# Some Applications of New Modifiedq-Integral Type Operators

Authors: R. P. Pathak; Shiv Kumar Sahoo
Journal: International Journal of Mathematics and Mathematical Sciences
 (2015)

---

## Abstract

We introduce a new sequence ofq-integral operators. We show that it is a weighted
approximation process in the polynomial space of continuous functions defined on unit interval. Weighted
statistical approximation theorem, Korovkin type theorems for fuzzy continuous functions, and an estimate for
the rate of convergence for these operators.


---

## Body

## 1. Introduction

The study ofq-Calculus is a generalization of any subjects, such as hyper geometric series, complex analysis, and particle physics. Currently it continues being an important subject of study. It has been shown that positive linear operators constructed by q-numbers are quite effective as far as the rate of convergence is concerned and we can have some unexpected results, which are not observed for classical case. In the last decade, some new generalizations of well-known positive linear operators, based on q-integers, were introduced and studied by several authors. For example, q-Meyer Konig and Zeller operators were studied by Trif [1], Dogru and Duman [2], Aral and Gupta [3], and so forth. In 20011, Aral and Gupta [4, 5] introduced a q-generalization of the classical Baskakov operators. In 2012, Sharma [6, 7] introduced the q-Durrmeyer type operators. Orkcu and Dogru [8] introduced Kantorovich type generalization of q-Szasz-Mirakjan operators and discussed their A-statistical approximation properties. In this paper motivated by Sharma we introduced a q-analogue of the q-Durrmeyer operators and we study better rate of convergence and statistical approximation properties.We mention some important definitions ofq-Calculus.Definition 1.
For any fixed real numbersq
>
0 and k
∈
N, the q-integers are defined by(1)
k
q
=
k
,
if
q
=
1
,
1
+
q
+
q
1
+
q
2
+
⋯
+
q
k
-
1
,
if
q
≠
1
.
In this way for real numbern one may write(2)
n
q
=
1
-
q
n
1
-
q
;
q
≠
1
.Definition 2.
Theq-factorial is defined by(3)
k
q
!
=
1
,
if
k
=
0
,
1
q
·
2
q
·
⋯
·
k
q
,
if
k
=
1,2
,
…
.Definition 3.
For any numberk
∈
(
0
,
n
), the q-binomial coefficient is defined by(4)
n
k
q
=
n
q
!
k
q
!
n
-
k
q
!
