# Rate of Convergence of Modified Baskakov-Durrmeyer Type Operators for Functions of Bounded Variation

Authors: Prashantkumar Patel; Vishnu Narayan Mishra
Journal: Journal of Difference Equations
 (2014)

---

## Abstract

We study a certain integral modification of well-known Baskakov operators with weight function of beta basis function. We establish rate of convergence for these operators for functions having derivative of bounded variation. Also, we discuss Stancu type generalization of these operators.


---

## Body

## 1. Introduction

The integral modification of Baskakov operators having weight function of some beta basis function are defined as the following: forx
∈
[
0
,
∞
), γ
>
0,
(1)
B
n
,
γ
(
f
,
x
)
=
∑
k
=
1
∞
p
n
,
k
,
γ
(
x
)
∫
0
∞
b
n
,
k
,
γ
(
t
)
f
(
t
)
d
t
+
(
1
+
γ
x
)
-
n
f
(
0
)
=
∫
0
∞
W
n
,
γ
(
x
,
t
)
f
(
t
)
d
t
,
where
(2)
p
n
,
k
,
γ
(
x
)
=
Γ
(
n
/
γ
+
k
)
Γ
(
k
+
1
)
Γ
(
n
/
γ
)
·
(
γ
x
)
k
(
1
+
γ
x
)
(
n
/
γ
)
+
k
,
b
n
,
k
,
γ
(
t
)
=
γ
Γ
(
n
/
γ
+
k
+
1
)
Γ
(
k
)
Γ
(
n
/
γ
+
1
)
·
(
γ
t
)
k
-
1
(
1
+
γ
t
)
(
n
/
γ
)
+
k
+
1
,
W
n
,
γ
(
x
,
t
)
=
∑
k
=
1
∞
p
n
,
k
,
γ
(
x
)
∫
0
∞
b
n
,
k
,
γ
(
t
)
d
t
+
(
1
+
γ
x
)
-
n
/
γ
δ
(
t
)
,
δ
(
t
) being the Dirac delta function.The operators defined by (1) were introduced by Gupta [1]; these operators are different from the usual Baskakov-Durrmeyer operators. Actually these operators satisfy condition B
n
,
γ
(
a
t
+
b
,
x
)
=
a
x
+
b, where a and b are constants. In [1], the author estimated some direct results in simultaneous approximation for these operators (1). In particular case γ
=
1, the operators (1) reduce to the operators studied in [2, 3].In recent years a lot of work has been done on such operators. We refer to some of the important papers on the recent development on similar type of operators [4–9]. The rate of convergence for certain Durrmeyer type operators and the generalizations is one of the important areas of research in recent years. In present article, we extend the studies and here we estimate the rate of convergence for functions having derivative of bounded variation.We denoteϕ
n
,
γ
(
x
,
t
)
=
∫
0
t
‍
W
n
,
γ
(
x
,
s
)
d
s; then, in particular, we have
(3)
ϕ
n
,
γ
(
x
,
∞
)
=
∫
0
∞
W
n
,
γ
(
x
,
s
)
d
s
=
1
.ByD
B
r
(
0
,
∞
)
,
r
≥
0 we denote the class of absolutely continuous functions f defined on the interval (
0
,
∞
) such that,(i)
f
(
t
)
=
O
(
t
r
), t
→
∞.
(ii)
having a derivativef
′ 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 functionf
∈
B
D
r
(
0
,
∞
) possess for each c
>
0 a representation
(4)
f
(
x
)
=
f
(
c
)
+
∫
c
x
ψ
(
t
)
d
t
,
x
≥
c
.


