Journal of Inequalities in Pure and
Applied Mathematics
RATE OF CONVERGENCE FOR A GENERAL SEQUENCE
OF DURRMEYER TYPE OPERATORS
volume 5, issue 3, article 79,
2004.
NIRAJ KUMAR
School of Applied Sciences
Netaji Subhas Institute of Technology
Sector 3, Dwarka
New Delhi 110045, India.
Received 21 January, 2004;
accepted 14 June, 2004.
Communicated by: C.P. Niculescu
EMail: neeraj@nsit.ac.in
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
016-04
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Abstract
In the present paper we give the rate of convergence for the linear combinations of the generalized Durrmeyer type operators which includes the well
known Szasz-Durrmeyer operators and Baskakov-Durrmeyer operators as special cases.
2000 Mathematics Subject Classification: 41A16, 41A25.
Key words: Szasz-Durrmeyer operators, Baskakov-Durrmeyer operators, Rate of
convergence, Order of approximation.
The author is extremely grateful to the referee for making valuable suggestions leading to the better presentation of the paper.
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
Niraj Kumar
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Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2
Auxiliary Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3
Rate Of Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
References
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1.
Introduction
Durrmeyer [1] introduced the integral modification of Bernstein polynomials so
as to approximate Lebesgue integrable functions on the interval [0, 1]. We now
consider the general family of Durrmeyer type operators, which is defined by
(1.1)
Sn (f (t), x) = (n − c)
∞
X
Z
pn,v (x)
v=0
∞
pn,v (t) f (t) dt,
x∈I
0
v
(v)
where n ∈ N , n > max {0, −c} and pn,v (x) = (−1)v xv! ϕn (x). Also {φn }
is a sequence of real functions having the following properties on [0, a], where
a > 0 and for all n ∈ N, v ∈ N ∪ {0}, we have
(I) φn ∈ C ∞ [0, a], φn (0) = 1.
(III) There exist c ∈ N :
Contents
=
(v)
−nφn+c , n
> max {0, −c} .
Some special cases of the operators (1.1) are as follows:
1. If c = 0,φn (x) = e−nx , we get the Szasz-Durrmeyer operator.
2. If c = 1, φn (x) = (1 + x)−n , we obtain the Baskakov-Durrmeyer operator.
3. If c > 1, φn (x) = (1 + cx)
operator.
Niraj Kumar
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(II) φn is complete monotonic.
(v+1)
φn
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
−n
c
, we obtain a general Baskakov-Durrmeyer
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4. If c = −1, φn (x) = (1 − x)n , we obtain the Bernstein-Durrmeyer operator.
Very recently Srivastava and Gupta [9] studied a similar type of operators
and obtained the rate of convergence for functions of bounded variation. It is
easily verified that the operators (1.1) are linear positive operators and these operators reproduce the constant ones, while the operators studied in [9] reproduce
every linear functional for the case c = 0. Several researchers studied different
approximation properties on the special cases of the operators (1.1), the pioneer
work on Durrmeyer type operators is due to S. Guo [3], Vijay Gupta (see e. g.
[4], [5]), R. P. Sinha et al. [8] and Wang and Guo [11], etc. It turns out that the
order of approximation for such type of Durrmeyer operators is at best O (n−1 ),
how so ever smooth the function may be. In order to improve the order of approximation, we have to slacken the positivity condition of the operators, for
this we consider the linear combinations of the operators (1.1). The technique
of linear combinations is described as follows:
−2
−k −1 −1
1 d−1
d
.....
d
d−2
..... d−k
0
0
0
0
0
0
Sd0 n (f, x) d−1
−1
−2
−k
−2
1 d1 d1 ..... d1 Sd1 n (f, x) d1 d1 ..... d−k
1
Sn,k (f, x) = ... ..
..
..... .. ...
..
..
..... ..
... ..
..
..... .. ...
..
..
..... ..