.Aral and Gupta [4] introduced a q-generalization of the classical Baskakov operators. For all f
∈
C
[
0
,
∞
), q
∈
(
0,1
), and each positive integer n, the operators are defined as (5)
B
n
,
q
f
x
=
∑
k
=
0
∞
n
+
k
-
1
k
q
q
k
k
-
1
/
2
k
x
k
1
+
q
x
q
n
+
k
f
k
q
q
q
-
1
n
q
.For q
=
1, the above operators become classical Baskakov operators [9].Deo et al. [10] introduced new version of Bernstein-Durrmeyer type operators defined as follows: for f
∈
C
I
n, where I
n
=
[
0
,
n
/
n
+
1
], (6)
B
n
,
q
f
x
=
n
1
+
1
n
2
∑
k
=
0
n
p
n
,
k
x
∫
0
n
/
n
+
1
p
n
,
k
t
f
t
d
t
,where(7)
p
n
,
k
x
=
1
+
1
n
n
n
k
x
k
n
n
+
1
-
x
n
-
k
,and they established some approximation results on it.Sharma [6] introduced the following q-Durrmeyer type operators defined as follows: for f
∈
C
I
n
,
q, where I
n
,
q
=
0
,
[
n
]
q
/
[
n
+
1
]
q,(8)
M
n
,
q
⋆
f
x
=
n
+
1
q
2
n
q
∑
k
=
0
n
q
-
k
p
n
,
k
⋆
q
;
x
×
⋯
×
∫
0
n
q
/
n
+
1
q
p
n
,
k
⋆
q
;
q
t
f
t
d
q
t
,where(9)
p
n
,
k
⋆
q
;
x
=
n
k
q
n
+
1
q
n
q
x
k
1
-
n
+
1
q
n
q
x
q
n
-
k
,and they established some approximation results on it.In this paper motivated by Sharma [6, 7, 10–12] we introduce a q-analogue of the q-Baskakov-Durrmeyer type operators defined as follows: for f
∈
C
I
n
,
q, (10)
M
n
,
q
f
x
=
n
+
1
q
2
n
q
·
∑
k
=
0
∞
b
n
,
k
q
;
x
∫
0
n
q
/
n
+
1
q
p
n
,
k
⋆
q
;
q
t
f
t
d
q
t
,where we set b
n
,
k
(
q
;
x
)
=
n
+
k
-
1
k
q
q
k
2
-
k
-
2
/
2
x
k
/
(
1
+
x
)
q
n
+
k.Kasana et al. [13] obtained a sequence of modified Szâsz type operators for integrable function on [
0
,
∞
) defined as(11)
M
n
,
x
f
x
≡
M
n
,
x
f
y
;
t
=
n
∑
k
=
0
∞
b
n
,
k
t
∫
0
∞
b
n
,
k
y
f
x
+
y
d
y
,where x and t belong to [
0
,
∞
) and x is fixed.In this paper, motivated by Kasana and Sharma, we introduce aq-analogue of the q-Baskakov-Durrmeyer type operators defined as follows: for f
∈
C
I
n
,
q, (12)
M
n
,
q
,
x
⋆
f
t
=
n
+
1
q
2
n
q
∑
k
=
0
∞
b
n
,
k
q
;
t
×
⋯
×
∫
0
n
q
/
n
+
1
q
p
n
,
k
⋆
q
;
q
y
f
x
+
y
d
q
y
,where x and t belong to I
n
,
q and x is fixed.The aim of this paper is to study some approximation properties of a new generalization of operators based onq-integers. We estimate moments for these operators. Also, we study statistical convergence and Korovkin type theorems for fuzzy continuous functions. Finally, we give better error estimations for operators (10) and (12).


