Journal of Inequalities in Pure and
Applied Mathematics
ON SIMULTANEOUS APPROXIMATION FOR CERTAIN BASKAKOV
DURRMEYER TYPE OPERATORS
VIJAY GUPTA, MUHAMMAD ASLAM NOOR
AND MAN SINGH BENIWAL
School of Applied Sciences
Netaji Subhas Institute of Technology
Sector 3 Dwarka
New Delhi 110075, India
EMail: vijay@nsit.ac.in
Mathematics Department
COMSATS Institute of Information Technology
Islamabad, Pakistan
EMail: noormaslam@hotmail.com
Department of Applied Science
Maharaja Surajmal Institute of Technology
C-4, Janakpuri, New Delhi - 110058, India
EMail: man_s_2005@yahoo.co.in
Department of Mathematics
Ch Charan Singh University
Meerut 250004, India
EMail: mkgupta2002@hotmail.com
volume 7, issue 4, article 125,
2006.
Received 06 January, 2006;
accepted 16 August, 2006.
Communicated by: N.E. Cho
Abstract
Contents
JJ
J
II
I
Home Page
Go Back
Close
c 2000 Victoria University
ISSN (electronic): 1443-5756
009-06
Quit
Abstract
In the present paper, we study a certain integral modification of the well known
Baskakov operators with the weight function of Beta basis function. We establish pointwise convergence, an asymptotic formula an error estimation and an
inverse result in simultaneous approximation for these new operators.
2000 Mathematics Subject Classification: 41A30, 41A36.
Key words: Baskakov operators, Simultaneous approximation, Asymptotic formula,
Pointwise convergence, Error estimation, Inverse theorem.
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
The work carried out when the second author visited Department of Mathematics
and Statistics, Auburn University, USA in fall 2005.
The authors are thankful to the referee for making many valuable suggestions, leading to the better presentation of the paper.
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Contents
Title Page
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3
Direct Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4
Inverse Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
References
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 2 of 33
1.
Introduction
For
f ∈ Cγ [0, ∞) ≡ {f ∈ C[0, ∞) : |f (t)| ≤ M tγ
for some M > 0, γ > 0} we consider a certain type of Baskakov-Durrmeyer
operator as
Z ∞
∞
X
(1.1)
Bn (f (t), x) =
pn,k (x)
bn,k (t)f (t)dt + (1 + x)−n f (0)
0
Zk=1∞
=
Wn (x, t)f (t)dt
0
where
n+k−1
xk
pn,k (x) =
,
k
(1 + x)n+k
tk−1
1
bn,k (t) =
·
B(n + 1, k) (1 + t)n+k+1
and
Wn (x, t) =
∞
X
pn,k (x)bn,k (t) + (1 + x)−n δ(t),
k=1
δ(t) being the Dirac delta function. The norm- || · ||γ on the class Cγ [0, ∞) is
defined as ||f ||γ = sup |f (t)|t−γ .
0≤t<∞
The operators defined by (1.1) are the integral modification of the well known
Baskakov operators with weight functions of some Beta basis functions. Very
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 3 of 33
recently Finta [2] also studied some other approximation properties of these operators. The behavior of these operators is very similar to the operators recently
introduced in [6], [9] and also studied in [8]. These operators reproduce not
only the constant functions but also the linear functions, which is the interesting
property of such operators. The other usual Durrmeyer type integral modification of the Baskakov operators [5] reproduce only the constant functions, so one
can not apply the iterative combinations easily to improve the order of approximation for the usual Baskakov Durrmeyer operators. For recent work in this
area we refer to [7]. In the present paper we study some direct results which
include pointwise convergence, asymptotic formula, error estimation and inverse theorem in the simultaneous approximation for the unbounded functions
of growth of order tγ .
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 4 of 33
2.
Basic Results
In this section we mention certain lemmas which will be used in the sequel.
Lemma 2.1 ([3]). For m ∈ N ∪ {0}, if the mth order moment be defined as
Un,m (x) =
∞
X
k=0
pn,k (x)
k
−x
n
m
,
then Un,0 (x) = 1, Un,1 (x) = 0 and
(1)
nUn,m+1 (x) = x(1 + x)(Un,m
(x) + mUn,m−1 (x)).
Consequently we have Un,m (x) = O n−[(m+1)/2] .
Lemma 2.2. Let the function Tn,m (x), m ∈ N ∪ {0}, be defined as
Tn,m (x) = Bn (t − x)m x
Z ∞
∞
X
pn,k (x)
bn,k (t)(t − x)m dt + (1 + x)−n (−x)m .
=
k=1
0
2x(1+x)
n−1
Then Tn,0 (x) = 1, Tn,1 = 0, Tn,2 (x) =
and also there holds the recurrence relation
(1)
(n − m)Tn,m+1 (x) = x(1 + x) Tn,m
(x) + 2mTn,m−1 (x) + m(1 + 2x)Tn,m (x).
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 5 of 33
Proof. By definition, we have
(1)
Tn,m
(x)
=
∞
X
(1)
pn,k (x)
Z
∞
bn,k (t)(t − x)m dt
0
k=1
−m
∞
X
Z
pn,k (x)
∞
bn,k (t)(t − x)m−1 dt
0
k=1
− n(1 + x)−n−1 (−x)m − m(1 + x)−n (−x)m−1 .