## 2. Rate of Convergence forB
n
,
γ

Lemma 1 (see [1]).
Let the functionT
n
,
m
,
γ
(
x
), m
∈
N
∪
{
0
}, be defined as
(5)
T
n
,
m
,
γ
(
x
)
=
B
n
,
γ
(
(
t
-
x
)
m
,
x
)
=
∑
k
=
1
∞
p
n
,
k
,
γ
(
x
)
∫
0
∞
b
n
,
k
,
γ
(
t
)
(
t
-
x
)
m
d
t
+
(
1
+
γ
x
)
-
n
/
γ
(
-
x
)
m
.
Then it is easily verified that, for eachx
∈
(
0
,
∞
)
,
T
n
,
0
,
γ
(
x
)
=
1, T
n
,
1
,
γ
(
x
)
=
0, and T
n
,
2
,
γ
(
x
)
=
2
x
(
1
+
γ
x
)
/
(
n
-
γ
), and also the following recurrence relation holds:
(6)
(
n
-
γ
m
)
T
n
,
m
+
1
,
γ
(
x
)
=
x
(
1
+
γ
x
)
×
[
T
n
,
m
,
γ
(
1
)
(
x
)
+
2
m
T
n
,
m
-
1
,
γ
(
x
)
]
+
m
(
1
+
2
γ
x
)
T
n
,
m
,
γ
(
x
)
.
From the recurrence relation, it can be easily be verified that for allx
∈
[
0
,
∞
), we have T
n
,
m
,
γ
(
x
)
=
O
(
n
-
[
(
m
+
1
)
/
2
]
).Remark 2.
From Lemma1, using Cauchy-Schwarz inequality, it follows that
(7)
B
n
,
γ
(
|
t
-
x
|
,
x
)
≤
[
B
n
,
γ
(
(
t
-
x
)
2
,
x
)
]
1
/
2
≤
2
x
(
1
+
γ
x
)
n
-
γ
.Lemma 3.
Letx
∈
(
0
,
∞
) and W
n
,
γ
(
x
,
t
) be the kernel defined in (1). Then for n being sufficiently large, one has (a)
ϕ
n
,
γ
(
x
,
y
)
=
∫
0
y
‍
W
n
,
γ
(
x
,
t
)
d
t
≤
2
x
(
1
+
γ
x
)
/
(
n
-
γ
)
(
x
-
y
)
2
,
0
≤
y
<
x.
(b)
1
-
ϕ
n
,
γ
(
x
,
z
)
=
∫
z
∞
‍
W
n
,
γ
(
x
,
t
)
d
t
≤
2
x
(
1
+
γ
x
)
/
(
n
-
γ
)
(
z
-
x
)
2
,
x
<
z
<
∞.Proof.
First we prove (a); by using Lemma1, we have
(8)
∫
0
y
‍
W
n
,
γ
(
x
,
t
)
d
t
≤
∫
0
y
(
x
-
t
)
2
(
x
-
y
)
2
W
n
,
γ
(
x
,
t
)
d
t
≤
(
x
-
y
)
-
2
T
n
,
2
,
γ
(
x
)
≤
2
x
(
1
+
γ
x
)
(
n
-
γ
)
(
x
-
y
)
2
.
The proof of (b) is similar; we omit the details.Theorem 4.
Letf
∈
D
B
r
(
0
,
∞
), r
∈
N, and x
∈
(
0
,
∞
). Then for n being sufficiently large, we have
(9)
|
B
n
,
γ
(
f
,
x
)
-
f
(
x
)
|
≤
2
(
1
+
γ
x
)
n
-
γ
∑
k
=
1
[
n
]
‍
⋁
x
-
x
/
k
x
+
x
/
k
(
(
f
′
)
x
)
+
x
n
⋁
x
-
x
/
n
x
+
x
/
n
(
(
f
′
)
x
)
+
2
(
1
+
γ
x
)
(
n
-
γ
)
x
×
[
|
f
(
2
x
)
-
f
(
x
)
-
x
f
′
(
x
+
)
|
+
|
f
(
x
)
|
]
+
2
x
(
1
+
γ
x
)
n
-
γ
[
M
2
r
O
(
n
-
r
/
2
)
+
|
f
′
(
x
+
)
|
]
+
x
(
1
+
γ
x
)
2
(
n
-
γ
)
|
f
′
(
x
+
)
-
f
′
(
x
-
)
|
,
where the auxiliary function f