## 2. Estimation of Moments

We use Lemma5 [6] for s
=
1,2
,
… and, by the definition of q-Beta function, we get ∫
0
[
n
]
q
/
[
n
+
1
]
q
p
n
,
k
⋆
(
q
;
q
t
)
t
s
d
q
t
=
[
n
]
q
s
+
1
/
[
n
+
1
]
q
s
+
1
q
k
[
n
]
q
!
[
k
+
s
]
q
!
/
[
k
]
q
!
[
s
+
n
+
1
]
q
!.Theorem 4.
Let the sequence of positive linear operators(
M
n
,
q
(
f
)
)
(
x
) be defined by (10). For all n
∈
N; q
∈
(
0,1
); f
∈
C
I
n
,
q; x
∈
I
n
,
q, one gets (13)
M
n
,
q
1
x
=
1
,
M
n
,
q
t
x
=
n
q
n
q
x
+
1
n
+
2
q
n
+
1
q
,
M
n
,
q
t
2
x
=
1
+
q
n
q
2
+
q
1
+
q
2
x
n
q
3
n
+
3
q
n
+
2
q
n
+
1
q
2
+
q
3
x
2
n
q
3
n
+
1
q
n
+
3
q
n
+
2
q
n
+
1
q
2
.Proof.
By using Lemma5 and letting f
(
t
)
=
1 in the operators M
n
,
q be defined by (10), we get (14)
M
n
,
q
1
x
=
n
+
1
q
2
n
q
∑
k
=
0
∞
b
n
,
k
q
;
x
∫
0
n
q
/
n
+
1
q
p
n
,
k
q
;
q
t
1
d
q
t
=
n
+
1
q
2
n
q
∑
k
=
0
∞
b
n
,
k
q
;
x
n
q
n
+
1
q
q
k
n
q
!
n
+
1
q
!
=
∑
k
=
0
∞
q
k
k
-
2
/
2
n
+
k
-
1
k
q
x
k
1
+
x
q
-
n
+
k
=
1
.Again, we set f
(
t
)
=
t in the operators M
n
,
q; we get (15)
M
n
,
q
t
x
=
n
+
1
q
2
n
q
∑
k
=
0
∞
b
n
,
k
q
;
x
n
q
2
n
+
1
q
2
q
k
n
q
!
k
+
1
q
!
k
q
!
n
+
2
q
!
=
n
q
n
+
2
q
n
+
1
q
∑
k
=
0
∞
b
n
,
k
q
;
x
k
+
1
q
=
n
q
n
q
x
+
1
n
+
2
q
n
+
1
q
.Similarly, we setf
(
t
)
=
t
2 in the operators M
n
,
q; we get (16)
M
n
,
q
t
2
x
=
n
+
1
q
2
n
q
∑
k
=
0
∞
b
n
,
k
q
;
x
n
q
3
n
+
1
q
3
q
k
n
q
!
k
+
2
q
!
k
q
!
n
+
3
q
!
=
n
q
2
∑
k
=
0
∞
b
n
,
k
q
;
x
k
+
1
q
k
+
2
q
n
+
3
q
n
+
2
q
n
+
1
q
2
=
n
q
2
1
+
q
+
q
1
+
q
2
n
q
x
n
+
3
q
n
+
2
q
n
+
1
q
2
+
n
q
2
q
4
n
+
1
q
n
q
x
2
/
q
+
n
q
x
-
n
q
x
n
+
3
q
n
+
2
q
n
+
1
q
2
=
1
+
q
n
q
2
+
q
1
+
q
2
x
n
q
3
+
q
3
x
2
n
q
3
n
+
1
q
n
+
3
q
n
+
2
q
n
+
1
q
2
.This completes the proof.Lemma 5.
For the special caseq
=
1, one gets [14] (17)
M
n
,
1
1
x
=
1
;
M
n
,
1
t
x
=
n
2
x
+
n
n
+
2
n
+
1
;
M
n
,
1
t
2
x
=
n
2
n
2
x
2
+
4
n
x
+
x
x
+
2
n
+
3
n
+
2
n
+
1
2
.Lemma 6.
For the sequence of positive linear operatorsM
n
,
q, one gets the following central moments: let ϕ
i
=
(
t
-
x
)
i, i
=
1,2
,
…; (18)
M
n
,
q
ϕ
1
x
=
M
n
,
q
t
x
-
x
M
n
,
q
1
x
=
n
q
n
q
x
+
1
n
+
2
q
n
+
1
q
-
x
·
1
=
1
-
3
x
n
q
-
2
x
n
+
2
q
n
+
1
q
;
M
n
,
q
ϕ
2
x
=
M
n
,
q
t
2
x
-
2
x
M
n
,
q
t
x
+
x
2
M
n
,
q
1
x
=
1
+
q
n
q
2
+
q
1
+
q
2
x
n
q
3
+
q
3
x
2
n
q
3
n
+
1
q
n
+
3
q
n
+
2
q
n
+
1
q
2
-
2
x
n
q
n
q
x
+
1
n
+
2
q
n
+
1
q
+
x
2
·
1
=
x
2
1
-
2
n
q
2
n
+
3
q
+
q
3
n
q
3
n
+
3
q
n
+
2
q
n
+
1
q
+
x
q
1
+
q
2
n
q
3
-
2
n
q
n
+
3
q
n
+
3
q
n
+
2
q
n
+
1
q
2
+
1
+
q
n
q
2
n
+
3
q
n
+
2
q
n
+
1
q
2
.Lemma 7.
For the special caseq
=
1, one has the following central moment [14]: (19)
M
n
,
q
ϕ
1
x
=
n
1
-
3
x
-
2
x
n
+
2
n
+
1
M
n
,
q
ϕ
2
x
=
2
x
n
3
+
n
2
11
x
2
-
8
x
+
2
n
+
3
n
+
2
n
+
1
2
+
n
17
x
2
-
6
x
+
2
+
6
x
2
n
+
3
n
+
2
n
+
1
2
.