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Using the identities
(1)
x(1 + x)pn,k (x) = (k − nx)pn,k (x)
and
(1)
t(1 + t)bn,k (t) = [(k − 1) − (n + 2)t]bn,k (t),
we have
Title Page
(1)
x(1 + x) Tn,m
(x) + mTn,m−1 (x)
Z ∞
∞
X
pn,k (x)
(k − nx)bn,k (t)(t − x)m dt + n(1 + x)−n (−x)m+1
=
=
k=1
∞
X
k=1
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
0
Z
pn,k (x)
Contents
JJ
J
II
I
∞
(k − 1) − (n + 2)t + (n + 2)(t − x)
0
+ (1 + 2x) bn,k (t)(t − x)m dt + n(1 + x)−n (−x)m+1
Go Back
Close
Quit
Page 6 of 33
=
∞
X
Z
pn,k (x)
∞
(1)
t(1 + t)bn,k (t)(t − x)m dt
0
k=1
+ (n + 2)[Tn,m+1 (x) − (1 + x)−n (−x)m+1 ]
+ (1 + 2x)[Tn,m (x) − (1 + x)−n (−x)m ] + n(1 + x)−n (−x)m+1
= −(m + 1)(1 + 2x)[Tn,m (x) − (1 + x)−n (−x)m ]
− (m + 2)[Tn,m+1 − (1 + x)−n (−x)m+1 ]
− mx(1 + x)[Tn,m−1 (x) − (1 + x)−n (−x)m−1 ]
+ (n + 2)[Tn,m+1 − (1 + x)−n (−x)m+1 ]
+ (1 + 2x)[Tn,m (x) − (1 + x)−n (−x)m ] + n(1 + x)−n (−x)m+1 .
Thus, we get
(1)
(n − m)Tn,m+1 (x) = x(1 + x)[Tn,m
(x) + 2mTn,m−1 (x)] + m(1 + 2x)Tn,m (x).
This completes the proof of recurrence relation. From the above recurrence
relation, it is easily verified for all x ∈ [0, ∞) that
Tn,m (x) = O n−[(m+1)/2] .
Remark 1. It is easily verified from Lemma 2.1 that for each x ∈ (0, ∞)
Bn (ti , x) =
(n + i − 1)!(n − i)! i
(n + i − 2)!(n − i)! i−1
x +i(i−1)
x +O(n−2 ).
n!(n − 1)!
n!(n − 1)!
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 7 of 33
Corollary 2.3. Let δ be a positive number. Then for every γ > 0, x ∈ (0, ∞),
there exists a constant M (s, x) independent of n and depending on s and x such
that
Z
Wn (x, t)tγ dt
≤ M (s, x)n−s ,
s = 1, 2, 3, . . .
|t−x|>δ
C[a,b]
Lemma 2.4. There exist the polynomials Qi,j,r (x) independent of n and k such
that
X
{x(1 + x)}r Dr pn,k (x) =
ni (k − nx)j Qi,j,r (x)pn,k (x),
2i+j≤r
i,j≥0
where D ≡
d
.
dx
By C0 , we denote the class of continuous functions on the interval (0, ∞)
having a compact support and C0r is the class of r times continuously differentiable functions with C0r ⊂ C0 . The function f is said to belong to the generalized Zygmund class Liz(α, 1, a, b), if there exists a constant M such that
ω2 (f, δ) ≤ M δ α , δ > 0, where ω2 (f, δ) denotes the modulus of continuity of
2nd order on the interval [a, b]. The class Liz(α, 1, a, b) is more commonly denoted by Lip∗ (α, a, b). Suppose G(r) = {g : g ∈ C0r+2 , supp g ⊂ [a0 , b0 ] where
[a0 , b0 ] ⊂ (a, b)}. For r times continuously differentiable functions f with supp
f ⊂ [a0 , b0 ] the Peetre’s K-functionals are defined as
h
n
oi
(r)
(r)
(r)
(r+2)
Kr (ξ, f ) = inf
f −g
+ξ g
+ g
,
C[a0 ,b0 ]
C[a0 ,b0 ]
C[a0 ,b0 ]
g∈G(r)
0 < ξ ≤ 1.
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 8 of 33
For 0 < α < 2, C0r (α, 1, a, b) denotes the set of functions for which
sup ξ −α/2 Kr (ξ, f, a, b) < C.
0<ξ≤1
Lemma 2.5. Let 0 < a0 < a00 < b00 < b0 < b < ∞ and f (r) ∈ C0 with
supp f ⊂ [a00 , b00 ] and if f ∈ C0r (α, 1, a0 , b0 ), we have f (r) ∈ Liz(α, 1, a0 , b0 ) i.e.
f (r) ∈ Lip∗ (α, a0 , b0 ) where Lip∗ (α, a0 , b0 ) denotes the Zygmund class satisfying
Kr (δ, f ) ≤ Cδ α/2 .
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Proof. Let g ∈ G(r) , then for f ∈ C0r (α, 1, a0 , b0 ), we have
42δ f (r) (x) ≤ 42δ (f (r) − g (r) )(x) + 42δ g (r) (x)
≤ 42δ (f (r) − g (r) )
≤ 4M1 Kr (δ 2 , f ) ≤
C[a0 ,b0 ]
M2 δ α .
+ δ 2 g (r+2)
C[a0 ,b0 ]
Lemma 2.6. If f is r times differentiable on [0, ∞), such that f (r−1) = O(tα ), α >
0 as t → ∞, then for r = 1, 2, 3, . . . and n > α + r we have
Z ∞
∞
(n + r − 1)!(n − r)! X
(r)
Bn (f, x) =
pn+r,k (x)
bn−r,k+r (t)f (r) (t)dt.
n!(n − 1)!