1 d−1 d−2 ..... d−k Sd n (f, x) d−1 d−2 ..... d−k
k
k
k
k
k
k
k
Such types of linear combinations were first considered by May [7] to improve
the order of approximation of exponential type operators. In the alternative form
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
Niraj Kumar
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the above linear combinations can be defined as
Sn,k (f, x) =
k
X
c(j, k)Sdj n (f, x),
j=0
where
C(j, k) =
k
Y
i=0
i6=j
dj
, k 6= 0
dj − di
C(0, 0) = 1.
and
If f ∈ Lp [0, ∞), 1 ≤ p < ∞ and 0 < a1 < a3 < a2 < b2 < b3 < b1 < ∞
and Ii = [ai , bi ], i = 1, 2, 3, the Steklov mean fη,m of mth order corresponding
to f is defined as
fη,m (t) = η
−m
Z
η
2
−η
2
Z
···
η
2
m−1
f (t) + (−1)
−η
2
m
Y
f (t)
dti ;
i=1 ti
Pm
∆m
t ∈ I1 .
i=1
It can be verified [6, 10] that
(m−1)
(m)
(i) fη,m has derivative up to order m, fη,m ∈ AC(I1 ) and fη,m exist almost
every where and belongs to Lp (I1 );
(r) (ii) fη,m ≤ K1 η −r ωr (f, η, p, I1 ), r = 1, 2, 3, . . . , m;
Lp (I1 )
(iii) kf − fη,m kLp (I2 ) ≤ K2 ωm (f, η, p, I1 );
m (iv) fη,m
≤ K3 η m kf kLp (I1 )
L (I )
p
2
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
Niraj Kumar
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(m) (v) fη,m Lp (I1 )
≤ K4 η −r ωr (f, η, p, I1 )
where Ki0 s, i = 1, 2, 3, 4 are certain constants independent of f and η.
In the present paper we establish the rate of convergence for the combinations of the generalized Durrmeyer type operators in Lp −norm.
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
Niraj Kumar
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2.
Auxiliary Results
To prove the rate of convergence we need the following lemmas:
Lemma 2.1. Let m ∈ N ∪ {0}, also µn,m (x) is the mth order central moment
defined by
Z ∞
∞
X
m
µn,m ≡ Sn ((t − x) , x) = (n − c)
pn,v (x)
pn,v (t)(t − x)m dt,
v=0
0
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
then
(i) µn,m (x) is a polynomial in x of degree m.
(ii) µn,m (x) is a rational
function in n and for each 0 ≤ x < ∞
O n−[(m+1)/2] .
Niraj Kumar
µn,m (x) =
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r
Remark 2.1. Using Hölder’s inequality, it can be easily verified that Sn (|t − x| , x)
= O(n−r/2 ) for each r > 0 and for every fixed 0 ≤ x < ∞.
Lemma 2.2. For sufficiently large n and q ∈ N, there holds
Sn,k ((t − x)q , x) = n−(k+1) {F (q, k, x) + o(1)},
where F (q, k, x) are certain polynomials in x of degree q and 0 ≤ x < ∞ is
arbitrary but fixed.
Proof. For sufficiently large values of n we can write, from Lemma 2.1
Sn ((t − x)q , x) =
Q[q/2] (x)
Q0 (x)
Q1 (x)
+
+
·
·
·
+
,
n[(q+1)/2] n[(q+1)/2]+1
nq
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where Qi (x),i = 0, 1, 2, . . . are certain polynomials in x of at most degree q.
Therefore Sn,k ((t − x)q , x) is given by
1
1
...
...
1
d−1
0
d−1
1
..
..
d−1
k
×
d−2
0
d−2
1
..
..
d−2
k
.....
.....
.....
.....
.....
Q0 (x)
(d0 n)[(q+1)/2]
Q0 (x)
(d1 n)[(q+1)/2]
d−k
0
d−k
1
..
..
d−k
k
−1
Q1 (x)
(d0 n)[(q+1)/2]+1
Q1 (x)
(d1 n)[(q+1)/2]+1
+ .... d−1
d−2
..... d−k
0
0
0
−2
−k +
+ .... d−1
d
.....
d
1
1
1
............................
..
..
..... .. ............................
..
..
..... .. Q1 (x)
Q0 (x)
+ (dk n)[(q+1)/2]+1 + .... d−1
d−2
..... d−k
k
k
k
(dk n)[(q+1)/2]
+
= n−(k+1) {F (q, k, x) + o(1)} ,
0 ≤ x < ∞.