## 3. Weighted Statistical Approximation Theorem

The aim of this section is to use statistical convergence to study Korovkin type approximation of functionf by means of sequence of positive linear operators from a weighted space into a weighted subspace. Let C
ρ
0
I
n
,
q be the space of the continuous and bounded functions defined on I
n
,
q such that |
f
|
≤
C
(
1
+
x
2
), where C is a constant depending on f. Our operators acting from similar methods of [3]; we obtain the following results.Theorem 8.
Let sequence(
q
n
)
n, q
n
∈
(
0,1
), such that s
t
-
l
i
m
n
→
∞
q
n
=
1, and the sequence of positive linear operators M
n
,
q
n, n
∈
N, be defined by (10). Then, for all x
∈
I
n
,
q and f
∈
C
ρ
0
I
n
,
q, one gets(20)
s
t
-
lim
n
→
∞
⁡
M
n
,
q
n
f
x
-
f
x
ρ
α
=
0
;
α
>
0
.Proof.
The weight functionsρ
0
(
x
) and weighted subspace C
ρ
0
I
n
,
q are defined by x
∈
I
n
,
q; α
>
0; ρ
0
(
x
)
=
1
+
x
2; ρ
α
(
x
)
=
1
+
x
2
+
α; and f
∈
C
ρ
0
I
n
,
q such that f is continuous on I
n
,
q with norm; here B
ρ
(
I
n
,
q
) and C
ρ
(
I
n
,
q
) are Banach space. By using Theorem 4, we get(21)
st
-
lim
n
→
∞
⁡
M
n
,
q
n
1
x
-
1
ρ
0
=
0
.Since(22)
M
n
,
q
n
t
x
-
x
1
+
x
2
=
n
q
n
q
x
+
1
/
n
+
2
q
n
+
1
q
-
x
1
+
x
2
≤
1
n
q
nand s
t
-
l
i
m
n
→
∞
q
n
=
1, this implies s
t
-
l
i
m
n
→
∞
1
/
[
n
]
q
n
=
0, and we get(23)
st
-
lim
n
→
∞
⁡
M
n
,
q
n
t
x
-
x
ρ
0
=
0
.Again, since(24)
M
n
,
q
n
t
2
x
-
x
2
1
+
x
2
≤
1
+
q
n
n
q
n
2
+
q
n
1
+
q
n
2
n
q
n
+
q
3
n
q
n
,we get(25)
st
-
lim
n
→
∞
⁡
M
n
,
q
n
t
2
x
-
x
2
ρ
0
=
0
.By using A-statistical convergence theorem given by Duman and Orhan [15], here we let A
=
C
1 ((21), (23), and (25)), and we get s
t
-
l
i
m
n
→
∞
|
(
M
n
,
q
n
t
k
)
(
x
)
-
x
k
|
ρ
0
=
0 for k
=
0,1
,
2 if and only if(26)
st
-
lim
n
→
∞
⁡
M
n
,
q
n
f
x
-
f
x
ρ
α
=
0
.This completes the proof.Theorem 9.
Let sequence(
q
n
)
n, q
n
∈
(
0,1
), such that s
t
-
l
i
m
n
→
∞
q
n
=