0
k=0
Title Page
Contents
JJ
J
II
I
Go Back
Proof. First
Bn(1) (f, x) =
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
∞
X
k=1
Close
(1)
Z
pn,k (x)
0
∞
bn,k (t)f (t)dt − n(1 + x)−n−1 f (0).
Quit
Page 9 of 33
Now using the identities
(1)
(2.1)
pn,k (x) = n[pn+1,k−1 (x) − pn+1,k (x)],
(2.2)
bn,k (t) = (n + 1)[bn+1,k−1 (t) − bn+1,k (t)].
(1)
for k ≥ 1, we have
Bn(1) (f, x)
Z
∞
X
=
n[pn+1,k−1 (x) − pn+1,k (x)]
∞
bn,k (t)f (t)dt − n(1 + x)−n−1 f (0)
0
k=1
Z
∞
bn,1 (t)f (t)dt − n(1 + x)−n−1 f (0)
= npn+1,0 (x)
+n
0
∞
X
Z
[bn,k+1 (t) − bn,k (t)]f (t)dt
∞
Z
(n + 1)(1 + t)−n−2 f (t)dt − n(1 + x)−n−1 f (0)
= n(1 + x)
0
+n
∞
X
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
0
k=1
−n−1
∞
pn+1,k (x)
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Contents
Z
pn+1,k (x)
k=1
Title Page
0
∞
1 (1)
− bn−1,k+1 (t)f (t)dt.
n
Integrating by parts, we get
Bn(1) (f, x) = n(1 + x)−n−1 f (0) + n(1 + x)−n−1
JJ
J
II
I
Go Back
Z
0
∞
(1 + t)−n−1 f (1) (t)dt
Close
Quit
Page 10 of 33
−n−1
− n(1 + x)
f (0) +
∞
X
=
k=0
Z
pn+1,k (x)
bn−1,k+1 (t)f (1) (t)dt
pn+1,k (x)
0
k=1
∞
X
∞
Z
∞
bn−1,k+1 (t)f (1) (t)dt.
0
Thus the result is true for r = 1. We prove the result by induction method.
Suppose that the result is true for r = i, then
∞
Bn(i) (f, x)
(n + i − 1)!(n − i)! X
pn+i,k (x)
=
n!(n − 1)!
k=0
Z
∞
(i)
bn−i,k+i (t)f (t)dt.
0
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Thus using the identities (2.1) and (2.2), we have
Bn(i+1) (f, x)
(n + i − 1)!(n − i)!
=
n!(n − 1)!
Z
∞
X
×
(n + i)[pn+i+1,k−1 (x) − pn+i+1,k (x)]
k=1
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Title Page
∞
bn−i,k+i (t)f (i) (t)dt
0
Z ∞
(n + i − 1)!(n − i)!
−n−i−1
+
(−(n + i)(1 + x)
)
bn−i,i (t)f (i) (t)dt
n!(n − 1)!
0
Z ∞
(n + i)!(n − i)!
pn+i+1,0 (x)
bn−i,i+1 (t)f (i) (t)dt
=
n!(n − 1)!
0
Z ∞
(n + i)!(n − i)!
−
pn+i+1,0 (x)
bn−i,i (t)f (i) (t)dt
n!(n − 1)!
0
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 11 of 33
Z ∞
∞
(n + i)!(n − i)! X
+
pn+i+1,k (x)
[bn−i,k+i+1 (t) − bn−i,k+i (t)]f (i) (t)dt
n!(n − 1)! k=1
0
Z ∞
(n + i)!(n − i)!
1
(1)
=
pn+i+1,0 (x)
−
bn−i−1,i+1 (t)f (i) (t)dt
n!(n − 1)!
(n − i)
0
Z ∞
∞
X
(n + i)!(n − i)!
1
(1)
+
pn+i+1,k (x)
−
bn−i−1,k+i+1 (t)f (i) (t)dt.
n!(n − 1)! k=1
(n
−
i)
0
Integrating by parts, we obtain
∞
Bn(i+1) (f, x)
(n + i)!(n − i − 1)! X
=
pn+i+1,k (x)
n!(n − 1)!
k=0
This completes the proof of the lemma.
Z
0
∞
bn−i−1,k+i+1 (t)f (i+1) (t)dt.
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 12 of 33
3.
Direct Theorems
In this section we present the following results.
Theorem 3.1. Let f ∈ Cγ [0, ∞) and f (r) exists at a point x ∈ (0, ∞). Then we
have
Bn(r) (f, x) = f (r) (x)
as n → ∞.
Proof. By Taylor expansion of f , we have
f (t) =
r
X
f (i) (x)
i=0
i!
(t − x)i + ε(t, x)(t − x)r ,
where ε(t, x) → 0 as t → x. Hence
Z ∞
(r)
Bn (f, x) =
Wn(r) (t, x)f (t)dt
0
r
X
Z
f (i) (x) ∞ (r)
Wn (t, x)(t − x)i dt
=
i!
0
i=0
Z ∞
+
Wn(r) (t, x)ε(t, x)(t − x)r dt
0
=: R1 + R2 .
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
First to estimate R1 , using the binomial expansion of (t − x)i and Remark 1, we
Quit
Page 13 of 33
have
R1 =
r
i X
f (i) (x) X i
i=0
(r)
i!
v=0
v
i−v
(−x)
∂r
∂xr
Z
∞
Wn (t, x)tv dt
0
(x) dr (n + r − 1)!(n − r)! r
=
x + terms containing lower powers of x
r! dxr
n!(n − 1)!
(n + r − 1)!(n − r)!