Lemma 2.3 ([4]). Let 1 ≤ p < ∞, f ∈ Lp [a, b], f (m) ∈ AC [a, b] and f (m+1) ∈
Lp [a, b], then
n
o
(j) (m+1) f ≤C f
+ kf kLp [a,b] ,
Lp [a,b]
Lp [a,b]
j = 1, 2, 3, . . . , m and C is a constant depending on j, p, m, a, b.
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
Niraj Kumar
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Lemma 2.4. Let f ∈ Lp [0, ∞), p > 1. If f (2k+1) ∈ AC(I1 ) and f (2k+2) ∈
Lp (I1 ), then for all n sufficiently large
n
o
(2.1) kSn,k (f, ·) − f kLp (I2 ) ≤ C1 n−(k+1) f (2k+2) Lp (I2 ) + kf kLp [0,∞) .
Also if f ∈ L1 [0, ∞), f (2k+1) ∈ L1 (I1 ) with f (2k) ∈ AC(I1 ) and f (2k+1) ∈
BV (I1 ), then for all n sufficiently large
(2.2)
kSn,k (f, ·) − f kLp (I2 )
n
o
≤ C2 n−(k+1) f (2k+1) BV (I1 ) + f (2k+1) L1 (I2 ) + kf kL1 [0,∞) .
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
Niraj Kumar
Proof. First let p > 1. By the hypothesis, for all t ∈ [0, ∞) and x ∈ I2 , we have
Title Page
2k+1
X
f (i) (x)
Sn,k (f, x) − f (x) =
Sn,k ((t − x)i , x)
i!
i=1
Z t
1
+
Sn,k (φ(t)
(t − w)2k+1 f (2k+2) (w)dw, x)
(2k + 1)!
x
+ Sn,k (F (t, x)(1 − φ(t)), x)
= E1 + E2 + E3 , say,
where φ(t) denotes the characteristic function of I1 and
F (t, x) = f (t) −
2k+1
X
i=0
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i
(t − x) (i)
f (x).
i!
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Applying Lemma 2.2 and Lemma 2.3, we have
kE1 kLp (I2 ) ≤ C3 n
−(k+1)
2k+1
X
(i) f Lp (I2 )
ni=1
o
≤ C4 n−(k+1) kf kLp (I2 ) + f (2k+2) Lp (I2 ) .
Next we estimate E2 . Let Hf be the Hardy Littlewood maximal function [10]
of f (2k+2) on I1 , using Hölder’s inequality and Lemma 2.1, we have
Z t
(2k+2)
1
2k+1
f
|E2 | ≤
Sn,k φ(t) |t − w|
(w)dw , x
(2k + 2)!
x
Z t
(2k+2)
1
2k+1 ≤
Sn,k φ(t) |t − x|
f
(w) dw , x
(2k + 2)!
x
1
2k+2
Sn,k φ(t) |t − x|
|Hf (t)| , x
≤
(2k + 2)!
n
o 1q
1
1
{Sn,k (φ(t) |Hf (t)|p , x)} p
≤
Sn,k φ(t) |t − x|q(2k+2) , x
(2k + 2)!
! p1
Z b1
k
∞
X
X
≤ C5 n−(k+1)
C(j, k)
(dj n − c)
pdj n,v (x)pdj n,v (t) |Hf (t)|p dt
.
a1
j=0
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Next applying Fubini’s theorem, we have
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kE2 kpLp (I2 )
≤ C6 n−p(k+1)
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
k
X
j=0
Z
C(j, k)
b2 Z b1
a2
a1
∞
X
(dj n − c)
pdj n,v (x)pdj n,v (t) |Hf (t)|p dtdx
v=0
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≤ C6 n−p(k+1)
k
X
Z
C(j, k)
Z
a1
j=0
≤
b1
C7 n−p(k+1) kHf kpLp (I1 )
b2
!
∞
X
(dj n − c)
pdj n,v (x)pdj n,v (t)dx |Hf (t)|p dt
a2
v=0
p
≤ C8 n−p(k+1) f (2k+2) Lp (I1 ) .
Therefore
kE2 kLp (I2 ) ≤ C8 n−(k+1) f (2k+2) Lp (I1 ) .
