# On(

Authors: Wen-Tao Cheng; Wen-Hui Zhang
Journal: Journal of Function Spaces
 (2019)

---

## Abstract

In this paper, a kind of new analogue of Gamma type operators based on(
p
,
q
)-integers is introduced. The Voronovskaja type asymptotic formula of these operators is investigated. And some other results of these operators are studied by means of modulus of continuity and Peetre K
-functional. Finally, some direct theorems concerned with the rate of convergence and the weighted approximation for these operators are also obtained.


---

## Body

## 1. Introduction

In recent years, with the rapid development ofq-calculus, the study of approximation theory with q-integer has been discussed widely. Afterwards, with the generalization from q-calculus to (
p
,
q
)-calculus, it has been used efficiently in many areas of sciences such as algebras [1, 2] and CAGD [3]. And, recently, approximation by sequences of linear positive operators has been transferred to operators with (
p
,
q
)-integer. Some useful notations and definitions about q-calculus and (
p
,
q
)-calculus in this paper are reviewed in [4–6].Let0
<
q
<
p
≤
1. For each nonnegative integer n, the (
p
,
q
)-integer [
n
]
p
,
q and (
p
,
q
)-factorial [
n
]
p
,
q
! are defined as(1)
n
p
,
q
=
p
n
-
q
n
p
-
q
n
=
1,2
,
…and(2)
n
p
,
q
!
=
1
p
,
q
2
p
,
q
⋯
n
p
,
q
,
n
≥
1
1
,
n
=
0Further, the(
p
,
q
)
-power basis is defined as(3)
x
⊕
y
p
,
q
n
=
x
+
y
p
x
+
q
y
p
2
x
+
q
2
y
⋯
p
n
-
1
x
+
q
n
-
1
y
.And(4)
x
⊖
y
p
,
q
n
=
x
-
y
p
x
-
q
y
p
2
x
-
q
2
y
⋯
p
n
-
1
x
-
q
n
-
1
y
.Letn be a nonnegative integer; the (
p
,
q
)-Gamma function is defined as(5)
Γ
p
,
q
n
+
1
=
p
⊖
q
p
,
q
n
p
-
q
n
=
n
p
,
q
!
,
0
<
q
<
p
≤
1
.Aral and Gupta [7] proposed (
p
,
q
)-Beta function of second kind for m
,
n
∈
N as(6)
B
p
,
q
m
,
n
=
∫
0
∞
x
m
-
1
1
⊕
p
x
p
,
q
m
+
n
d
p
,
q
x
.And the relationship between (
p
,
q
)-analogues of Beta and Gamma functions is as follows:(7)
B
p
,
q
m
,
n
=
q
Γ
p
,
q
m
Γ
p
,
q
n
p
m
+
1
q
m
-
1
m
/
2
Γ
p
,
q
m
+
n
.Particularly, when p
=
q
=
1, B
(
m
,
n
)
=
Γ
(
m
)
Γ
(
n
)
/
Γ
(
m
+
n
). It may be observed that, in (
p
,
q
)-setting, order is important, which is the reason why (
p
,
q
)-variant of Beta function does not satisfy commutativity property; that is, B
p
,
q
(
m
,
n
)
≠
B
p
,
q
(
n
,
m
).In [8], Mazhar studied some approximation properties of the Gamma operators as follows:(8)
F
-
n
f
;
x
=
2
n
!
x
n
+
1
n
!
n
-
1
!
∫
0
∞
t
n
-
1
x
+
t
2
n
+
1
f
t
d
t
,
n
>
1
,
x
>
0
.Recently, Mursaleen first applied(
p
,
q
)-calculus in approximation theory and introduced the (
p
,
q
)-analogue of Bernstein operators [9], (
p
,
q
)-Bernstein-Stancu operators [10], and (
p
,
q
)-Bernstein-Schurer operators [11] and investigated their approximation properties. And many well-known approximation operators with (
p
,
q
)-integer have been introduced, such as (
p
,
q
)-Bernstein-Stancu-Schurer-Kantorovich operators [12], (
p
,
q
)-Szász-Baskakov operators [13], and (
p
,
q
)-Baskakov-Beta operators [14]. As we know, many researchers have studied approximation properties of the Gamma operators and their modifications (see [15–21], etc.). All this achievement motivates us to construct the (
p
,
q
)-analogue of the Gamma operators (8). First, we introduce (
p
,
q
)-analogue of Gamma operators as follows.Definition 1.
Forn
∈
N, n
>
1, x
∈
(
0
,
∞
), and 0
<
q
<
p
≤
1, the (
p
,
q
)-Gamma operators can be defined as(9)
F
n
p
,
q
f
;
x
=
x
n
+
1
p
n
2
q
n
2
+
n
B
p
,
q
n
,
n
+
1
∫
0
∞
t
n
-
1
p
n
q
n
x
⊕
t
p
,
q
2
n
+
1
f
t
d
p
,
q
tThe paper is organized as follows. In the first section, we give the basic notations and the definition of(
p
,
q
)-Gamma operators. In the second section, we present the moments of the operators. In the third section, we obtain Voronovskaja type asymptotic formula. In the fourth section, we present a direct result of (
p
,
q
)-Gamma operators in terms of first- and second-order modulus of continuity. In the last section, we study the rate of convergence and the weighted approximation of the (
p
,
q
)-Gamma operators.