x is given by
(10)
f
x
(
t
)
=
{
f
(
t
)
-
f
(
x
-
)
,
0
≤
t
<
x
;
0
,
t
=
x
;
f
(
t
)
-
f
(
x
+
)
,
x
<
t
<
∞
.
⋁
a
b
f
(
x
) denotes the total variation of f
x on [
a
,
b
].Proof.
By the application of mean value theorem, we have(11)
B
n
,
γ
(
f
,
x
)
-
f
(
x
)
=
∫
0
∞
W
n
,
γ
(
x
,
t
)
(
f
(
t
)
-
f
(
x
)
)
d
t
=
∫
0
∞
∫
x
t
W
n
,
γ
(
x
,
t
)
(
f
′
(
u
)
d
u
)
d
t
.
Also, using the identity(12)
f
′
(
u
)
=
f
′
(
x
+
)
+
f
′
(
x
-
)
2
+
(
f
′
)
x
(
u
)
+
f
′
(
x
+
)
-
f
′
(
x
-
)
2
sgn
⁡
(
u
-
x
)
+
[
f
′
(
x
)
-
f
′
(
x
+
)
+
f
′
(
x
-
)
2
]
χ
x
(
u
)
,
where
(13)
χ
x
(
u
)
=
{
1
,
u
=
x
;
0
,
u
≠
x
,
we can see that(14)
∫
0
∞
(
∫
x
t
‍
(
f
′
(
x
)
-
f
′
(
x
+
)
+
f
′
(
x
-
)
2
)
χ
x
(
u
)
d
u
)
×
W
n
,
γ
(
t
,
x
)
d
t
=
0
.
Also,(15)
∫
0
∞
‍
(
∫
x
t
‍
(
f
′
(
x
+
)
-
f
′
(
x
-
)
2
)
sgn
⁡
(
u
-
x
)
d
u
)
W
n
,
γ
(
t
,
x
)
d
t
=
f
′
(
x
+
)
-
f
′
(
x
-
)
2
B
n
,
γ
(
|
t
-
x
|
,
x
)
,
∫
0
∞
(
∫
x
t
(
f
′
(
x
+
)
+
f
′
(
x
-
)
2
)
d
u
)
W
n
,
γ
(
t
,
x
)
d
t
=
f
′
(
x
+
)
+
f
′
(
x
-
)
2
B
n
,
γ
(
t
-
x
,
x
)
.
Substitute value off
′
(
u
) from (12) in (11) and using (14) and (15), we get
(16)
|
B
n
,
γ
(
f
,
x
)
-
f
(
x
)
|
≤
|
∫
x
∞
(
∫
x
t
(
f
′
)
x
(
u
)
d
u
)
W
n
,
γ
(
x
,
t
)
d
t
|
+
|
∫
0
x
(
∫
x
t
(
f
′
)
x
(
u
)
d
u
)
W
n
,
γ
(
x
,
t
)
d
t
|
+
|
f
′
(
x
+
)
-
f
′
(
x
-
)
|
2
B
n
,
γ
(
|
t
-
x
|
,
x
)
+
|
f
′
(
x
+
)
+
f
′
(
x
-
)
|
2
B
n
,
γ
(
t
-
x
,
x
)
.
Using Lemma1 and Remark 2, we obtain
(17)
|
B
n
,
γ
(
f
,
x
)
-
f
(
x
)
|
≤
|
∫
x
∞
‍
(
∫
x
t
‍
(
f
′
)
x
(
u
)
d
u
)
W
n
,
γ
(
x
,
t
)
d
t
|
+
|
∫
0
x
‍
(
∫
x
t
‍
(
f
′
)
x
(
u
)
d
u
)
W
n
,
γ
(
x
,
t
)
d
t
|
+
|
f
′
(
x
+
)
-
f
′
(
x
-
)
|
2
2
x
(
1
+
γ
x
)
n
-
γ
=
|
P
n
,
γ
(
f
,
x
)
|
+
|
Q
n
,
γ
(
f
,
x
)
|
+
|
f
′
(
x
+
)
-
f
′
(
x
-
)
|
2
2
x
(
1
+
γ
x
)
n
-
γ
.
On applying Lemma3 with y
=
x
-
x
/
n and integrating by parts, we have
(18)
|
Q
n
,
γ
(
f
,
x
)
|
=
|
∫
0
x
‍
∫
x
t
‍
(
f
′
)
x
(
u
)
d
u
d
t
(
ϕ
n
,
γ
(
x
,
y
)
)
|
=
|
∫
0
x
‍
ϕ
n
,
γ
(
x
,
y
)
(
f
′
)
x
(
t
)
d
t
|
≤
(
∫
0
y
‍
+
∫
y
x
‍
)
|
(
f
′
)
x
(
t
)
|
|
ϕ
n
,
γ
(
x
,
t
)
|
d
t