1, and the sequence of positive linear operators M
n
,
x
,
q
n
⋆, n
∈
N, be defined by (12). Then, for all functions f
∈
C
ρ
0
I
n
,
q, one gets(27)
s
t
-
lim
n
→
∞
⁡
M
n
,
x
,
q
n
⋆
f
t
-
f
t
ρ
α
=
0
;
α
>
0
,where x and t belong to I
n and x is fixed.Proof.
The proof is analogous as Theorem8.Theorem 10.
Let sequence(
q
n
)
n, q
n
∈
(
0,1
), such that s
t
-
l
i
m
n
→
∞
q
n
=
1, and the sequence of positive linear operators M
n
,
q
n, n
∈
N, be defined by (10). Then, for all x
∈
I
n and functions f
∈
C
x
2
I
n
,
q, we get(28)
lim
n
→
∞
⁡
M
n
,
q
n
f
x
-
f
x
x
2
=
0
.Proof.
To prove the theorem, we use modulus of continuity off on closed interval I
n given by(29)
ω
f
,
δ
=
sup
t
-
x
≤
δ
⁡
sup
t
∈
I
n
,
q
⁡
f
t
-
f
x
.We see that f
∈
C
x
2
I
n; the modulus of continuity ω
(
f
,
δ
) tends to zero. Consider (30)
M
n
,
q
n
t
x
-
x
x
2
≤
n
q
n
-
q
n
-
1
q
n
n
+
2
q
n
n
+
1
q
n
sup
x
∈
I
n
,
q
⁡
x
1
+
x
2
+
n
q
n
n
+
2
q
n
n
+
1
q
n
sup
x
∈
I
n
,
q
⁡
1
1
+
x
2
,and we get(31)
lim
n
→
∞
⁡
M
n
,
q
n
t
x
-
x
x
2
=
0
.Again (32)
M
n
,
q
n
t
2
x
-
x
2
x
2
≤
n
q
n
3
+
q
3
n
q
n
4
n
+
3
q
n
n
+
2
q
n
n
+
1
q
n
2
-
1
sup
x
∈
I
n
,
q
⁡
x
2
1
+
x
2
+
q
1
+
q
2
x
n
q
n
3
n
+
3
q
n
n
+
2
q
n
n
+
1
q
n
2
sup
x
∈
I
n
,
q
⁡
x
1
+
x
2
+
1
+
q
n
q
n
2
n
+
3
q
n
n
+
2
q
n
n
+
1
q
n
2
sup
x
∈
I
n
,
q
⁡
1
1
+
x
2
,and we get(33)
lim
n
→
∞
⁡
M
n
,
q
n
t
2
x
-
x
2
x
2
=
0
.
By (31) and (33), we get l
i
m
n
→
∞
|
(
M
n
,
q
n
t
k
)
(
x
)
-
x
k
|
x
2
=
0 for k
=
0,1
,
2 if and only if(34)
lim
n
→
∞
⁡
M
n
,
q
n
f
x
-
f
x
x
2
=
0
.This completes the proof.Theorem 11.
Let sequence(
q
n
)
n, q
n
∈
(
0,1
), such that s
t
-
l
i
m
n
→
∞
q
n
=
1, and the sequence of positive linear operators M
n
,
x
,
q
n
⋆, n
∈
N, be defined by (12). Then, for all functions f
∈
C
t
2
I
n
,
q, one gets(35)
lim
n
→
∞
⁡
M
n
,
x
,
q
n
⋆
f
t
-
f
t
t
2
=
0
,where x and t belong to I
n
,
q and x is fixed.Proof.
The proof is analogous as Theorem10.