## 2. Auxiliary Results

In order to obtain the approximation properties of the operatorsF
n
p
,
q
(
f
;
x
), we need the following lemma and remarks.Lemma 2.
The following equalities hold:(1)
F
n
p
,
q
(
1
;
x
)
=
1.
(2)
F
n
p
,
q
(
t
;
x
)
=
x, for n
>
1.
(3)
F
n
p
,
q
(
t
2
;
x
)
=
[
n
+
1
]
p
,
q
x
2
/
p
q
[
n
-
1
]
p
,
q, for n
>
2.
(4)
F
n
p
,
q
(
t
3
;
x
)
=
[
n
+
1
]
p
,
q
[
n
+
2
]
p
,
q
x
3
/
(
p
q
)
3
[
n
-
1
]
p
,
q
[
n
-
2
]
p
,
q, for n
>
3.
(5)
F
n
p
,
q
(
t
4
;
x
)
=
[
n
+
1
]
p
,
q
[
n
+
2
]
p
,
q
[
n
+
3
]
p
,
q
x
4
/
(
p
q
)
6
[
n
-
1
]
p
,
q
[
n
-
2
]
p
,
q
[
n
-
3
]
p
,
q, for n
>
4.Proof.
According to the properties of(
p
,
q
)-Beta function and (
p
,
q
)-Gamma function, we have(10)
F
n
t
k
;
x
=
x
n
+
1
p
n
2
q
n
2
+
n
B
p
,
q
n
,
n
+
1
∫
0
∞
t
n
+
k
-
1
p
n
q
n
x
⊕
t
p
,
q
2
n
+
1
d
p
,
q
t
=
x
n
+
1
p
n
2
q
n
2
+
n
B
p
,
q
n
,
n
+
1
∫
0
∞
1
x
2
n
+
1
p
n
2
n
+
1
q
n
2
n
+
1
t
n
+
k
-
1
1
⊕
p
t
/
x
p
n
+
1
q
n
p
,
q
2
n
+
1
d
p
,
q
t
=
x
n
+
1
p
n
2
q
n
2
+
n
B
p
,
q
n
,
n
+
1
∫
0
∞
x
n
+
k
p
n
+
1
n
+
k
q
n
n
+
k
x
2
n
+
1
p
n
2
n
+
1
q
n
2
n
+
1
t
/
x
p
n
+
1
q
n
n
+
k
-
1
1
⊕
p
t
/
x
p
n
+
1
q
n
p
,
q
2
n
+
1
d
p
,
q
t
x
p
n
+
1
q
n
=
x
k
p
n
k
+
k
q
n
k
B
p
,
q
n
+
k
,
n
-
k
+
1
B
p
,
q
n
,
n
+
1
=
x
k
p
n
k
+
k
q
n
k
q
n
-
1
p
n
+
1
n
/
2
q
n
+
k
-
1
p
n
+
k
+
1
n
+
k
/
2
Γ
p
,
q
n
+
k
Γ
p
,
q
n
-
k
+
1
Γ
p
,
q
n
Γ
p
,
q
n
+
1
=
x
k
p
q
-
k
k
-
1
/
2
n
+
k
-
1
p
,
q
!
n
-
k
p
,
q
!
n
-
1
p
,
q
!
n
p
,
q
!This proves Lemma 2.Remark 3.
Letn
>
2, and x
∈
(
0
,
∞
); then, for 0
<
q
<
p
≤
1, we have the central moments as follows:(1)
F
n
p
,
q
(
t
-
x
;
x
)
=
0.
(2)
A
(
x
)
:
=
F
n
p
,
q
(
(
t
-
x
)
2
;
x
)
=
q
/
p
-
1
+
p
n
-
2
[
2
]
p
,
q
/
q
[
n
-
1
]
p
,
q
x
2.Remark 4.
The sequences(
p
n
) and (
q
n
) satisfy 0
<
q
n
<
p
n
<
1 such that p
n
→
1, q
n
→
1, and p
n
n
→
a, q
n
n
→
b, and [
n
]
p
n
,
q
n
→
∞ as n
→
∞, where 0
≤
a
,
b
<
1, and x
∈
(
0
,
∞
); then(1)
lim
n
→
∞
⁡
[
n
-
1
]
p
n
,
q
n
F
n
p
n
,
q
n
(
(
t
-
x
)
2
;
x
)