≤
2
x
(
1
+
γ
x
)
n
-
γ
∫
0
y
‍
⋁
t
x
(
(
f
′
)
x
)
1
(
x
-
t
)
2
d
t
+
∫
y
x
‍
⋁
t
x
(
(
f
′
)
x
)
d
t
≤
2
x
(
1
+
γ
x
)
n
-
γ
∫
0
y
‍
⋁
t
x
(
(
f
′
)
x
)
1
(
x
-
t
)
2
d
t
+
x
n
⋁
x
-
x
/
n
x
(
(
f
′
)
x
)
=
2
x
(
1
+
γ
x
)
n
-
γ
∫
1
n
‍
⋁
x
-
x
/
u
x
(
(
f
′
)
x
)
d
u
+
x
n
⋁
x
-
x
/
n
x
(
(
f
′
)
x
)
≤
2
x
(
1
+
γ
x
)
n
-
γ
∑
k
=
1
[
n
]
‍
⋁
x
-
x
/
k
x
(
(
f
′
)
x
)
+
x
n
⋁
x
-
x
/
n
x
(
(
f
′
)
x
)
,
where u
=
x
/
(
x
-
t
).
On the other hand, we have(19)
|
P
n
,
γ
(
f
,
x
)
|
=
|
∫
x
∞
(
∫
x
t
(
f
′
)
x
(
u
)
d
u
)
W
n
,
γ
(
x
,
t
)
d
t
|
=
|
∫
2
x
∞
(
∫
x
t
(
f
′
)
x
(
u
)
d
u
)
W
n
,
γ
(
x
,
t
)
d
t
+
∫
x
2
x
(
∫
x
t
‍
(
f
′
)
x
(
u
)
d
u
)
d
t
(
1
-
ϕ
n
,
γ
(
x
,
t
)
)
|
≤
|
∫
2
x
∞
(
f
(
t
)
-
f
(
x
)
)
W
n
,
γ
(
x
,
t
)
d
t
|
+
|
f
′
(
x
+
)
|
|
∫
2
x
∞
(
t
-
x
)
W
n
,
γ
(
x
,
t
)
d
t
|
+
|
∫
x
2
x
(
f
′
)
x
(
u
)
d
u
|
|
1
-
ϕ
x
,
γ
(
x
,
2
x
)
|
+
∫
x
2
x
|
(
f
′
)
x
(
t
)
|
|
1
-
ϕ
x
,
γ
(
x
,
t
)
|
d
t
≤
[
|
f
(
x
)
|
x
2
M
x
∫
2
x
∞
W
n
,
γ
(
x
,
t
)
t
r
|
t
-
x
|
d
t
+
|
f
(
x
)
|
x
2
∫
2
x
∞
W
n
,
γ
(
x
,
t
)
(
t
-
x
)
2
d
t
]
+
[
|
f
′
(
x
+
)
|
∫
2
x
∞
W
n
,
γ
(
x
,
t
)
|
t
-
x
|
d
t
]
+
2
(
1
+
γ
x
)
(
n
-
γ
)
x
|
f
(
2
x
)
-
f
(
x
)
-
x
f
′
(
x
+
)
|
+
2
(
1
+
γ
x
)
n
-
γ
∑
k
=
1
[
n
]
⋁
x
x
+
x
/
k
(
(
f
′
)
x
)
+
x
n
⋁
x
x
+
x
/
n
(
(
f
′
)
x
)
=
R
n
,
γ
(
f
,
x
)
+
S
n
,
γ
(
f
,
x
)
+
2
(
1
+
γ
x
)
(
n
-
γ
)
x
|
f
(
2
x
)
-
f
(
x
)
-
x
f
′
(
x
+
)
|
+
2
(
1
+
γ
x
)
n
-
γ
∑
k
=
1
[
n
]
⋁
x
x
+
x
/
k
(
(
f
′
)
x
)
+
x
n
⋁
x
x
+
x
/
n
(
(
f
′
)
x
)
.
Applying Holder’s inequality, Remark2, and Lemma 1, we have
(20)
R
n
,
γ
(
f
,
x
)
≤
M
x
(
∫
2
x
∞
W
n
,
γ
(
x
,
t
)
t
2
r
d
t
)
1
/
2
×
(
∫
0
∞
W
n
,
γ
(
x
,
t
)
(
t
-
x
)
2
d
t
)
1
/
2
+
|
f
(
x
)
|
x
2
∫
2
x
∞
W
n
,
γ
(
x
,
t
)
(
t
-
x
)
2
d
t
≤
M
2
r
O
(
n
-
r
/
2
)
2
x
(
1
+
γ
x
)
n
-
γ
+
|
f
(
x
)
|
2
(
1
+
γ
x
)
(
n
-
γ
)
x
.
Also,(21)
S
n
,
γ
(
f
,
x
)
≤
|
f
(
x
+
)
|
∫
0
∞
W
n
(
t
,
x
)
|
t
-
x
|
d
t
≤
|
f
(
x
+
)
|
2
x
(
1
+
γ
x
)
n
-
γ
.
Combining the estimates (17)–(21), we get the desired results.
This completes the proof of Theorem.


