1. Introduction
The study of
q
-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 of
q
-Calculus.
Definition 1.
For any fixed real numbers
q
>
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 number
n
one may write
(2)
n
q
=
1
-
q
n
1
-
q
;
q
≠
1
.
Definition 2.
The
q
-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 number
k
∈
(
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 a
q
-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 on
q
-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 Lemma 5 [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 Lemma 5 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 set
f
(
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 case
q
=
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 operators
M
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 case
q
=
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 function
f
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 functions
ρ
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
n
and
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 Theorem 8.
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 of
f
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 Theorem 10.
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 an
r
-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 sequence
S
that has an
r
-limit is called
r
-convergent and it is said that
S
,
r
-converges to its
r
-limit
a
. It is denoted by
S
→
r
a
.
Definition 14.
A function
f
:
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 functions
f
(
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 functions
e
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 Theorem 15.
Theorem 17.
Let
f
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 if
f
′
′
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.
Let
f
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 Theorem 17.