(r)
= f (x)
→ f (r) (x)
n!(n − 1)!
f
as n → ∞. Next applying Lemma 2.4, we obtain
Z ∞
R2 =
Wn(r) (t, x)ε(t, x)(t − x)r dt,
0
∞
|Qi,j,r (x)| X
|R2 | ≤
n
|k − nx|j pn,k (x)
r
{x(1
+
x)}
2i+j≤r
k=1
X
0
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
i
i,j≥0
Z
∞
bn,k (t)|ε(t, x)||t − x|r dt +
(n + r + 1)!
(1 + x)−n−r |ε(0, x)|xr .
(n − 1)!
The second term in the above expression tends to zero as n → ∞. Since
ε(t, x) → 0 as t → x for a given ε > 0 there exists a δ such that |ε(t, x)| < ε
whenever 0 < |t − x| < δ. If α ≥ max{γ, r}, where α is any integer, then we
can find a constant M3 > 0, |ε(t, x)(t − x)r | ≤ M3 |t − x|α , for |t − x| ≥ δ.
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 14 of 33
Therefore
|R2 | ≤ M3
X
n
i
∞
X
pn,k (x)|k − nx|j
k=0
2i+j≤r
i,j≥0
Z
× ε
bn,k (t)|t − x| dt
Z
r
α
bn,k (t)|t − x| dt +
|t−x|<δ
|t−x|≥δ
=: R3 + R4 .
Applying the Cauchy-Schwarz inequality for integration and summation respectively, we obtain
R3 ≤ εM3
X
2i+j≤r
i,j≥0
ni
(∞
X
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
) 12
pn,k (x)(k − nx)2j
k=1
×
(∞
X
k=1
Z
pn,k (x)
) 12
∞
bn,k (t)(t − x)2r dt
.
0
Using Lemma 2.1 and Lemma 2.2, we get
R3 = ε · O(nr/2 )O(n−r/2 ) = ε · o(1).
Title Page
Contents
JJ
J
II
I
Go Back
Again using the Cauchy-Schwarz inequality, Lemma 2.1 and Corollary 2.3, we
Close
Quit
Page 15 of 33
get
X
R4 ≤ M4
n
i
≤ M4
j
n
i
∞
X
Z
|t−x|≥δ
j
Z
pn,k (x)|k − nx|
12
bn,k (t)dt
|t−x|≥δ
k=1
2i+j≤r
i,j≥0
bn,k (t)|t − x|α dt
pn,k (x)|k − nx|
k=1
2i+j≤r
i,j≥0
X
∞
X
12
bn,k (t)(t − x) dt
Z
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
2α
×
|t−x|≥δ
X
≤ M4
2i+j≤r
i,j≥0
×
ni
(∞
X
=
pn,k (x)(k − nx)2j
k=1
(∞
X
k=1
X
) 12
Z
pn,k (x)
∞
) 12
bn,k (t)(t − x)2α dt
0
ni O(nj/2 )O(n−α/2 ) = O(n(r−α)/2 ) = o(1).
2i+j≤r
i,j≥0
Collecting the estimates of R1 − R4 , we obtain the required result.
Theorem 3.2. Let f ∈ Cγ [0, ∞). If f (r+2) exists at a point x ∈ (0, ∞). Then
lim n Bn(r) (f, x) − f (r) (x)
n→∞
= r(r − 1)f (r) (x) + r(1 + 2x)f (r+1) (x) + x(1 + x)f (r+2) (x).
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 16 of 33
Proof. Using Taylor’s expansion of f, we have
f (t) =
r+2 (i)
X
f (x)
i=0
i!
(t − x)i + ε(t, x)(t − x)r+2 ,
where ε(t, x) → 0 as t → x and ε(t, x) = O((t − x)γ ), t → ∞ for γ > 0.
Applying Lemma 2.2, we have
n Bn(r) (f (t), x) − f (r) (x)
#
" r+2
X f (i) (x) Z ∞
=n
Wn(r) (t, x)(t − x)i dt − f (r) (x)
i!
0
i=0
Z ∞
+n
Wn(r) (t, x)ε(t, x)(t − x)r+2 dt
0
=: E1 + E2 .
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
First, we have
E1 = n
r+2 (i)
i X
f (x) X i
i=0
=
f
(r)
i!
j=0
j
(−x)i−j
Z
Title Page
∞
Wn(r) (t, x)tj dt − nf (r) (x)
0
(x) (r) r
n Bn (t , x) − (r!)
r!
f (r+1) (x) +
n (r + 1)(−x)Bn(r) (tr , x) + Bn(r) (tr+1 , x)
(r + 1)!
"
f (r+2) (x) (r + 2)(r + 1) 2 (r) r
+
n
x Bn (t , x)
(r + 2)!
2
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 17 of 33
#
+ (r + 2)(−x)Bn(r) (tr+1 , x) + Bn(r) (tr+2 , x) .
Therefore, by Remark 1, we have
(n + r − 1)!(n − r)!
(r)
E1 = nf (x)
−1
n!(n − 1)!
nf (r+1) (x)
(n + r − 1)!(n − r)!
+
(x − 1)(−x)
(r + 1)!
n!(n − 1)!
(n + r − 1)!(n − r − 1)!
(n + r)!(n − r − 1)!
+
(r + 1)!x + (r + 1)r
r!
n!(n − 1)!
n!(n − 1)!
nf (r+2) (x) (r + 2)(r + 1) 2 (n + r − 1)!(n − r)!
+
x
r!
(r + 2)!
2
n!(n − 1)!
(n + r)!(n − r − 1)!
