# On the Rate of Convergence by Generalized Baskakov Operators

Authors: Yi Gao; Wenshuai Wang; Shigang Yue
Journal: Advances in Mathematical Physics
 (2015)

---

## Abstract

We firstly construct generalized Baskakov operatorsV
n
,
α
,
q
(
f
;
x
) and their truncated sum B
n
,
α
,
q
(
f
;
γ
n
,
x
). Secondly, we study the pointwise convergence and the uniform convergence of the operators V
n
,
α
,
q
(
f
;
x
), respectively, and estimate that the rate of convergence by the operators V
n
,
α
,
q
(
f
;
x
) is 1
/
n
q
/
2. Finally, we study the convergence by the truncated operators B
n
,
α
,
q
(
f
;
γ
n
,
x
) and state that the finite truncated sum B
n
,
α
,
q
(
f
;
γ
n
,
x
) can replace the operators V
n
,
α
,
q
(
f
;
x
) in the computational point of view provided that l
i
m
n
→
∞
n
γ
n
=
∞.


---

## Body

## 1. Introduction

LetN
=
{
1,2
,
…
}, N
0
=
N
∪
{
0
}, R
+
=
(
0
,
+
∞
), and R
0
=
R
+
∪
{
0
}. For a fixed q
∈
N
0, we introduce the weighted function w
q on R
0 by(1)
w
q
x
=
1
,
q
=
0
,
1
+
x
q
-
1
,
q
∈
N
.Associated with the above weighted function, we also introduce the polynomial weighted space S
q of all real-valued continuous functions f on R
0 for which w
q
f is uniformly continuous and bounded on R
0, and the norm on S
q is defined by the formula(2)
f
q
,
∞
=
sup
⁡
x
∈
R
0
w
q
(
x
)
f
(
x
)
.Obviously, when q
=
0, then the above norm is the ordinary norm f
∞. Furthermore, for fixed q
∈
N
0, let S
q
q be the set of all functions f
∈
S
q for which w
q
-
k
(
x
)
f
(
k
)
(
x
) (k
=
0,1
,
2
,
…
,
q) are continuous and bounded on R
0 and f
(
q
) is uniformly continuous on R
0, where f
(
k
)
(
x
) (k
=
0,1
,
2
,
…
,
q) denote the kth order derivative of f on R
0.Letf be a function defined on R
0; Baskakov [1] introduced the sequence of linear positive operators V
n
(
f
;
x
) as follows:(3)
V
n
f
;
x
=
∑
k
=
0
∞
‍
b
n
,
k
x
f
k
n
,where b
n
,
k
(
x
) is called a Baskakov operator’s kernel, which is defined by(4)
b
n
,
k
(
x
)
=
n
(
n
+
1
)
⋯
(
n
+
k
-
1
)
k
!
x
k
1
+
x
-
n
-
k
.Based on the Baskakov operators, many Baskakov-type operators [2–13] and their multivariate Baskakov operators [11, 14–18] were discussed. Particularly, Gupta and Agarwal studied the Baskakov-Kantorovich operators, Szász-Baskakov operators, and so forth in their recent book [6]. One of the most famous Baskakov-type operators is called generalized Baskakov operators [19–22]. One has(5)
V
n
,
α
f
;
x
=
∑
k
=
0
∞
‍
b
n
,
k
,
α
x
f
k
n
,where(6)
b
n
,
k
,
α
x
=
n
n
+
α
⋯
n
+
k
-
1
α
k
!
·
x
k
1
+
α
x
-
n
/
α
-
k
,
α
>
0
.Other modified Baskakov operators are defined as follows [10]:(7)
V
n
,
q
f
;
x
=
∑
k
=
0
∞
‍
b
n
,
k
x
∑
i
=
0
q
‍
f
i
k
/
n
i
!
x
-
k
n
i
,
x
∈
R
0, q
∈
N.By combining the above operators (5) with (7), we introduce the following class of operators.Definition 1.
Forx
∈
R
0 and q
∈
N, other generalized Baskakov-type operators are defined by(8)
V
n
,
α
,
q
f
;
x
=