## 4. Korovkin Type Theorems for Fuzzy Continuous Functions

In this section we mention some important definitions given by Burgin and Duman [16].Definition 12.
A number a is called anr-limit of a sequence S, if for any ϵ
∈
R, the inequality |
a
-
a
i
|
<
r
+
ϵ is valid for almost all a
i, that is, there is such n that for any i
>
n, we have |
a
-
a
i
|
<
r
+
ϵ
. It is denoted by a
=
r
-
l
i
m
S.Definition 13.
A sequenceS that has an r-limit is called r-convergent and it is said that S, r-converges to its r-limita. It is denoted by S
→
r
a.Definition 14.
A functionf
:
R
→
R is called r-continuous in X
⊂
R if γ
(
f
,
X
)
≤
r and is called fuzzy continuous in X if γ
(
f
,
X
)
≤
∞, where γ
(
f
,
X
) defined as: (36)
γ
f
,
X
≥
inf
⁡
sup
⁡
f
x
-
g
x
:
x
∈
X
:
g
x
∈
C
X
.For example the functionsf
(
x
)
=
x
n when x
∈
[
n
,
n
+
1
), n
∈
Z and g
(
x
)
=
[
x
]
n are fuzzy continuous in each finite interval of the real line R, but they are not continuous in any interval with the length larger than 1. To define the Riemann integral for a continuous function f
(
x
), step functions are utilized. If the integral of f
(
x
) exists, then any such step function is fuzzy continuous.Theorem 15.
Let a sequence(
q
n
)
n; q
n
∈
(
0,1
) such that r
-
l
i
m
n
→
∞
q
n
=
1 and let the sequence of positive linear operators M
n
,
q
n; n
∈
N be defined by (10). If r
i
-
l
i
m
n
→
∞
|
(
M
n
,
q
n
e
i
)
(
x
)
-
e
i
|
=
0 for i
=
0,1
,
2
. Then for all functions f
∈
C
I
n
,
q, we get(37)
r
-
lim
n
→
∞
⁡
M
n
,
q
n
f
x
-
f
=
0
,where r is any real number such that r
≥
K
3
(
r
0
+
r
1
+
r
2
) for some K
3
>
0
.Proof.
Let the functionse
i defined as: e
i
(
x
)
=
t
i for all x
∈
I
n
,
q. Now, for each ϵ
>
0, there corresponds δ
>
0 such that |
λ
(
t
-
x
)
|
≤
ϵ whenever |
t
-
x
|
≤
δ. Again for |
t
-
x
|
>
δ, then there exist a positive number M such that |
λ
(
t
-
x
)
|
≤
M
≤
M
(
t
-
x
)
2
/
δ
2. Thus for all t and x
∈
I
n
,
q, we get(38)
λ
t
-
x
≤
ϵ
+
M
t
-
x
2
δ
2
.Applying M
n
,
q
n on (38), we get (39)
M
n
,
q
n
f
x
-
f
x
≤
ϵ
M
n
,
q
n
e
0
x
+
M
δ
2
M
n
,
q
n
t
-
x
2
x
≤
ϵ
+
ϵ
M
n
,
q
n
e
0
x
-
e
0
x
+
K
3
∑
i
=
0
2
M
n
,
q
n
e
i
x
-
e
i
x
,where K
3
=
max
⁡
{
M
/
δ
2
,
2
M
x
/
δ
2
,
M
x
2
/
δ
2
}. Then for every ɛ
>
0 there exist N
=
N
(
ϵ
)
>
0 such that for all n
∈
N, we get (40)
M
n
,
q
n
f
x
-
f
x
≤
ϵ
+
ϵ
r
0
+
ϵ
+
K
3
3
ϵ
+
r
0
+
r
1
+
r
2
≤
r
+
ɛ
1
,here, ɛ
1
=
ϵ
(
1
+
r
0
+
ϵ
+
3
K
3
)
. Since ϵ is arbitrary and small, r
-
l
i
m
n
→
∞
q
n
=
1, we get(41)
r
-
lim
n
→
∞
⁡
M
n
,
q
n
,
c
f
x
-
f
=
0
.This completes the proof.Theorem 16.
Let a sequence(
q
n
)
n; q
n
∈
(
0,1
) such that r
-
l
i
m
n
→
∞
q
n
=
1 and let the sequence of positive linear operators M
n
,
x
,
q
n
⋆; n
∈
N be defined by (12). If r
i
-
l
i
m
n
→
∞
|
(
M
n
,
x
,
q
n
⋆
e
i
)
(
x
)
-
e
i
|
=
0 for i
=
0,1
,
2
. Then for all functions f
∈
C
I
n
,
q, we get(42)
r
-
lim
n
→
∞
⁡
M
n
,
x
,
q
n
⋆
f
t
-
f
t
=
0
,where r is any real number such that r
≥
K