=
(
a
+
b
)
x
2.
(2)
lim
n
→
∞
⁡
[
n
-
3
]
p
n
,
q
n
F
n
p
n
,
q
n
(
(
t
-
x
)
4
;
x
)
=
0.Proof.
(1) Using Remark3,(11)
l
i
m
n
→
∞
n
-
1
p
n
,
q
n
F
n
p
n
,
q
n
t
-
x
2
;
x
=
l
i
m
n
→
∞
p
n
n
-
2
2
p
n
,
q
n
q
n
-
p
n
-
q
n
n
-
1
p
n
,
q
n
p
n
x
2
=
l
i
m
n
→
∞
p
n
n
-
2
2
p
n
,
q
n
q
n
-
p
n
n
-
1
-
q
n
n
-
1
p
n
x
2
=
2
a
-
a
-
b
x
2
=
a
+
b
x
2
(2) Letk
=
n
-
3; we have(12)
n
+
1
p
n
,
q
n
n
+
2
p
n
,
q
n
n
+
3
p
n
,
q
n
=
q
n
4
k
p
n
,
q
n
+
p
n
k
4
p
n
,
q
n
q
n
5
k
p
n
,
q
n
+
p
n
k
5
p
n
,
q
n
q
n
6
k
p
n
,
q
n
+
p
n
k
6
p
n
,
q
n
~
q
n
15
k
p
n
,
q
n
3
+
p
n
k
q
n
9
6
p
n
,
q
n
+
q
n
10
5
p
n
,
q
n
+
q
n
11
4
p
n
,
q
n
k
p
n
,
q
n
2
.Similarly, we have(13)
n
+
1
p
n
,
q
n
n
+
2
p
n
,
q
n
n
-
3
p
n
,
q
n
~
q
n
9
k
p
n
,
q
n
3
+
p
n
k
q
n
5
4
p
n
,
q
n
+
q
n
4
5
p
n
,
q
n
k
p
n
,
q
n
2
.
(14)
n
+
1
p
n
,
q
n
n
-
2
p
n
,
q
n
n
-
3
p
n
,
q
n
~
q
n
5
k
p
n
,
q
n
3
+
p
n
k
q
n
4
p
n
,
q
n
+
q
n
4
k
p
n
,
q
n
2
(15)
n
-
1
p
n
,
q
n
n
-
2
p
n
,
q
n
n
-
3
p
n
,
q
n
~
q
n
3
k
p
n
,
q
n
3
+
p
n
k
q
n
2
p
n
,
q
n
+
q
n
2
k
p
n
,
q
n
2Using Lemma 2, we obtain(16)
F
n
p
n
,
q
n
t
-
x
4
;
x
~
A
n
+
1
k
p
n
,
q
n
B
n
x
4where A
n
=
q
n
15
-
4
p
n
3
q
n
12
+
6
p
n
5
q
n
10
-
3
p
n
6
q
n
q and(17)
B
n
=
p
n
k
q
n
9
6
p
n
,
q
n
+
q
n
10
5
p
n
,
q
n
+
q
n
11
4
p
n
,
q
n
-
4
p
n
3
q
n
3
q
n
5
4
p
n
,
q
n
+
q
n
4
5
p
n
,
q
n
+
6
p
n
5
q
n
5
q
n
4
p
n
,
q
n
+
q
n
4
-
3
p
n
6
q
n
6
q
n
2
p
n
,
q
n
+
q
n
2
.Combining with(18)
k
p
n
,
q
n
A
n
~
k
p
n
,
q
n
q
n
6
-
4
p
n
3
q
n
3
+
6
p
n
5
q
n
-
3
p
n
6
=
k
p
n
,
q
n
4
p
n
3
p
n
3
-
q
n
3
-
6
p
n
5
p
n
-
q
n
-
p
n
6
-
q
n
6
=
k
p
n
,
q
n
4
p
n
3
3
p
n
,
q
n
p
n
-
q
n
-
6
p
n
5
p
n
-
q
n
-
6
p
n
,
q
n
p
n
-
q
n
=
k
p
n
,
q
n
p
n
n
-
q
n
n
n
p
n
,
q
n
4
p
n
3
3
p
n
,
q
n
-
6
p
n
5
-
6
p
n
,
q
n
~
p
n
n
-
q
n
n
4
p
n
3
3
p
n
,
q
n
-
6
p
n
5
-
6
p
n
,
q
n
~
a
-
b
12
-
6
-
6
=
0and(19)
B
n
~
4
+
5
+
6
-
4
×
4
+
5
+
6
×
4
+
1
-
3
×
2
+
1
=
0
,we can obtain lim
n
→
∞
⁡
[
n
-
3
]
p
n
,
q
n
F
n
p
n
,
q
n
(
(
t
-
x
)
4
;
x
)
=
0.