∑
k
=
0
∞
‍
b
n
,
k
,
α
(
x
)
∑
i
=
0
q
‍
f
(
i
)
(
k
/
n
)
i
!
x
-
k
n
i
.The actual construction of Baskakov operator and its various modifications requires estimations of infinite series which in a certain sense restrict their usefulness from the computational point of view. A question naturally arises of whether the Baskakov operators can be replaced by a finite sum. In connection with this question we construct a new family of linear positive operators as follows:(9)
B
n
,
α
,
q
(
f
;
γ
n
,
x
)
=
∑
k
=
0
[
n
(
x
+
γ
n
)
]
b
n
,
k
,
α
(
x
)
∑
i
=
0
q
f
i
k
/
n
i
!
x
-
k
n
i
,where {
γ
n
}
n
=
1
∞ is a sequence of positive numbers such that lim
n
→
∞
n
γ
n
=
∞ and [
n
(
x
+
γ
n
)
] denotes the integral part of n
(
x
+
γ
n
).Obviously, whenα
=
1 and q
=
0, the operators (8) are (5), while the operators (9) are degenerated as follows, which are firstly proposed by Walczak [11]:(10)
A
n
(
f
;
γ
n
,
x
)
=
∑
k
=
0
[
n
(
x
+
γ
n
)
]
‍
b
n
,
k
x
f
k
n
.And when α
=
1, the operators (8) are (7), while the operators (9) can be represented by [12](11)
F
n
,
q
(
f
;
γ
n
,
x
)
=
∑
k
=
0
[
n
(
x
+
γ
n
)
]
b
n
,
k
(
x
)
∑
i
=
0
q
‍
f
i
k
/
n
i
!
x
-
k
n
i
.For the convenience of discussion in the rest of paper, we use the notation thatK
n
,
α
,
q
(
f
;
γ
n
,
x
) denotes the remainder term of operators V
n
,
α
,
q
(
f
;
x
) associated with the truncated sum B
n
,
α
,
q
(
f
;
γ
n
,
x
). Consider(12)
K
n
,
α
,
q
f
;
γ
n
,
x
=
∑
k
=
[
n
(
x
+
γ
n
)
]
+
1
∞
‍
b
n
,
k
,
α
(
x
)
∑
i
=
0
q
‍
f
i
k
/
n
i
!
x
-
k
n
i
.This paper focuses on convergence of the operatorsV
n
,
α
,
q
(
f
;
x
) and their truncated sum B
n
,
α
,
q
(
f
;
γ
n
,
x
). The rest of the paper is organized as follows. In Section 2, we give main lemmas and prove that the remainder term K
n
,
α
,
q
(
f
;
γ
n
,
x
) of the operators V
n
,
α
,
q
(
f
;
x
) associated with the truncated sum B
n
,
α
,
q
(
f
;
γ
n
,
x
) is convergent to 0 provided that lim
n
→
∞
n
γ
n
=
∞. In Section 3, we state the pointwise convergence and the uniform convergence of the operators V
n
,
α
,
q
(
f
;
x
) on the polynomial weighted space S
q, respectively, which indicate that the rate of convergence by the operators V
n
,
α
,
q
(
f
;
x
) is 1
/
n
q
/
2. Finally, we study the convergence by the truncated operators B
n
,
α
,
q
(
f
;
γ
n
,
x
) and state that the finite truncated sum B
n
,
α
,
q
(
f
;
γ
n
,
x
) can replace the operators V
n
,
α
,
q
(
f
;
x
) in the computational point of view.In this paper, for better characterizing the degree of approximation by the generalized Baskakov operatorsV
n
,
α
,
q
(
f
;
x
), we introduce the classical modulus of continuity of a function f
∈
S
q, defined by [23](13)
w
(
f
;
t
)
:
=
sup
⁡
0
≤
h
≤
t
f
(
·
+
h
)
-
f
(
·
)
∞
,
t
∈
R
0
.Here, we give an important property of modulus of continuity, which will be used in the proof of Theorem 6. One has(14)
w
(
f
;
t
)
≤
(
1
+
t
δ
)
w
(
f
;
δ
)
,
δ
∈
R
0
.