4
(
r
0
+
r
1
+
r
2
) for some K
4
>
0
.Proof.
The proof is analogous as Theorem15.Theorem 17.
Letf be the integrable and bounded in the interval I
n
,
q and let if f
′
′ exists at a point x
∈
I
n
,
q. Let a sequence (
q
n
)
n; q
n
∈
(
0,1
) such that l
i
m
n
→
∞
q
n
=
1 and let the sequence of positive linear operators M
n
,
q
n; n
∈
N be defined by (10). Then, one gets that (43)
lim
n
→
∞
⁡
n
q
n
M
n
,
q
n
f
x
-
f
x
=
1
-
3
x
f
′
x
+
x
2
f
′
′
x
.Proof.
Let iff
′
′ exists at a point x
∈
I
n
,
q, then by using Taylor’s expansion, we write (44)
f
t
=
f
x
+
t
-
x
f
′
x
+
t
-
x
2
2
f
′
′
x
+
t
-
x
2
λ
t
-
x
,where λ
(
t
-
x
)
→
0 as t
→
x. Applying M
n
,
q
n, we get (45)
M
n
,
q
n
f
x
=
f
x
M
n
,
q
n
1
x
+
f
′
x
M
n
,
q
n
t
-
x
x
+
f
′
′
x
2
M
n
,
q
n
t
-
x
2
x
+
M
n
,
q
n
t
-
x
2
λ
t
-
x
x
.By using Theorem 4, and multiplying [
n
]
q
n both sides, we get (46)
n
q
n
M
n
,
q
n
f
x
-
f
x
=
f
′
x
n
q
n
1
-
3
x
n
q
n
-
2
x
n
+
2
q
n
n
+
1
q
n
+
⋯
+
f
′
′
x
n
q
n
2
M
n
,
q
n
,
c
ϕ
2
x
+
n
q
n
R
n
q
n
t
,
x
.Here we write,(47)
n
q
n
R
n
q
n
t
,
x
=
n
+
1
q
n
2
∑
k
=
0
∞
b
n
,
k
q
;
x
⋯
∫
0
n
q
n
/
n
+
1
q
n
p
n
,
k
⋆
q
n
;
q
n
t
ϕ
2
λ
ϕ
d
q
t
,
n
q
n
R
n
q
n
t
,
x
≤
n
+
1
q
n
2
∑
k
=
0
∞
b
n
,
k
q
;
x
⋯
∫
0
n
q
n
/
n
+
1
q
n
p
n
,
k
⋆
q
n
;
q
n
t
ϕ
2
λ
ϕ
d
q
t
≤
n
q
n
·
ϵ
M
n
,
q
n
t
-
x
2
x
+
n
q
n
M
δ
2
M
n
,
q
n
,
c
t
-
x
4
x
≤
n
q
n
·
ϵ
o
1
n
q
n
+
n
q
n
M
δ
2
o
1
n
q
n
2
≤
ϵ
+
M
n
q
n
-
1
/
2
o
1
n
q
n
≤
ϵ
+
M
o
1
√
n
q
n
;
for
δ
=
n
q
n
-
1
/
4
.Since ϵ is arbitrary and small, l
i
m
n
→
∞
q
n
=
1 and whenever n
→
∞, we get(48)
n
q
n
R
n
q
n
t
,
x
⟶
0
.By using (48), in (46), we get (49)
lim
n
→
∞
⁡
n
q
n
M
n
,
q
n
f
x
-
f
x
=
1
-
3
x
f
′
x
+
x
2
f
′
′
x
.This completes the proof.Theorem 18.
Letf be the integrable and bounded in the interval I
n
,
q and let if f
′
′ exists at points x; t
∈
I
n
,
q. Let a sequence (
q
n
)
n; q
n
∈
(
0,1
) such that l
i
m
n
→
∞
q
n
=
1 and let the sequence of positive linear operators M
n
,
x
,
q
n
⋆; n
∈
N be defined by (12). Then, one gets that (50)
lim
n
→
∞
⁡
n
q
n
M
n
,
x
,
q
n
⋆
f
t
-
f
t
=
1
-
3
t
f
′
x
+
t
+
t
2
f
′
′
x
+
t
.Proof.
The proof is analogous as Theorem17.


## 5. Conclusion

In this way we conclude that weighted statistical approximation theorem, Korovkin type theorems for fuzzy continuous functions, an estimate for the rate of convergence, and some properties are obtained for these operators. Forq
=
1, the operators defined by (8) reduce generalized Baskakov operators. For q
=
1, the operators defined by (6) reduce generalized Durrmeyer operators (5). By using maple programming, Honey Sharma concludes that Deo’s modified operator [10, 11] does not improve the approximation process. In this paper we conclude that, by Deo’s modified operators, (10) and (12) improve the approximation process when the value of n is very large, that is, when n tends to infinity.



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*Source: 874178-2015-08-03.xml*