+ (r + 2)(−x)
x(r + 1)!
2
(n − r − 1)!(n − r − 1)!
+(r + 1)r
r!
n!(n − 1)!
(n + r + 1)!(n − r − 2)! (r + 2)! 2
+
x
n!(n − 1)!
2
(n + r)!(n − r − 2)!
+ (r + 2)(r + 1)
(r + 1)!x
+ O(n−2 ).
n!(n − 1)!
In order to complete the proof of the theorem it is sufficient to show that E2 → 0
as n → ∞ which easily follows proceeding along the lines of the proof of
Theorem 3.1 and by using Lemma 2.1, Lemma 2.2 and Lemma 2.4.
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 18 of 33
Lemma 3.3. Let 0 < α < 2, 0 < a < a0 < a00 < b00 < b0 < b < ∞. If f ∈ C0
(r)
with supp f ⊂ [a00 , b00 ] and Bn (f, ·) − f (r)
= O(n−α/2 ), then
C[a,b]
Kr (ξ, f ) = M5 n−α/2 + nξKr (n−1 , f ) .
Consequently Kr (ξ, f ) ≤ M6 ξ α/2 , M6 > 0.
Proof. It is sufficient to prove
Kr (ξ, f ) = M7 {n−α/2 + nξKr (n−1 , f )},
for sufficiently large n. Because supp f ⊂ [a00 , b00 ] therefore by Theorem 3.2
there exists a function h(i) ∈ G(r) , i = r, r + 2, such that
Bn(r) (f, •) − h(i)
C[a0 ,b0 ]
≤ M8 n−1 .
Therefore,
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
Kr (ξ, f ) ≤ 3M9 n−1 + Bn(r) (f, •) − f (r)
n
+ ξ Bn(r) (f, •)
Contents
C[a0 ,b0 ]
+ Bn(r+2) (f, •)
C[a0 ,b0 ]
o
C[a0 ,b0 ]
JJ
J
.
Next, it is sufficient to show that there exists a constant M10 such that for each
g ∈ G(r)
II
I
Go Back
Close
(3.1)
Bn(r+2) (f, •) C[a0 ,b0 ]
≤ M10 n{ f
(r)
−g
(r)
C[a0 ,b0 ]
+n
−1
g
(r+2)
C[a0 ,b0 ]
.
Quit
Page 19 of 33
Also using the linearity property, we have
(3.2)
Bn(r+2) (f, •)
C[a0 ,b0 ]
≤ Bn(r+2) (f − g, •)
Applying Lemma 2.4, we get
Z ∞ r+2
∂
Wn (x, t) dt ≤
∂xr+2
0
∞
X X
2i+j≤r+2
i,j≥0
C[a0 ,b0 ]
+ Bn(r+2) (g, •)
ni |k − nx|j
k=1
Z
× pn,k (x)
C[a0 ,b0 ]
.
|Qi,j,r+2 (x)|
{x(1 + x)}r+2
∞
bn,k (t)dt +
0
dr+2
[(1 + x)−n ].
dxr+2
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Therefore by the Cauchy-Schwarz inequality and Lemma 2.1, we obtain
(3.3)
Bn(r) (f − g, •)
C[a0 ,b0 ]
≤ M11 n f (r) − g (r)
C[a0 ,b0 ]
,
where the constant M11 is independent of f and g. Next by Taylor’s expansion,
we have
r+1 (i)
X
g (x)
g (r+2) (ξ)
g(t) =
(t − x)i +
(t − x)r+2 ,
i!
(r
+
2)!
i=0
where
R ∞ ∂ mξ lies between ti and x. Using the above expansion and the fact that
Wn (x, t)(t − x) dt = 0 for m > i, we get
0 ∂xm
(3.4)
Bn(r+2) (g, •)
≤ M12 g
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Title Page
Contents
JJ
J
II
I
Go Back
C[a0 ,b0 ]
Z
(r+2)
C[a0 ,b0 ]
·
0
∞
∂ r+2
Wn (x, t)(t − x)r+2 dt
r+2
∂x
Close
.
Quit
C[a0 ,b0 ]
Page 20 of 33
Also by Lemma 2.4 and the Cauchy-Schwarz inequality, we have
Z ∞ r+2
∂
Wn (x, t) (t − x)r+2 dt
E≡
r+2
∂x
0
Z ∞
∞
X X
i
j |Qi,j,r+2 (x)|
≤
n pn,k (x)|k − nx|
bn,k (t)(t − x)r+2 dt
r+2
{x(1
+
x)}
0
2i+j≤r+2 k=1
i,j≥0
dr+2
[(−x)r+2 (1 + x)−n ]
dxr+2
! 12
∞
X
X
|Qi,j,r+2 (x)|
pn,k (x)(k − nx)2j
≤
r+2
{x(1 + x)}
2i+j≤r+2
k=1
+
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
i,j≥0
×
! 12 Z
Z ∞
∞
X
pn,k (x)
bn,k (t)(t − x)2r+4 dt
k=1
r+2
∞
12
bn,k (t)dt
0
0
d
+ r+2 [(−x)r+2 (1 + x)−n ]
dx
X
−(1+ r2 )
i |Qi,j,r+2 (x)|
j/2
.
=
n
O(n )O n
{x(1 + x)}r+2
2i+j≤r+2
i,j≥0
Hence
(3.5)
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
Contents
JJ
J
II
I
Go Back
||Bn(r+2) (g, •)||C[a0 ,b0 ]
≤ M13 ||g
(r+2)
||C[a0 ,b0 ] .