## 3. Voronovskaja Type Theorem

We give a Voronovskaja type asymptotic formula forF
n
p
,
q
(
f
;
x
) by means of the second and fourth central moments.Theorem 5.
Letf be bounded and integrable on the interval x
∈
(
0
,
∞
); second derivative of f exists at a fixed point x
∈
(
0
,
∞
); the sequences (
p
n
) and (
q
n
) satisfy 0
<
q
n
<
p
n
<
1 such that p
n
→
1, q
n
→
1, and p
n
n
→
a, q
n
n
→
b, and [
n
]
p
n
,
q
n
→
∞ as n
→
∞, where 0
≤
a
,
b
<
1; then(20)
l
i
m
n
→
∞
n
-
3
p
n
,
q
n
F
n
p
n
,
q
n
f
;
x
-
f
x
=
a
+
b
2
x
2
f
′
′
x
.Proof.
Letx
∈
(
0
,
∞
) be fixed. In order to prove this identity, we use Taylor’s expansion:(21)
f
t
-
f
x
=
t
-
x
f
′
x
+
t
-
x
2
f
′
′
x
2
+
Φ
p
n
,
q
n
t
,
x
,where Φ
p
n
,
q
n
(
x
,
t
) is bounded and lim
t
→
x
⁡
Φ
p
n
,
q
n
(
t
,
x
)
=
0. By applying the operator F
n
p
n
,
q
n
(
f
;
x
) to the equality above, we obtain(22)
F
n
p
n
,
q
n
f
;
x
-
f
x
=
f
′
x
F
n
p
n
,
q
n
t
-
x
;
x
+
1
2
f
′
′
x
F
n
p
n
,
q
n
t
-
x
2
;
x
+
F
n
p
n
,
q
n
Φ
p
n
,
q
n
t
,
x
t
-
x
2
;
x
=
1
2
f
′
′
x
F
n
p
n
,
q
n
t
-
x
2
;
x
+
F
n
p
n
,
q
n
Φ
p
n
,
q
n
t
,
x
t
-
x
2
;
x
.Since lim
t
→
x
⁡
Φ
p
n
,
q
n
(
t
,
x
)
=
0, for all ϵ
>
0, there exists δ
>
0 such that |
t
-
x
|
<
δ and it will imply |
Φ
p
n
,
q
n
(
t
,
x
)
|
<
ϵ for all fixed x
∈
(
0
,
∞
) as n is sufficiently large. Meanwhile, if |
t
-
x
|
≥
δ, then |
Φ
p
n
,
q
n
(
t
,
x
)
|
≤
C
/
δ
2
(
t
-
x
)
2, where C
>
0 is a constant. Using Remark 4, we have(23)
l
i
m
n
→
∞
n
-
3
F
n
p
n
,
q
n
t
-
x
2
;
x
=
l
i
m
n
→
∞
n
-
1
F
n
p
n
,
q
n
t
-
x
2
;
x
=
a
+
b
x
2and(24)
n
-
3
p
n
,
q
n
F
n
p
n
,
q
n
Φ
p
n
,
q
n
t
,
x
t
-
x
2
;
x
≤
ϵ
n
-
3
p
n
,
q
n
F
n
p
n
,
q
n
t
-
x
2
;
x
+
C
δ
2
n
-
3
p
n
,
q
n
F
n
p
n
,
q
n
t
-
x
4
;
x
→
0
n
→
∞
.The proof is completed.