## 2. Main Lemmas

In this section, we give some properties of the above operators, which will be used to prove the main theorems.Lemma 2 (see [22]).
IfV
n
,
α
(
f
;
x
) is defined by formula (5) then(15)
V
n
,
α
(
1
;
x
)
=
1
;
V
n
,
α
(
t
;
x
)
=
x
;
V
n
,
α
t
-
x
2
;
x
=
x
1
+
α
x
n
.From the first equality in Lemma2, for all f
(
x
), x
∈
R
0, we have f
(
x
)
=
V
n
,
α
(
f
(
x
)
;
x
).Lemma 3 (see [19]).
IfV
n
,
α
(
f
;
x
) is defined by formula (5), for fixed 2
≤
q
∈
N, there exist m
≤
q-order algebraic polynomials P
i
,
q
,
α, 0
≤
i
≤
q, with coefficients depending only on i
,
q
,
α, such that(16)
V
n
,
α
t
-
x
q
;
x
=
∑
i
=
0
[
q
/
2
]
‍
P
i
,
q
,
α
(
x
)
n
q
-
i
,where x
∈
R
0 and [
q
/
2
] denotes the integral part of q
/
2. Moreover,(17)
V
n
,
α
t
-
x
2
m
;
x
≤
C
x
1
+
α
x
n
+
1
n
2
m
,
m
∈
N
.Here and in the rest of the paper, C denotes a positive absolute constant, whose value may change from line to line but is independent of n.For example, whenq
=
4, we have the following 4-order algebraic polynomial: (18)
V
n
,
α
t
-
x
4
;
x
=
3
α
n
2
+
2
α
n
2
x
4
+
6
n
α
n
+
2
α
n
2
x
3
+
1
n
2
3
+
α
n
x
2
+
1
n
3
x
.For fixed x
0
∈
R
+, obviously, we have (19)
V
n
,
α
t
-
x
4
;
x
=
O
x
0
1
n
2
.Furthermore, with respect to the above weighted functionw
q
(
x
), the generalized Baskakov operators (5) have the following results, which demonstrate that the weighted function w
q
(
x
) is also important to the generalized Baskakov operators.Lemma 4 (see [15, 21]).
IfV
n
,
α
(
f
;
x
) and weighted function w
q
(
x
) are defined by formula (5) and (1), respectively, for x
∈
R
+, then there exist positive absolute constants C, such that(20)
w
q
x
V
n
,
α
1
w
q
t
;
x
≤
C
;
w
q
x
V
n
,
α
t
-
x
2
w
q
(
t
)
;
x
≤
C
x
1
+
α
x
n
.Now we will give the estimation ofK
n
,
α
,
q
(
f
;
γ
n
,
x
).Lemma 5.
Forf
∈
S
q
q, q
∈
N, K
n
,
α
,
q
(
f
;
γ
n
,
x
) is defied by (12), then(21)
K
n
,
α
,
q
f
;
γ
n
,
x
≤
2
q
C
x
(
1
+
α
x
)
n
+
1
n
2
q
/
2
+
C
∑
i
=
0
q
‍
1
+
2
q
-
i
-
1
x
q