Combining the estimates of (3.2)-(3.5), we get (3.1). The other consequence
follows form [1]. This completes the proof of the lemma.
Close
Quit
Page 21 of 33
Theorem 3.4. Let f ∈ Cγ [0, ∞) and suppose 0 < a < a1 < b1 < b < ∞.
Then for all n sufficiently large, we have
1
Bn(r) (f, •) − f (r) C[a1 ,b1 ] ≤ max M14 ω2 f (r) , n− 2 , a, b + M15 n−1 kf kγ ,
where M14 = M14 (r), M15 = M15 (r, f ).
Proof. For sufficiently small δ > 0, we define a function f2,δ (t) corresponding
to f ∈ Cγ [0, ∞) by
f2,δ (t) = δ
−2
Z
δ
2
− 2δ
Z
δ
2
− 2δ
f (t) − ∆2η f (t) dt1 dt2 ,
2
where η = t1 +t
, t ∈ [a, b] and ∆2η f (t) is the second forward difference of f
2
with step length η. Following [4] it is easily checked that:
(i) f2,δ has continuous derivatives up to order 2k on [a, b],
(r)
c1 δ −r ω2 (f, δ, a, b),
(ii) kf2,δ kC[a1 ,b1 ] ≤ M
c2 ω2 (f, δ, a, b),
(iii) kf − f2,δ kC[a1 ,b1 ] ≤ M
c3 kf kγ ,
(iv) kf2,δ kC[a1 ,b1 ] ≤ M
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
Contents
JJ
J
II
I
Go Back
ci , i = 1, 2, 3 are certain constants that depend on [a, b] but are indewhere M
pendent of f and n [4].
Close
Quit
Page 22 of 33
We can write
Bn(r) (f, •) − f (r)
C[a1 ,b1 ]
≤ Bn(r) (f − f2,δ , •)
(r)
C[a1 ,b1 ]
+ Bn(r) (f2,δ , •) − f2,δ
(r)
C[a1 ,b1 ]
+ f (r) − f2,δ
C[a1 ,b1 ]
=: H1 + H2 + H3 .
(r)
Since f2,δ = f (r)
2,δ
(t), by property (iii) of the function f2,δ , we get
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
c4 ω2 (f (r) , δ, a, b).
H3 ≤ M
Next on an application of Theorem 3.2, it follows that
c5 n−1
H2 ≤ M
r+2
X
j=r
(j)
f2,δ
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
.
C[a,b]
Using the interpolation property due to Goldberg and Meir [4], for each j =
r, r + 1, r + 2, it follows that
(j)
(r+2)
c
f2,δ
≤ M6 ||f2,δ ||C[a,b] + f2,δ
.
C[a1 ,b1 ]
C[a,b]
Therefore by applying properties (iii) and (iv) of the of the function f2,δ , we
obtain
c7 4 · n−1 ||f ||γ + δ −2 ω2 (f (r) , δ) .
H2 ≤ M
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 23 of 33
Finally we shall estimate H1 , choosing a∗ , b∗ satisfying the conditions 0 < a <
a∗ < a1 < b1 < b∗ < b < ∞. Suppose ψ(t) denotes the characteristic function
of the interval [a∗ , b∗ ], then
H1 ≤ Bn(r) (ψ(t)(f (t) − f2,δ (t)), •)
+
Bn(r) ((1
C[a1 ,b1 ]
− ψ(t))(f (t) − f2,δ (t)), •)
C[a1 ,b1 ]
=: H4 + H5 .
Using Lemma 2.6, it is clear that
Bn(r) ψ(t)(f (t) − f2,δ (t)), x
Z ∞
∞
(n + r − 1)!(n − r)! X
(r)
=
pn+r,k (x)
bn−r,k+r (t)ψ(t)(f (r) (t)−f2,δ (t))dt.
n!(n − 1)!
0
k=0
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Hence
Title Page
Bn(r) (ψ(t)(f (t)
− f2,δ (t)), •)
C[a1 ,b1 ]
c8 f (r) − f (r)
≤M
2,δ
C[a∗ ,b∗ ]
.
Next for x ∈ [a1 , b1 ] and t ∈ [0, ∞) \ [a∗ , b∗ ], we choose a δ1 > 0 satisfying
|t − x| ≥ δ1 .
Therefore by Lemma 2.4 and the Cauchy-Schwarz inequality, we have
I ≡ Bn(r) ((1 − ψ(t))(f (t) − f2,δ (t), x)
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 24 of 33
and
∞
|Qi,j,r (x)| X
|I| ≤
n
pn,k (x)|k − nx|j
r
{x(1
+
x)}
2i+j≤r
k=1
X
i
i,j≥0
∞
Z
×
bn,k (t)(1 − ψ(t))|f (t) − f2,δ (t)|dt
0
(n + r − 1)!
(1 + x)−n−r (1 − ψ(0))|f (0) − f2,δ (0)|.
(n − 1)!
+
For sufficiently large n, the second term tends to zero. Thus
Z
∞
X
X
i
j
c
|I| ≤ M9 ||f ||γ
n
pn,k (x)|k − nx|
bn,k (t)dt
X
c9 ||f ||γ δ −2m
≤M
1
|t−x|≥δ1
k=1
2i+j≤r
i,j≥0
ni
∞
X
×
∞
21
bn,k (t)dt
0
k=1
2i+j≤r
i,j≥0
Z
pn,k (x)|k − nx|j
Z
Contents
JJ
J
0
X
c9 ||f ||γ δ −2m
≤M
1
ni
×
(∞
X
k=1
) 12
pn,k (x)(k − nx)2j
Z
) 12
∞
4m
bn,k (t)(t − x)
pn,k (x)
II
I
Go Back
k=1
2i+j≤r
i,j≥0
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
12
∞
4m
bn,k (t)(t − x) dt
(∞
X
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
dt
Close
.