## 4. Local Approximation

We denote the space of all real valued continuous bounded functionsf defined on the interval [
0
,
+
∞
) by C
B
[
0
,
+
∞
). The norm · on the space C
B
[
0
,
+
∞
) is given by(25)
f
=
s
u
p
f
x
:
x
∈
0
,
+
∞
.Let us consider the followingK-functional:(26)
K
f
,
δ
=
i
n
f
g
∈
C
B
2
0
,
∞
f
-
g
+
δ
g
′
′
,where δ
>
0 and C
B
2
0
,
∞
=
g
∈
C
B
0
,
∞
:
g
′
,
g
′
′
∈
C
B
0
,
∞. By [22] (p. 177, Theorem 2.4), there exists an absolute constant C
>
0 such that(27)
K
f
,
δ
≤
C
ω
2
f
,
δwhere(28)
ω
2
f
,
δ
=
s
u
p
0
<
h
≤
δ
s
u
p
x
∈
0
,
∞
f
x
+
2
h
-
2
f
x
+
h
+
f
xis the second-order modulus of smoothness of f. By(29)
ω
f
,
δ
=
s
u
p
0
<
h
≤
δ
s
u
p
x
∈
0
,
∞
f
x
+
h
-
f
x
,we denote the usual modulus of continuity of f
∈
C
B
[
0
,
∞
).Our first result is a direct local approximation theorem for the operatorsF
n
p
,
q
(
f
;
x
).Theorem 6.
Letf
∈
C
B
[
0
,
+
∞
); 0
<
q
<
p
≤
1; then, for every x
∈
(
0
,
∞
) and n
>
2, we have(30)
F
n
p
,
q
f
;
x
-
f
x
≤
C
ω
2
f
,
A
x
,where C is some positive constant.Proof.
For allg
∈
C
B
2
[
0
,
∞
), using Taylor’s expansion for x
∈
(
0
,
∞
), we have(31)
g
t
=
g
x
+
g
′
x
t
-
x
+
∫
x
t
t
-
u
g
′
′
u
d
u
.Applying the operators F
n
p
,
q to both sides of the equality above and using Remark 3, we get(32)
F
n
p
,
q
g
;
x
-
g
x
=
F
n
p
,
q
∫
x
t
t
-
u
g
′
′
u
d
u
;
x
≤
F
n
p
,
q
∫
x
t
t
-
u
g
′
′
u
d
u
;
x
≤
F
n
p
,
q
g
′
′
t
-
x
2
;
x
≤
A
x
g
′
′
.Using F
n
p
,
q
f
;
x
≤
f, we have(33)
F
n
p
,
q
f
;
x
-
f
x
≤
F
n
p
,
q
f
-
g
;
x
-
f
-
g
x
+
F
n
p
,
q
g
;
x
-
g
x
≤
2
f
-
g
+
A
x
g
′
′Lastly, taking infimum on both sides of the inequality above over all g
∈
C
B
2
[
0
,
∞
),(34)
F
n
p
,
q
f
;
x
-
f
x
≤
2
K
f
;
A
xfor which we have the desired result by (27).Theorem 7.
Let0
<
γ
≤
1 and let E be any bounded subset of the interval [
0
,
∞
). If f
∈
C
B
[
0
,
∞
) is locally in L
i
p
(
γ
), i
.
e
., the condition(35)
f
x
-
f
t
≤
L
x
-
t
γ
,
t
∈
E
a
n
d
x
∈
0
,
∞holds, then, for each x
∈
(
0
,
∞
), we have(36)
F
n
p
,
q
f
;
x
-
f
x
≤
L
A
x
γ
/
2
+
2
d
x
;
E
γ
,where L is a constant depending on γ and f; and d
(
x
;
E
) is the distance between x and E defined by(37)
d
x
;
E
=
i
n
f
t
-
x
:
t
∈
E
.Proof.
From the properties of infimum, there is at least one pointt
0 in the closure of E; that is, t
0
∈
E
¯, such that(38)
d
x
;
E
=
t
0
-
x
.Using the triangle inequality, we have(39)
F
n
p
,
q
f
;
x
-
f
x
≤
F
n
p
,
q
f
t
-
f
x
;
x
≤
F
n
p
,
q
f
t
-
f
t
0
;
x
+
F
n
p
,
q
f
t
0
-
f
x
;
x
≤
L
F
n
p
,
q
t
-
t
0
γ
;
x
+
F
n
p
,
q
t
0
-
x
γ
;
x
≤
L
F
n
p
,
q
t
-
x
γ
;
x
+
2
t
0
-
x
γChoosing a
1
=
2
/
γ and a
2
=
2
/
2
-
γ and using the well-known Hölder inequality,(40)
F
n
p
,
q
f
;
x
-
f
x
≤
L
F
n
p
,
q
t
-
x
γ
a
1
;
x
1
/
a
1
F
n
p
,
q
1
a
2
;
x
1
/
a
2
+
2
t
0
-
x
γ
≤
L
F
n
p
,
q
t
-
x
2
;
x
γ
/
2
+
2
t
0
-
x
γ
≤
L
A
γ
/
2
x
+
2
d
x
;
E
γThis completes the proof.