-
i
γ
n
q
x
(
1
+
α
x
)
n
+
1
n
2
(
q
+
i
)
/
2
.Furthermore, one has(22)
lim
n
→
∞
K
n
,
α
,
q
f
;
γ
n
,
x
=
0
.Proof.
By assumptionf
∈
S
q
q, there is a positive absolute constant C, such that |
f
(
i
)
(
t
)
|
≤
C
(
1
+
t
q
-
i
), i
=
0,1
,
…
,
q. With the elementary inequality (
a
+
b
)
k
≤
2
k
-
1
(
a
k
+
b
k
) for a
,
b
∈
R
+, k
∈
N, we get (23)
f
i
t
≤
C
1
+
t
-
x
+
x
q
-
i
≤
C
1
+
2
q
-
i
-
1
t
-
x
q
-
i
+
x
q
-
i
.So we have (24)
K
n
,
α
,
q
f
;
γ
n
,
x
≤
C
∑
k
=
[
n
(
x
+
γ
n
)
]
+
1
∞
b
n
,
k
,
α
x
·
∑
i
=
0
q
‍
1
i
!
1
+
2
q
-
i
-
1
k
n
-
x
q
-
i
+
x
q
-
i
k
n
-
x
i
≤
C
∑
i
=
0
q
2
q
-
i
-
1
∑
k
=
[
n
(
x
+
γ
n
)
]
+
1
∞
b
n
,
k
,
α
x
k
n
-
x
q
+
C
∑
i
=
0
q
‍
1
+
2
q
-
i
-
1
x
q
-
i
∑
k
=
[
n
(
x
+
γ
n
)
]
+
1
∞
b
n
,
k
,
α
x
k
n
-
x
i
≤
C
2
q
V
n
,
α
t
-
x
q
;
x
+
C
∑
i
=
0
q
‍
1
+
2
q
-
i
-
1
x
q
-
i
·
∑
k
=
[
n
(
x
+
γ
n
)
]
+
1
∞
b
n
,
k
,
α
x
k
n
-
x
i
.Next, we estimate the sum of the last term, since k
>
n
(
x
+
γ
n
) in the last term; for i
=
0,1
,
…
,
q, we remark that (25)
∑
k
=
[
n
(
x
+
γ
n
)
]
+
1
∞
b
n
,
k
,
α
x
k
n
-
x
i
≤
∑
γ
n
<
|
k
/
n
-
x
|
∞
b
n
,
k
,
α
(
x
)
k
n
-
x
i
≤
1
γ
n
q
∑
γ
n
<
|
k
/
n
-
x
|
∞
b
n
,
k
,
α
(
x
)
k
n
-
x
q
+
i
≤
1
γ
n
q
V
n
,
α
t
-
x
q
+
i
;
x
.Finally, using Hölder inequality with Lemmas 2 and 3, we get the following inequality: (26)
K
n
,
α
,
q
f
;
γ
n
,
x
≤
2
q
C
V
n
,
α
t
-
x
2
q
;
x
1
/
2
+
C
∑
i
=
0
q
‍
1
+
2
q
-
i
-
1
x
q
-
i
γ
n
q
V
n
,
α
t
-
x
2
q
+
i
;
x
1
/
2
≤
2
q
C
x
(
1
+
α
x
)
n
+
1
n
2
q
/
2
+
C
∑
i
=
0
q
‍
1
+
2
q
-
i
-
1
x
q
-
i
γ
n
q
x
(
1
+
α
x
)
n
+
1
n
2
(
q
+
i
)
/
2
.Fixing x
∈
R
0, there exist constants C
(
x
) that maybe depend on x and constants α
,
q but are independent of n, such that (27)
K
n
,
α
,
q
f
;
γ
n
,
x
≤
2
q
C
x
(
1
+
α
x
)
n
+
1
n
2
q
/
2
+
C
γ
n
q
∑
i
=
0
q
‍
1
+
2
q
-
i
-
1
x
q
-
i
x
1
+
α
x
n
+
1
n
2
(
q
+
i
)
/
2
≤
C
(
x
)
n
q
/
2
+
C
(
x
)
n
q
/
2
γ
n
q
,and noticing that lim
n
→
∞
n
γ
n
=
∞, then we can get K
n
,
α
,
q
(
f
;
γ
n
,
x
)
=
o
(
1
), n
→
∞.