For t ∈ [0, ∞)\[a1 , b1 ], x ∈ I2 there exists a δ > 0 such that |t − x| ≥ δ. Thus
E3 ≡ Sn,k (F (t, x)(1 − φ(t)), x))
≤ δ −(2k+2)
k
X
C(j, k)Sdj n |F (t, x)| (t − x)2k+2 , x
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
Niraj Kumar
j=0
≤δ
−(2k+2)
k
X
Title Page
"
2k+2
C(j, k) Sdj n |f (t)| (t − x)
,x
j=0
+
2k+1
X
(i) f (x)
i=0
i!
2k+2+i
Sdj n |t − x|
,x
#
= E31 + E32 , say.
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Applying Hölder’s inequality and Lemma 2.1, we have
|E31 | ≤ δ −(2k+2)
k
X
j=0
o 1q
p1 n
p
q(2k+2)
C(j, k) Sdj n (|f (t)| , x)
Sdj n (|t − x|
, x)
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≤ C9
k
X
1
C(j, k) Sdj n (|f (t)|p , x) p
j=0
1
.
(dj n)k+1
Finally by Fubini’s theorem, we obtain
Z b2
p
kE31 kLp (I2 ) =
|E31 |p dx
a2
≤ C9
k
X
C(j, k)(dj n)−p(k+1)
j=0
Z
b2
Z
∞
(dj n − c)pdj n,v (x)pdj n,v (t) |f (t)|p dtdx
×
≤ C10 n
a2
−p(k+1)
0
kf kpLp [0,∞)
.
−(k+1)
Thus
kE3 kLp (I2 ) ≤ C12 n
−(k+1)
Niraj Kumar
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Again by Lemma 2.3, we have
kE32 kLp (I2 ) ≤ C11 n
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
n
o
(2k+2) kf kLp (I2 ) + f
.
Lp (I2 )
n
o
(2k+2) kf kLp [0,∞) + f
.
Lp (I2 )
Combining the estimates of E1 , E2 , E3 , we get (2.1).
Next suppose p = 1. By the hypothesis, for almost all x ∈ I2 and for all
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t ∈ [0, ∞), we have
2k+1
X
f (i) (x)
Sn,k ((t − x)i , x)
i!
i=1
Z t
1
+
Sn,k (φ(t)
(t − w)2k+1 f (2k+1) (w)dw, x)
(2k + 1)!
x
+ Sn,k (F (t, x)(1 − φ(t)), x)
= M1 + M2 + M3 ,
say,
Sn,k (f, x) − f (x) =
whereφ(t) denotes the characteristic function of I1 and
F (t, x) = f (t) −
2k+1
X
i=0
(t − x)i (i)
f (x),
i!
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
Niraj Kumar
Title Page
for almost all x ∈ I2 and t ∈ [0, ∞).
Applying Lemma 2.2 and Lemma 2.3, we have
n
o
kM1 kL1 (I2 ) ≤ C13 n−(k+1) kf kL1 (I2 ) + f (2k+1) L1 (I2 ) .
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Next, we have
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kM2 kL1 (I2 )
k
X
1
≤
|C(j, k)|
(2k + 1)! j=0
Z
b2
Z
b1
(dj n − c)
a2
a1
∞
X
Close
2k+1
pdj n,v (x)pdj n,v (t) |t − x|
v=0
Z t
(2k+1)
× df
(w) dtdx.
x
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For each dj n there exists a non negative integer r = r(dj n) satisfying
r(dj n)−1/2 < max(b1 − a2 , b2 − a1 ) ≤ (r + 1)(dj n)−1/2 .
Thus
(Z
k
r Z b2
x+(l+1)(dj n)−1/2
X
X
1
kM2 kL1 (I2 ) ≤
|C(j, k)|
φ(t)(dj n − c)
(2k + 1)! j=0
x+(l)(dj n)−1/2
l=0 a2
) Z
!