## 3. Rate of Convergence for Stancu Type Generalization ofB
n
,
γ

In 1968, Stancu introduces Bernstein-Stancu operators in [10], a sequence of the linear positive operators depending on two parameters α and β satisfying the condition 0
≤
α
≤
β. Recently many researchers applied this approach to many operators; for details see [11–17]. For f
∈
C
[
0
,
∞
), Stancu generalization of operators (1) is as follows:
(22)
B
n
,
γ
α
,
β
(
f
(
t
)
,
x
)
=
∑
k
=
1
∞
p
n
,
k
,
γ
(
x
)
∫
0
∞
b
n
,
k
,
γ
(
t
)
f
(
n
t
+
α
n
+
β
)
d
t
+
p
n
,
0
,
γ
(
x
)
f
(
α
n
+
β
)
=
∫
0
∞
W
n
,
γ
(
x
,
t
)
f
(
n
t
+
α
n
+
β
)
d
t
,
where p
n
,
k
,
γ
(
x
)
,
b
n
,
k
,
γ
(
x
), and W
n
,
γ
(
x
,
t
)are as defined in (1).Lemma 5 (see [18]).
If we define the central moments, for everym
∈
N as
(23)
μ
n
,
m
,
γ
α
,
β
(
x
)
=
B
n
,
γ
α
,
β
(
(
t
-
x
)
m
,
x
)
=
∑
k
=
1
∞
p
n
,
k
,
γ
(
x
)
∫
0
∞
b
n
,
k
,
γ
(
t
)
(
n
t
+
α
n
+
β
-
x
)
m
d
t
+
p
n
,
0
,
γ
(
x
)
(
α
n
+
β
-
x
)
m
,
thenμ
n
,
0
,
γ
α
,
β
(
x
)
=
1, μ
n
,
1
,
γ
α
,
β
(
x
)
=
(
α
-
β
x
)
/
(
n
+
β
), and
(24)
μ
n
,
2
,
γ
α
,
β
(
x
)
=
α
2
(
n
+
β
)
2
+
x
(
2
n
2
-
2
n
α
β
+
2
α
β
γ
)
(
n
+
β
)
2
(
n
-
γ
)
+
x
2
(
n
β
2
+
2
n
2
γ
-
β
2
γ
)
(
n
+
β
)
2
(
n
-
γ
)
.
Forn
>
m we have the following recurrence relation:
(25)
(
n
-
γ
m
)
(
n
+
β
)
μ
n
,
m
+
1
,
γ
α
,
β
(
x
)
=
n
x
(
1
+
γ
x
)
[
(
μ
n
,
m
,
γ
α
,
β
)
(
1
)
(
x
)
+
m
μ
n
,
m
-
1
,
γ
α
,
β
(
x
)
]
+
[
m
n
+
n
2
x
-
(
2
γ
m
-
n
)
(
α
-
(
n
+
β
)
x
)
]
μ
n
,
m
,
γ
α
,
β
(
x
)
+
[
m
γ
(
n
+
β
)
(
α
n
+
β
-
x
)
2
-
m
n
(
α
n
+
β
-
x
)
]
×
μ
n
,
m
-
1
,
γ
α
,
β
(
x
)
.
From the recurrence relation, it can be easily verified that for allx
∈
[
0
,
∞
), we have μ
n
,
m
,
γ
α
,
β
(
x
)
=
O
(
n
-
[
(
m
+
1
)
/
2
]
).Remark 6.
Observe thatB
n
,
γ
α
,
β preserve constant functions but not linear functions. If α
=
β
=
