − r − 2m − 2 +
∈ #0, ∞ , we have
n+β , − 1N mI6, ,% n
−
+ < + 1 Q I6,
,
,%
(2.4)
,% .
To prove recurrence formula,
% O Q /
=
,%
,%
+ mI6,
, ,′ ,%
Further, for all I6,
,
where #S0 is the integral part of S.
Proof: We Have Some Identities to Use in Our Proof
• •
1+
t 1+t
′ ′
,
,
=# −
t =# −
+1 0
0
,
t
,
,
= 0,1,2 respectively in I6,
Obviously (2.1)-(2.3) can be obtained by taking
taking first derivative of I6, , ,′ I6, ,%
∴
, ,′ I6, ,%
=
+
,
,%
with respect to x,
−<−1 & +< I
∞
'(
, ,6,%
=
′
6,
E
∞
(
6,
−<−1 & +< ∞
'(
+ +
6
′
6,
E
∞
(
− 6,
%
6
Multiplying by x 1 + x on both sides and using identity (1), we have 1+
OI
, ,′ ,6,%
+
I
, ,6,%
8 −
,
I
+ +
, ,6,%
−
%
8
Q
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99
On Certain Generalized Modified Beta Operators
−<−1 & +<
=
1+
∞
'(
−r−1 &# − +<
=
+<+1 0
∞
'(
−r−1 & +<
=
∞
'(
−r−1 & +<
=
∞
'(
E#
6,
E
6,
×
(
−<
−r−1 & +<
= +
∞
+ < + 1 0I
Substituting +
+ +
W
E
−
−
+
6, 6,
(
, ,6,%
−
(
′
∞
6,
'(
E
∞
6,
1+
(
∞
−#< + =
E
−r−1 & +<
6 6,
6
6,
6
%
−
+ +
+ +
6
+ +
6
8
− −
%
%
−
6, %
%
−
+ +
+<+1 0
−< t−r−
∞
+ +
+<+1 x
+
6,
'(
−<
(
6,
(
−
∞
+<−
∞
E
∞
6,
′
8 8
+ +
6
8
−
%
8
8
X
and Using the Identity (2), We Have
=
1+
− + −2
+
+
+ +
OI
#
− /
+
, ,′ ,6,%
−
Q
, ,6,%
−<−1 & +<
. #
I
∞
6,
'(
∞
∞
.
E
6,
'(
E ∞
−<−1 & +< ∞
'(
(
6,
6,
′
(
6,
′
E
6
+ +
6
∞
(
+ +
6
∞
6,
'(
6,
′
(
−<−1 & +<
−<−1 & +<
E
∞
′
6,
6
−
+ +
−
+ +
%
−
8
%
% /
−
8 0
8
%
8
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100
Prerna Maheshwari Sharma & Sangeeta Garg
+
−
+
+
+ −
/
.
∞
−< # −
+
−#< +
−<−1 & +<
6,
'(
−<−1 & +<
(
∞
∞
+ < + 1 0I
'(
, ,6,%
E
E
6,
(
+ +
6
6,
(
∞
6,
6,
′
∞
6,
'(
−<−1 & +<
.
E
∞
6
+ +
6
+ +
%
− − −
8 0
% %
8
8 0
.
Now Solving the Integrals by Parts, We Get 1+
OI
=− + +
+1
+
+2 − + −
+
+
+
−#< + =
W1 − +)
I
−<−1 & +< ∞
'(
+2
−
'(
−<−1 & +<
∞
− < OI
, ,6,%
−
+
, ,6,%
'(
E
′
∞
(
E (
E 6,
′
∞
E
− 1X I
, ,6,%
(
6,
−
+ +
6 6,
6
6
6,
(
6,
′
(
E
∞
6,
∞
∞
6,
'(
6
6,
6,
−<−1 & +<
6,
′
6,
'(
'(
∞
+
E
+ +
6
∞
∞
−<−1 & +<
+ < + 1 0I
6,
′
(
−<−1 & +<
−<−1 & +<
.
−
6,
∞
−< # −
(
∞
+1 /
E
∞
6,
∞
−
+
Q
, ,6,%
−<−1 & +<
#−
+ 1 W2
*
+
'(
−
+
+
, ,′ ,6,%
6
+ +
+ +
6
+ +
%
8
−
+ +
%
+ +
−
%
−
−
−
8
%
% %
%
−
8 8
8 0 8
8 0
.
+
XI
, ,6,%
−)
− m+2
− *I
, ,6,%
+m
+
+
I
Q − #< +
−
, ,6,%
×
+ < + 1 0I
, ,6,%
.
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101
On Certain Generalized Modified Beta Operators
≥ 0.