∞
x+(l+1)(dj n)−1/2
X
.
pdj n,v (x)pdj n,v (t) |t − x|2k+1
φ(w) df (2k+1) (w) dtdx
x
v=0
k
X
1
+
|C(j, k)|
(2k + 1)! j=0
×
∞
X
r Z
X
)
pdj n,v (x)pdj n,v (t) |t − x|
x−(l)(dj n)−1/2
φ(t)(dj n − c)
Z
x
·
x−(l+1)(dj n)−1/2
v=0
!
(2k+1)
φ(w) df
(w) dtdx.
Suppose φx,c,s (w)
denotes
the
characteristic
function
of
the
interval
x − c(dj n)−1/2 ,
−1/2
x + s(dj n)
, where c, s are nonnegative integers. Then we have
k
r
X
X
1
|C(j, k)|
≤
(2k + 1)! j=0
l=1
×
v=0
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kM2 kL1 (I2 )
∞
X
Niraj Kumar
x+(l+1)(dj n)−1/2
a2
l=0
2k+1
(Z
b2
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
Z
b2
(Z
Close
x+(l+1)(dj n)−1/2
2
φ(t)(dj n) (dj n − c)
a2
Quit
x+(l)(dj n)−1/2
)
pdj n,v (x)pdj n,v (t)l−4 |t − x|2k+5
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x+(l+1)(dj n)−1/2
Z
×
!
(2k+1)
φ(w)φx,0,l+1 (w) df
(w) dtdx
x
k
r
X
X
1
|C(j, k)|
+
(2k + 1)! j=0
l=1
×
∞
X
Z
(Z
b2
a2
x−(l)(dj n)−1/2
φ(t)(dj n)2 (dj n − c).
x+(l+1)(dj
n)−1/2
)
bdj n,v (x)pdj n,v (t)l−4 |t − x|2k+5
v=0
Z
x
×
x−(l+1)(dj n)−1/2
!
(2k+1)
φ(w)φx,l+1,0 (w) df
(w) dtdx
k
X
1
×
|C(j, k)|
(2k + 1)! j=0
×
∞
X
Z
(Z
b2
Niraj Kumar
a1 +(dj n)−1/2
φ(t)(dj n − c)
(dj n)−1/2
a2
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)
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pdj n,v (x)pdj n,v (t) |t − x|2k+1
JJ
J
v=0
Z
x+(dj n)−1/2
×
x−(dj n)−1/2
k
!
(2k+1)
φ(w)φx,1,1 (w) df
(w) dtdx
r
X
X
1
|C(j, k)|
l−4
≤
(2k + 1)! j=0
l=1
×
∞
X
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
Z
b2
(Z
Go Back
x+(l+1)(dj n)−1/2
φ(t)(dj n − c)
a2
x+(l)(dj n)−1/2
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pdj n,v (x)pdj n,v (t) |t − x|2k+5
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v=0
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b1
(2k+1)
(w) dtdx
×
φx,0,l+1 (w) df
a1
Z b2 (Z x−(l)(dj n)−1/2
k
r
X
X
1
−4
+
|C(j, k)|
l
φ(t)(dj n − c)
(2k + 1)! j=0
a2
x+(l+1)(dj n)−1/2
l=1
)
∞
X
×
pdj n,v (x)pdj n,v (t) |t − x|2k+5
Z
v=0
b1
Z
(2k+1)
×
φx,l+1,0 (w) df
(w) dtdx
a1
Z b2 (Z a1 +(dj n)−1/2
k
X
1
+
|C(j, k)|
φ(t)(dj n − c)
(2k + 1)! j=0
a2
(dj n)−1/2
)
∞
X
×
pdj n,v (x)pdj n,v (t) |t − x|2k+1
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
Niraj Kumar
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Contents
v=0
Z
b1
JJ
J
φ(w)φx,1,1 (w) df (2k+1) (w) dtdx.