Quit
0
Page 25 of 33
Hence by using Lemma 2.1 and Lemma 2.2, we have
c10 ||f ||γ δ −2m O n(i+ 2j −m) ≤ M
c11 n−q ||f ||γ ,
I≤M
1
where q = m − 2r . Now choosing m > 0 satisfying q ≥ 1, we obtain I ≤
c11 n−1 kf kγ . Therefore by property (iii) of the function f2,δ (t), we get
M
(r)
c8 f (r) − f
H1 ≤ M
2,δ
C[a∗ ,b∗ ]
c11 n−1 ||f ||γ
+M
c12 ω2 (f (r) , δ, a, b) + M
c11 n−1 ||f |kγ .
≤M
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
1
Choosing δ = n− 2 , the theorem follows.
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 26 of 33
4.
Inverse Theorem
This section is devoted to the following inverse theorem in simultaneous approximation:
Theorem 4.1. Let 0 < α < 2, 0 < a1 < a2 < b2 < b1 < ∞ and suppose
f ∈ Cγ [0, ∞). Then in the following statements (i) ⇒ (ii)
(r)
(i) ||Bn (f, •)||C[a1 ,b1 ] = O(n−α/2 ),
(ii) f (r) ∈ Lip∗ (α, a2 , b2 ),
where Lip∗ (α, a2 , b2 ) denotes the Zygmund class satisfying ω2 (f, δ, a2 , b2 ) ≤
M δα.
Proof. Let us choose a0 , a00 , b0 , b00 in such a way that a1 < a0 < a00 < a2 < b2 <
b00 < b0 < b1 . Also suppose g ∈ C0∞ with suppg ∈ [a00 , b00 ] and g(x) = 1 on the
d
interval [a2 , b2 ]. For x ∈ [a0 , b0 ] with D ≡ dx
, we have
Bn(r) (f g, x)
(r)
− (f g) (x)
= Dr (Bn ((f g)(t) − (f g)(x)), x)
= Dr (Bn (f (t)(g(t) − g(x)), x)) + Dr (Bn (g(x)(f (t) − f (x)), x))
=: J1 + J2 .
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Using the Leibniz formula, we have
Z ∞
∂r
J1 =
Wn (x, t)f (t)[g(t) − g(x)]dt
∂xr 0
Close
Quit
Page 27 of 33
r Z
X
r
∞
∂ r−i
[f (t)(g(t) − g(x))]dt
r−i
i
∂x
0
i=0
Z ∞
r−1 X
r (r−i)
(i)
=−
g
(x)Bn (f, x) +
Wn(r) (x, t)f (t)(g(t) − g(x))dt
i
0
i=0
=
Wn(i) (x, t)
=: J3 + J4 .
Applying Theorem 3.4, we have
J3 = −
r−1 X
r
i=0
i
g
(r−i)
(i)
(x)f (x) + O n
−α
2
,
uniformly in x ∈ [a0 , b0 ]. Applying Theorem 3.2, the Cauchy-Schwarz inequality, Taylor’s expansions of f and g and Lemma 2.2, we are led to
r
1
X
g (i) (x)f (r−i) (x)
J4 =
r! + o n− 2
i!(r − i)!
i=0
r X
α
r (i)
=
g (x)f (r−i) (x) + o n− 2 ,
i
i=0
uniformly in x ∈ [a0 , b0 ]. Again using the Leibniz formula, we have
r Z ∞
X
r
∂ r−i
J2 =
Wn(i) (x, t) r−i [g(t)(f (t) − f (x))]dt
i 0
∂x
i=0
r X
r (r−i)
=
g
(x)Bn(i) (f, x) − (f g)(r) (x)
i
i=0
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 28 of 33
=
r X
r
i
i=0
g (r−i) (x)f (i) (x) − (f g)(r) (x) + o(n−α/2 )
α
= O n− 2 ,
uniformly in x ∈ [a0 , b0 ]. Combining the above estimates, we get
α
Bn(r) (f g, •) − (f g)(r) C[a0 ,b0 ] = O n− 2 .
Thus by Lemma 2.4 and Lemma 2.5, we have (f g)(r) ∈ Lip∗ (α, a0 , b0 ) also
g(x) = 1 on the interval [a2 , b2 ], it proves that f (r) ∈ Lip∗ (α, a2 , b2 ). This
completes the validity of the implication (i) ⇒ (ii) for the case 0 < α ≤ 1.
To prove the result for 1 < α < 2 for any interval [a∗ , b∗ ] ⊂ (a1 , b1 ), let a∗2 , b∗2
be such that (a2 , b2 ) ⊂ (a∗2 , b∗2 ) and (a∗2 , b∗2 ) ⊂ (a∗1 , b∗1 ). Letting δ > 0 we shall
prove the assertion α < 2. From the previous case it implies that f (r) exists and
belongs to Lip(1 − δ, a∗1 , b∗1 ). Let g ∈ C0∞ be such that g(x) = 1 on [a2 , b2 ] and
supp g ⊂ (a∗2 , b∗2 ). Then with χ2 (t) denoting the characteristic function of the
interval [a∗1 , b∗1 ], we have
Bn(r) (f g, •)
(r)
− (f g)
C[a∗2 ,b∗2 ]
r
≤ ||D [Bn (g(·)(f (t) − f (·)), •)]||C[a∗2 ,b∗2 ]
+ ||Dr [Bn (f (t)(g(t) − g(·)), •)]||C[a∗2 ,b∗2 ]
=: P1 + P2 .