On rearranging both sides, we get the required recurrence formula (2.4) for The Peetre’s K-Function The Peetre’s K-functional is defined by
Z/ ;, δ = F ;{‖; − \‖ + δ‖\"‖ ∶ \ ∈ _ / }
where
_ / = {\ ∈ `a #0, ∞ : \′ , \" ∈ `a #0, ∞ }, ∃ a positive constant ` > 0 such that Z/ ;, o ≤ `p/ ;, o , o > 0,
where the second order modulus of smoothness is given by p/ -;, δ
//
sup |;
1 = sup
+ 2ℎ − 2;
(qr9√δ (9tq∞
+ℎ +;
Corollary 1: Let o be a positive integer, then for all
constant x depending upon n−1
& ∞
and E
,
'(
such that y
,
|z {|:δ
8 ≤x
%
,
|.
> w > 0, and
∈ #0, ∞ , there exists a positive
∈ |.
3. MAIN RESULTS In this section we have some important theorems and obtain asymptotic formula. Theorem 1: Let ; ∈ } has a derivative of order (< + 2) at a fixed point
subinterval of @ . For some w > 0, ; → ∞, we have
lim €
;,
6 , ,
→∞
−;
×;
6
=
y
, as
•= r 2+r−β ;
6
+
1+‚ ;
6 /
6
Proof: Using Taylor Series Expansion, We Have 6 /
;
=& 4'(
where ƒ , €
=
6 , ,
6 , ,
;,
;
4
F!
−
−
−;
6 /
„& 4'(
;
4
6
F!
6 /
4
+ƒ ,
−
.
+# 1+<+
−
4
, …+
+ 3 + 2< −
‚0
6 /
is of exponential order as → ∞ and ƒ ,
•
∈ 0, ∞ . ; is bounded on every finite
6 , ,
ƒ ,
−
6 /
,
→ 0 as → . Now from Lemma 3, we have − ;
6
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102
Prerna Maheshwari Sharma & Sangeeta Garg
=
6 /
× †& 4'(
+ ˆ
=
×;
−<−2 ! +< ! n . ! −2 ! n+β
;
4
∞
−1
ˆ
6 /
&
E
6 ,
∞
'(
∞
(
−<−2 ! +< ! ! −2 ! +Š
= n−1 &
,6
'(
Tends to 0 as
ƒ
,
6
+
E
6 ,
∞
6
+
,6
E
∞
6,
'(
−<−2 ! +< ! . ! −2 !
where Š
−r−1 & +<
.
F!
C
∞
(
− 1‰ ;
6
− 1‰ ;
6
+ +
,
ƒ t,
,
(
6,
+ +
→ ∞ to show this we suppose ‹ =
+ + 6
∞
×E
∞
≤
≤
(
,
−1 Z
ƒ t,
−1 Z
× Ž& ∞
'(
,
where Z
'( /4 5q6; 4,5:(
&
/4 5q6; 4,5:(
&
•E
∞
(
,
6 /
−
4
∞
'(
„&
|ƒ ,
| –
,
∞
'(
6 /
−
1+
6
,
| –
Š
E
∞
(
+ 1 |/5 … /
|| − |6 / 8 • ‘
6
8 ,6
4
−
+
8 −;
n I 2!
6
‡
, ,6,/
, so that ‹ → 0 as
,
|5
+1
8
;
, ,6,
+ 1 |5 34,5,6
8
+1 & 4
+1
/4 5q6 4,5:(
= sup ’34,5,6 /4 596; 4,5:(
+ +
+1 4| −
+ +
6 /
−
+ nI
Hence using Lemma 2
|‹ | ≤ n − 1 & &
86 8 6
6
,
|ƒ ,
|| − |6 / 8
/
/
’
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3.1
→ ∞.
103
On Certain Generalized Modified Beta Operators
o ƒ,
Here 34,5,6
represents certain polynomials in
> 0 such that |ƒ , ∴ •E
∞
≤
(
1 −1
|| −
E
|6 /
,
|z {|q–
1 −1
+
E
,
|z {|:δ
&
'(
≤
•E
∞
,
1 & −1
(
∞
'(
= ƒ/ -
,
6 ›
E
∞
/
−
-ƒ ,
1
/
−
1+ -
−
1,
œ
/6 •
/6 •
/6 •
= €ϵ/ ≤ϵ
−1 Z
1 + -
1 .
&
6 ›
/4 5q6; 4,5:( œ
This shows that ‹ → 0 as
4 5
#
5
(
8
-ƒ ,
,
1 t−x /
/6 •
| ≤ x| − |y .