×
a1
Applying Lemma 2.1 and using Fubini’s theorem we get
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kM2 kL1 (I2 )
≤ C14
k
X
j=0
II
I
− 2k+1
2
|C(j, k)|(dj n)
r
X
l=1
l
−4
Z
b2
a2
Z
b1
φx,0,l+1 (w) df (2k+1) (w) dx
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a1
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+ C14
k
X
− 2k+1
2
|C(j, k)|(dj n)
j=0
Z
r
X
φx,1,1 (w) df (2k+1) (w) dx
b2
Z
a2
l=1
b1
+
l
Z
−4
b1
φx,l+1,0 (w) df (2k+1) (w) dx
a1
a1
≤ C14
k
X
− 2k+1
2
|C(j, k)|(dj n)
j=0
+ C14
l
k
X
− 2k+1
2
|C(j, k)|(dj n)
b1
Z
r
X
l
Z
−4
φx,0,l+1 (w)dx df (2k+1) (w)
a2
Z
a1
l=1
φx,1,1 (w) df (2k+1) (w) dx
b1
b2
Z
a1
b2
+
−4
l=1
j=0
Z
r
X
b2
φx,l+1,0 (w) df (2k+1) (w) dx
a2
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
Niraj Kumar
a2
≤ C15
k
X
− 2k+1
2
|C(j, k)|(dj n)
j=0
r
X
l
−4
l=1
(Z
b1
a1
Z
w
1
w−(l+1)(dj n)− 2
!
)
(2k+1)
dx df
(w)


Z b1 Z w+(l+1)(dj n)− 12
2k+1

+ C15 |C(j, k)|(dj n)− 2
l−4
dxdf (2k+1) (w)
a1  w
j=0
l=1



Z b1 Z w+(dj n)− 21
(2k+1)


dx df
(w)
+
−1

2
a1
w−(dj n)
≤ C16 n−(k+1) f (2k+1) B.V.(I1 ) .
k
X
r
X
Finally we estimate M3 . It is sufficient to choose the expression without the
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linear combinations. For all t ∈ [0, ∞)\[a1 , b1 ] and all x ∈ I2 , we choose a
δ > 0 such that |t − x| ≥ δ. Thus
kSn (F (t, x)(1 − φ(t)), x)kL1 (I2 )
Z b2 Z ∞
∞
X
≤
(n − c)
pn,v (x)pn,v (t) |f (t)| (1 − φ(t))dtdx
a2
≤
2k+1
X
i=0
0
1
i!
v=0
Z
b2
Z
∞
(n − c)
a2
0
= M4 + M5 ,
∞
X
pn,v (x)pn,v (t) f (i) (t) . |t − x|i (1 − φ(t))dtdx
v=0
say.
2k+2
For sufficiently large t there exist positive constants C17 , C18 such that (t−x)
>
t2k+2 +1
C17 for all t ≥ C18 , t ∈ I2 . Applying Fubini’s theorem and Lemma 2.1, we obtain
Z C18 Z b2 Z ∞ Z b2 X
∞
M4 =
+
(n − c)pn,v (x)pn,v (t) |f (t)| (1 − φ(t))dtdx
0
≤ C19 n
a2
−(k+1)
C18
Z
C20
a2
v=0
|f (t)| dt
∞
∞
b2 X
2k+2
(t − x)
1
pn,v (x)pn,v (t) 2k+2
|f (t)| dxdt
C17 C20 a2 v=0
(t
+ 1)
Z C20
Z ∞
−(k+1)
≤ C20 n
|f (t)| dt +
|f (t)| dt
+
Z
0
≤ C21 n
−(k+1)
Niraj Kumar
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Z
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
kf kL1 [0,∞) .
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C20
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Finally by the Lemma 2.3, we have
n
o
M5 ≤ C22 n−(k+1) kf kL1 (I2 ) + f (2k+1) L1 (I2 ) .
Combining M4 and M5 , we obtain
o
n
M3 ≤ C23 n−(k+1) kf kL1 [0,∞) + f (2k+1) L1 (I2 ) .
This completes the proof of (2.2) of the lemma.
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
Niraj Kumar
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3.
Rate Of Convergence
Theorem 3.1. Let f ∈ Lp [0, ∞), p ≥ 1. Then for n sufficiently large
n
o
kSn,k (f, ·) − f kLp (I2 ) ≤ C24 ω2k+2 (f, n−1/2 , p, I1 ) + n−(k+1) kf kLp [0,∞) ,
where C24 is a constant independent of f and n.