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 29 of 33
To estimate P1 , by Theorem 3.4, we have
P1 = Dr [Bn (g(·)(f (t), •)] − (f g)(r)
=
r X
r
i
i=0
=
r X
r
i
i=0
α
= O n− 2 .
C[a∗2 ,b∗2 ]
g (r−i) (·)Bn(i) (f, •) − (f g)(r)
C[a∗2 ,b∗2 ]
+ O(n−α/2 )
g (r−i) (·)f (i) − (f g)(r)
C[a∗2 ,b∗2 ]
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Also by the Leibniz formula and Theorem 3.2, have
P2 ≤
r X
r
i=0
i
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
g (r−i) (·)Bn (f, •) + Bn(r) (f (t)(g(t) − g(·))χ2 (t), •)
C[a∗2 ,b∗2 ]
−1
+ O(n )
=: ||P3 + P4 ||C[a∗2 ,b∗2 ] + O(n−1 ).
P3 =
i=0
i
Contents
JJ
J
Then by Theorem 3.4, we have
r−1 X
r
Title Page
α
g (r−i) (x)f (i) (x) + O n− 2 ,
II
I
Go Back
Close
Quit
Page 30 of 33
uniformly in x ∈ [a∗2 , b∗2 ]. Applying Taylor’s expansion of f, we have
Z ∞
P4 =
Wn(r) (x, t)[f (t)(g(t) − g(x))χ2 (t)dt
i=0
=
Z
r
X
f (i) (x)
i!
i=0
Z
∞
Wn(r) (x, t)(t − x)i (g(t) − g(x))dt
0
∞
Wn(r) (x, t)
+
0
(f (r) (ξ) − f (r) (x))
(t − x)r (g(t) − g(x))χ2 (t)dt,
r!
where ξ lying between t and x. Next by Theorem 3.4, the first term in the above
expression is given by
r X
α
r (m) (r−m)
g f
(x) + O n− 2 ,
m
m=0
uniformly in x ∈ [a∗2 , b∗2 ]. Also by mean value theorem and using Lemma 2.4,
we can obtain the second term as follows:
Z ∞
(f (r) (ξ) − f (r) (x))
(t − x)r (g(t) − g(x))χ2 (t)dt
Wn(r) (x, t)
r!
0
C[a∗2 ,b∗2 ]
Z
X
|Qm,s,r (x)| r ∞
≤
nm+s
Wn (x, t)|t − x|δ+r+1
x(1
+
x)
0
2m+s≤r
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
Contents
JJ
J
II
I
m,s≥0
×
δ
|f (r) (ξ) − f (r) (x)| 0
|g (η)|χ2 (t)dt
r!
= O n− 2 ,
Go Back
C[a∗2 ,b∗2 ]
Close
Quit
Page 31 of 33
choosing δ such that 0 ≤ δ ≤ 2 − α. Combining the above estimates we get
α
Bn(r) (f g, •) − (f g)(r) C[a∗ ,b∗ ] = O n− 2 .
2
2
Since suppf g ⊂ (a∗2 , b∗2 ), it follows from Lemma 2.4 and Lemma 2.5 that
(f g)(r) ∈ Liz(α, 1, a∗2 , b∗2 ). Since g(x) = 1 on [a2 , b2 ], we have f (r) ∈
Liz(α, 1, a∗2 , b∗2 ). This completes the proof of the theorem.
Remark 2. As noted in the first section, these operators also reproduce the linear functions so we can easily apply the iterative combinations to the operators
(1.1) to improve the order of approximation.
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 32 of 33
References
[1] H. BERENS AND G.G. LORENTZ, Inverse theorem for Bernstein polynomials, Indiana Univ. Math. J., 21 (1972), 693–708.
[2] Z. FINTA, On converse approximation theorems, J. Math. Anal. Appl.,
312(1) (2005), 159–180.
[3] G. FREUD AND V. POPOV, On approximation by Spline functions, Proceeding Conference on Constructive Theory Functions, Budapest (1969),
163–172.
[4] S. GOLDBERG AND V. MEIR, Minimum moduli of ordinary differential
operators, Proc. London Math. Soc., 23 (1971), 1–15.
[5] V. GUPTA, A note on modified Baskakov type operators, Approximation
Theory Appl., 10(3) (1994), 74–78.
[6] V. GUPTA, M.K. GUPTA AND V. VASISHTHA, Simultaneous approximation by summation-integral type operators, Nonlinear Funct. Anal. Appl.,
8(3), (2003), 399–412.
[7] V. GUPTA AND M.A. NOOR, Convergence of derivatves for certain mixed
Szasz Beta operators, J. Math. Anal. Appl., 321(1) (2006), 1–9.
[8] N. ISPIR AND I. YUKSEL, On the Bezier variant of Srivastava-Gupta operators, Applied Math. E Notes, 5 (2005), 129–137.
[9] H.M. SRIVASTAVA AND V. GUPTA, A certain family of summation integral type operators, Math. Comput. Modelling, 37 (2003), 1307–1315.
On Simultaneous
Approximation For Certain
Baskakov Durrmeyer Type
Operators
Vijay Gupta,
Muhammad Aslam Noor,
Man Singh Beniwal and
M. K. Gupta
Title Page
Contents
JJ
J
II
I
Go Back
Close
Quit
Page 33 of 33