8 •
8
/
8 +˜
Now using Lemma 1, taking 3 > < + 2 and |‹ | ≤
8 • •E
∞
,
(
|| − |6 / 8 • ƒ/
,
(
8 • ≤ •E
∞
1
|ƒ ,
,
/
-ƒ ,
From Corollary 1, we have ∞
7 8 . For a given ƒ > 0, there exists a
| < ƒ for 0 < | − | < o; and for | − | ≥ o, we observe that |ƒ ,
|ƒ ,
,
and independent of
M/ š& −1 ∞
'(
E
,
|z {|:δ
8
,
−
, ,6,/
from Lemma 4 in (3.1).
/6 /y •
to be large enough, we get
0/ €ƒ / -
6 ›
1+ -
œ
1•/
1•/ → ∞.
Hence in order to prove the theorem, we substitute the values of I
, ,6,
and I
Suppose that `a #0, ∞ be the space of real valued continuous bounded functions ; on the interval [0,∞ ,
and let the norm ‖. ‖ on the space `a #0, ∞ be defined as ‖;‖ = sup |; (9{q∞
|.
Also using K-functional, defined in Section 4 and for ; ∈ `a #0, ∞ the usual modulus of continuity is given by
p ;, o = •ž
•ž |;
(qr9– (9{q∞
+ℎ −;
|
Therefore, let us modify the operators 1.5 as Impact Factor(JCC): 1.4507 - This article can be downloaded from www.impactjournals.us
104
Prerna Maheshwari Sharma & Sangeeta Garg
Ÿ
;,
,
=
;,
,
−;˜ +
n−2
Theorem 2: Let ; ∈ `a #0, ∞ , then ∀ ’
;,
,
where δ
−2
=
/
’ ≤ `p/ -;, δB
−;
−
n−2
+2
+2
/
+
−
/
+n 1+3 š+; +
∈ #0, ∞ , and
1 + p ˜;,
− n−2
+n 1+3
+
− n−2
−
∈ |, there exists an absolute constant Z > 0 such that
n−2
1+3
− n−2 +
/
3.2
+n 1+3 š. +
2n/ # n + 7 x / + n + 5 x + 10 + / n−2 n−3
Proof: Let \ ∈ _ / . Then from Taylor expansion
\
=\
+ \′
−
+ ¢{
z
Therefore from Lemma 4, we have Ÿ
\,
,
− \
Ÿ
= \′
− ž \" ž 8ž, ƒ#0, ∞ \
− ,
,
+ Ÿ
z
•E
,
− ž \" ž 8ž, •
{
Hence ’Ÿ ≤ ≤€
\,
,
B /
{
=δ
, /
For Ÿ ∴’
− \
, ,
,
B /
E {
−
‖\"‖ ;,
;,
{
/
B
,
’≤£ ›{
•E {
¤ +
•+˜ +
, we have Ÿ
−;
+ ¤; ˜ +
≤ 3‖; − \‖ + o
z
,
/
n
’≤’ −
,
,
+
− ž \" ž 8ž, • £
n−2 n−2 ;,
− n−2
− n−2
+n 1+3 +
+n 1+3 š ‖\"‖ +
≤ 3‖;‖.
; − \,
1+3
‖\"‖ + ¤; ˜ +
− ;−\
š−;
n−2
− ž¤ ’\" ž ’8ž
¤
− n−2
Taking infimum on RHS over all \ ∈ _ / , we get
’+’
/
,
\,
−\
+n 1+3 š−; +
’
¤
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105
On Certain Generalized Modified Beta Operators
’
,
;,
−;
’ ≤ 3Z/ );, δ/B
* + p ˜;,
n−2
− n−2
+n 1+3 š +
− n−2
+n 1+3 š +
According to the property of Peetre’s K-functional, we get ’
,
;,
−;
’ ≤ `p/ -;, δ
1 + p ˜;,
n−2
which is the required result and hence completes the proof of theorem.
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V. Gupta and A. Ahmad, “Simultaneous approximation by modified Beta operators,” Istanbul Univ. Fen. Fak. Mat. Derg. 54 (1995), 11-22.
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P. Maheshwari, “An inverse result in simultaneous approximation by modified Beta operators,” Georgian Math. J. 13 (2), (2006), 297-306.
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P. Maheshwari, “Saturation theorem for the combinations of modified Beta operators in Lp-space,” Applied Mathematics E-notes 7 (2007), 32-37.
8.
P. Maheshwari and V. Gupta, “On the rate of approximation by modified Beta operators,” Int. J. of Mathematics and Mathematical Sciences Vol. 2009, Art ID 205649, 9 pages.
9.
P. Maheshwari and D. Sharma, “Rate of Convergence for certain mixed family of linear positive operators,” J. of Mathematics and Applications, No. 33, 2009, (2010), 00-06.
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