## 5. Rate of Convergence and Weighted Approximation

LetC
ρ
[
0
,
∞
) be the set of all functions defined on [
0
,
∞
) satisfying the condition |
f
(
x
)
|
≤
C
f
ρ
(
x
), where C
f
>
0 is a constant depending only on f and ρ
(
x
) is a weight function. Let C
ρ
[
0
,
∞
) be the space of all continuous functions in C
ρ
[
0
,
∞
) with the norm f
ρ
=
sup
x
∈
[
0
,
∞
)
⁡
|
f
(
x
)
|
/
ρ
(
x
) and C
ρ
0
[
0
,
∞
)
=
f
∈
C
ρ
[
0
,
∞
)
:
lim
x
→
∞
⁡
|
f
(
x
)
|
/
ρ
(
x
)
<
∞. We consider ρ
(
x
)
=
1
+
x
2 in the following two theorems. Meanwhile, we denote the modulus of continuity of f on the closed interval [
0
,
a
], a
>
0, by(41)
ω
a
f
,
δ
=
s
u
p
t
-
x
≤
δ
s
u
p
x
,
t
∈
0
,
a
f
t
-
f
x
.Obviously, for the function f
∈
C
ρ
[
0
,
∞
), the modulus of continuity ω
a
(
f
,
δ
) tends to zero. Then we establish the following theorem on the rate of convergence for the operators F
n
p
,
q
(
f
;
x
).Theorem 8.
Letf
∈
C
ρ
[
0
,
∞
), 0
<
q
<
p
≤
1, and ω
a
+
1
(
f
,
δ
) be its modulus of continuity on the finite interval [
0
,
a
+
1
]
⊂
[
0
,
∞
), where a
>
0. Then, for every n
>
2, x
>
0,(42)
F
n
p
,
q
f
;
x
-
f
x
C
0
,
a
≤
4
C
f
1
+
a
2
A
x
+
2
ω
a
+
1
f
,
A
xProof.
For allx
∈
[
0
,
a
] and t
>
a
+
1, we easily have (
t
-
x
)
2
≥
(
t
-
a
)
2
≥
1; therefore,(43)
f
t
-
f
x
≤
f
t
+
f
x
≤
C
f
2
+
x
2
+
t
2
=
C
f
2
+
x
2
+
x
-
t
-
x
2
≤
C
f
2
+
3
x
2
+
2
x
-
t
2
≤
C
f
4
+
3
x
2
t
-
x
2
≤
4
C
f
1
+
a
2
t
-
x
2
.And, for all x