0 these operators reduce to the operators defined in (1). Notice that
(26)
μ
n
,
2
,
γ
α
,
β
(
x
)
=
[
β
2
γ
(
n
+
β
)
2
+
2
n
2
(
n
+
β
)
2
(
n
-
γ
)
]
x
(
1
+
γ
x
)
+
β
[
(
2
α
+
β
)
γ
-
n
(
β
+
2
α
γ
)
]
γ
(
n
+
β
)
2
(
n
-
γ
)
x
+
α
2
(
n
+
β
)
2
.Remark 7.
From Lemma3, taking n to be sufficiently large and x
∈
(
0
,
∞
), we observe that
(27)
μ
n
,
2
,
γ
α
,
β
(
x
)
≤
C
x
(
1
+
γ
x
)
n
-
γ
,
where C is positive constant.Remark 8.
Applying the Cauchy-Schwarz inequality and keeping the same condition as in Remark7 for x, n, and C, we derive from Lemma 5 that
(28)
B
n
,
γ
α
,
β
(
|
t
-
x
|
,
x
)
≤
[
B
n
,
γ
α
,
β
(
(
t
-
x
)
2
,
x
)
]
1
/
2
≤
C
x
(
1
+
γ
x
)
n
-
γ
.Lemma 9.
Letx
∈
(
0
,
∞
) and W
n
,
γ
(
x
,
t
) be the kernel defined in (1). Then for n being sufficiently large, we have (c)
ϕ
n
,
γ
(
x
,
y
)
=
∫
0
y
‍
W
n
,
γ
(
x
,
t
)
d
t
=
C
x
(
1
+
γ
x
)
/
(
n
-
γ
)
(
x
-
y
)
2
,
0
≤
y
<
x.
(d)
1
-
ϕ
n
,
γ
(
x
,
z
)
=
∫
z
∞
‍
W
n
,
γ
(
x
,
t
)
d
t
=
C
x
(
1
+
γ
x
)
/
(
n
-
γ
)
(
z
-
x
)
2
,
x
<
z
<
∞.The proof is the same as Lemma3, thus we omit the details.Theorem 10.
Letf
∈
D
B
r
(
0
,
∞
), r
∈
N, and x
∈
(
0
,
∞
). Then for n being sufficiently large, we have
(29)
|
B
n
,
γ
α
,
β
(
f
,
x
)
-
f
(
x
)
|
≤
C
(
1
+
γ
x
)
n
-
γ
∑
k
=
1
[
n
]
‍
⋁
x
-
x
/
k
x
+
x
/
k
(
(
f
′
)
x
)
+
x
n
⋁
x
-
x
/
n
x
+
x
/
n
(
(
f
′
)
x
)
+
C
(
1
+
γ
x
)
(
n
-
γ
)
x
×
[
|
f
(
2
x
)
-
f
(
x
)
-
x
f
′
(
x
+
)
|
+
|
f
(
x
)
|
]
+
C
x
(
1
+
γ
x
)
n
-
γ
×
[
M
1
2
r
O
(
n
-
r
/
2
)
+
|
f
′
(
x
+
)
|
]
+
1
2
C
x
(
1
+
γ
x
)
(
n
-
γ
)
|
f
′
(
x
+
)
-
f
′
(
x
-
)
|
+
1
2
|
f
′
(
x
+
)
+
f
′
(
x
-
)
|
|
α
-
β
x
|
n
+
β
,
where the auxiliary functions f
x and ⋁
a
b
f
(
x
) were defined in Theorem 4.
The proof of the above theorem follows along the lines of Theorem4; thus we omit the details.



---

*Source: 235480-2014-07-02.xml*