Proof. We can write
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
kSn,k (f, ∗) − f kLp (I2 )
≤ kSn,k (f − fη,2k+2 , ∗)kLp (I2 )
Niraj Kumar
+ kSn,k (fη,2k+2 , ∗) − fη,2k+2 )kLp (I2 ) + k(fη,2k+2 − f )kL
= E1 + E2 + E3 ,
p (I2 )
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say.
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It is well known that
(2k+1) fη,2k+2 B.V.(I3 )
(2k+2) = fη,2k+2 .
L1 (I3 )
Therefore from Lemma 2.4 (p > 1) and (p = 1) we have
−(k+1) (2k+2) E2 ≤ C25 n
+ kfη,2k+2 kLp [0,∞)
fη,2k+2 Lp (I3 )
≤ C26 n−(k+1) n−(k+2) ω2k+2 (f, η, p, I1 ) + kf kLp [0,∞) ,
which follows from the properties of Steklov means.
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Let φ(t) be the characteristic function of I3 , we have
Sn ((f − fη,2k+2 )(t), x)
= Sn (φ(t)(f − fη,2k+2 )(t), x) + Sn ((1 − φ(t))(f − fη,2k+2 )(t), x)
= E4 + E5 ,
say.
By Hölder’s inequality
Z b2
Z b2 Z
p
|E4 | dx ≤
a2
a2
b1
(n − c)
a1
∞
X
pn,v (x)pn,v (t) |(f − fη,2k+2 )(t)|p dtdx.
v=0
Applying Fubini’s theorem, we have
Z b2
|E4 |p dx ≤ kf − fη,2k+2 kLp(I2 ) .
a2
Niraj Kumar
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Similarly, for all p ≥ 1
Contents
kE5 kLp(I ) ≤ C27 η
2
−(k+1)
kf − fη,2k+2 kLp(0,∞) .
Consequently, via the property of Steklov means, we find that
n
o
kSn (f − f η, 2k + 2, ·)kLp(I2) ≤ C28 ω2k+2 (f, η, p, I1 ) + η −(k+1) kf kLp[0,∞) .
Hence
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
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n
o
E1 ≤ C29 ω2k+2 (f, η, p, I1 ) + η −(k+1) kf kLp[0,∞) .
Thus, with η = n−1/2 , the result follows.
This completes the proof of the theorem.
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References
[1] J.L. DURRMEYER, Une Formula d’ inversion de la Transformee de
Laplace: Application a la Theorie des Moments, These de 3c cycle, Faculte des Science de l’ University de Paris, 1967.
[2] S. GOLDBERG AND V. MEIR, Minimum moduli of ordinary differential
operators, Proc. London Math. Soc., 23 (3) (1971), 1–15.
[3] S. GUO, On the rate of convergence of Durrmeyer operator for functions
of bounded variation, J. Approx. Theory, 51(1987), 183–192.
Rate Of Convergence For A
General Sequence Of
Durrmeyer Type Operators
[4] V. GUPTA, Simultaneous approximation by Szasz-Durrmeyers operators,
The Math. Student, 64 (1-4) (1995), 27–36.
Niraj Kumar
[5] V. GUPTA, A note on modified Baskakov operators, Approx.Theory and
its Appl., 10(3) (1994), 74–78.
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Contents
[6] E. HEWITT AND K. STROMBERG, Real and Abstract Analysis, McGraw
Hill, New York (1956).
[7] C.P. MAY, Saturation and inverse theorem for combination of a class of
exponential type operators, Canad. J. Math., 28 (1976), 1224–1250.
[8] R.P. SINHA, P.N. AGRAWAL AND V. GUPTA, On simultaneous approximation by modified Baskakov operators, Bull Soc. Math Belg. Ser. B., 44
(1991), 177–188.
[9] H.M. SRIVASTAVA AND V. GUPTA, A certain family of summation integral type operators, Math and Comput. Modelling, 37 (2003), 1307–1315.
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[10] A.F. TIMAN, Theory of Approximation of Functions of a Real Variable,
Hindustan Publ. Corp., Delhi 1966.
[11] Y. WANG AND S. GUO, Rate of approximation of function of bounded
variation by modified Lupas operators, Bull. Austral. Math. Soc., 44
(1991), 177–188.
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General Sequence Of
Durrmeyer Type Operators
Niraj Kumar
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