∈
[
0
,
a
], t
∈
[
0
,
a
+
1
], and δ
>
0, we have(44)
f
t
-
f
x
≤
ω
a
+
1
f
,
t
-
x
≤
1
+
t
-
x
δ
ω
a
+
1
f
,
δFrom (43) and (44), we get(45)
f
t
-
f
x
≤
4
C
f
1
+
a
2
t
-
x
2
+
1
+
t
-
x
δ
ω
a
+
1
f
,
δ
.By Schwarz’s inequality and Remark 3, we have(46)
F
n
p
,
q
f
;
x
-
f
x
≤
F
n
p
,
q
f
t
-
f
x
;
x
≤
4
C
f
1
+
a
2
F
n
p
,
q
t
-
x
2
;
x
+
F
n
p
,
q
1
+
t
-
x
δ
;
x
ω
a
+
1
f
,
δ
≤
4
C
f
1
+
a
2
F
n
p
,
q
t
-
x
2
;
x
+
ω
a
+
1
f
,
δ
1
+
1
δ
F
n
p
,
q
t
-
x
2
;
x
≤
4
C
f
1
+
a
2
A
x
+
ω
a
+
1
f
,
δ
1
+
1
δ
A
xBy taking δ
=
A
(
x
), we get the proof of Theorem 8.The following is a direct estimate in weighted approximation.Theorem 9.
Let(
p
n
) and (
q
n
) satisfy 0
<
q
n
<
p
n
≤
1 such that p
n
→
1, q
n
→
1, p
n
n
→
a, q
n
n
→
b, and [
n
]
p
n
,
q
n
→
∞. Then, for f
∈
C
ρ
0
[
0
,
∞
), we have(47)
l
i
m
n
→
∞
F
n
p
n
,
q
n
f
;
x
-
f
x
ρ
=
0
.Proof.
Using the Korovkin theorem in [23], we see that it is sufficient to verify the following three conditions:(48)
l
i
m
n
→
∞
F
n
p
n
,
q
n
t
k
;
x
-
x
k
ρ
=
0
,
k
=
0,1
,
2
.Since F
n
p
n
,
q
n
(
1
;
x
)
=
1 and F
n
p
n
,
q
n
(
t
;
x
)
=
x, (48) holds true for k
=
0,1. By Remark 3, for n
>
2, we have(49)
l
i
m
n
→
∞
F
n
p
n
,
q
n
f
;
x
-
f
x
ρ
=
s
u
p
x
∈
0
,
∞
q
n
p
n
-
1
+
p
n
n
-
2
2
p
n
,
q
n
q
n
n
-
1
p
n
,
q
n
x
2
1
+
x
2
≤
q
n
p
n
-
1
+
p
n
n
-
2
2
p
n
,
q
n
q
n
n
-
1
p
n
,
q
n
→
0
n
→
∞
.Thus the proof is completed.



---

*Source: 9607517-2019-02-12.xml*