## 3. Main Results

In this section, we will study the properties of the operatorsV
n
,
α
,
q
(
f
;
x
) and give the estimation of degree of approximation by these operators.Theorem 6.
Fixq
∈
N
0, for every f
∈
S
2
q
+
1
2
q
+
1; then there exists a positive absolute constant C, such that(28)
w
2
q
+
1
x
V
n
,
α
,
2
q
+
1
f
;
x
-
f
x
≤
C
(
2
q
+
1
)
!
1
n
q
+
1
/
2
+
1
+
α
x
n
q
ω
(
f
(
2
q
+
1
)
;
1
n
)
,where C
>
0 is dependent only on q and α but is independent of x and n.Proof.
By assumption, using the modified Taylor formula [10], (29)
f
x
=
∑
i
=
0
2
q
+
1
‍
f
i
k
/
n
i
!
x
-
k
n
i
+
x
-
k
/
n
2
q
+
1
(
2
q
)
!
·
∫
0
1
‍
1
-
t
2
q
·
f
2
q
+
1
k
n
+
t
x
-
k
n
-
f
2
q
+
1
k
n
d
t
,with Lemma 2 and inequality (14), we get (30)
V
n
,
α
,
2
q
+
1
f
;
x
-
f
x
=
∑
k
=
0
∞
b
n
,
k
,
α
x
x
-
k
/
n
2
q
+
1
2
q
!
k
n
+
t
x
-
k
n
·
∫
0
1
‍
1
-
t
2
q
f
2
q
+
1
k
n
+
t
x
-
k
n
k
n
+
t
x
-
k
n
-
f
2
q
+
1
k
n
d
t
≤
∑
k
=
0
∞
b
n
,
k
,
α
(
x
)
x
-
k
/
n
2
q
+
1
2
q
!
·
∫
0
1
‍
1
-
t
2
q
f
2
q
+
1
k
n
+
t
x
-
k
n
k
n
+
t
x
-
k
n
-
f
2
q
+
1
k
n
d
t
≤
∑
k
=
0
∞
b
n
,
k
,
α
(
x
)
x
-
k
/
n
2
q
+
1
2
q
!
·
∫
0
1
‍
1
-
t
2
q
ω
f
(
2
q
+
1
)
;
t
x
-
k
n
d
t
≤
∑
k
=
0
∞
b
n
,
k
,
α
(
x
)
x
-
k
/
n
2
q
+
1
2
q
!
·
∫
0
1
‍
1
-
t
2
q
1
+
n
t
x
-
k
n
ω
f
2
q
+
1
;
1
n
d
t
=
ω
f
2
q
+
1
;
1
/
n
2
q
+
1
!
V
n
,
α
x
-
t
2
q
+
1
;
x
+
n
ω
f
2
q
+
1
;
1
/
n
2
q
!
B
2,2
q
+
1
V
n
,
α
x
-
t
2
q
+
2
;
x
,where B
(
a
,
b
) (a
>
0
,
b
>
0) denotes the Beta function, B
(
2,2
q
+
1
)
=
1
/
(
(
2
q
+
1
)
(
2
q
+
2
)
). Using the Hölder inequality with Lemmas 2 and 3, we further have(31)
V
n
,
α
,
2
q
+
1
f
;
x
-
f
x
≤
ω
f
2
q
+
1
;
1
/
n
2
q
+
1
!
V
n
,
α
x
-
t
4
q
+
2
;
x
1
/
2
+
n
ω
(
f
(
2
q
+
1
)
;
1
/
n
)
(
2
q
+
1
)
!
V
n
,
α
(
x
-
t
2
q
+
2
;
x
)
=
ω
f
2
q
+
1
;
1
/
n
2
q
+
1
!
·
∑
j
=
0
2
q
+
1
‍
P
j
,
4
q
+
2
,
α
(
x
)
n
4
q
+
2
-
j
1
/
2
+
n
∑
j
=
0
q
+
1
‍
P
j
,
2
q
+
2
,
α
(
x
)
n
2
q
+
2
-
j
=
ω
f
2
q
+
1
;
1
/
n
2
q
+
1
!
·
1
n
q
+
1
/
2
∑
j
=
0
2
q
+
1
P
j
,
4
q
+
2
,
α
(
x
)
1
/
2
∑
j
=
0
2
q
+
1
‍
P
j
,
4
q
+
2
,
α
(
x
)
1
/
2
+
1
n
q
∑
j
=
0
q
+
1
P
j
,
2
q
+
2
,
α
(
x
)
.Thus, we obtain (32)
w
2
q
+
1
x
V
n
,
α
,
2
q
+
1
f
;
x
-
f
x
=
ω
f
2
q
+
1
;
1
/
n
2
q
+
1
!
1
n
q
+
1
/
2
∑
j
=
0
2
q
+
1
‍
P
j
,
4
q
+
2
,
α
x
1
+
x
2
q
+
1
2
1
/
2
∑
j
=
0
2
q
+
1
‍
P
j
,
4
q
+
2
,
α
x
1
+
x
2
q
+
1
2
1
/
2
+
1
n
q
∑
j
=
0
q
+
1
‍
P
j
,
2
q
+
2
,
α
x
1
+
x
2
q
+
1
.Because P
j
,
4
q
+
2
,
α
(
x
) denotes an algebraic polynomial with order at most 4
q
+
2, there exists a positive absolute constant C, such that P
j
,
4
q
+
2
,
α
(
x
)
/
(
1
+
x
2
q
+
1
)
2
≤
C, while P
j
,
2
q
+
2
,
α
(
x
)
/
(
1
+
x
2
q
+
1
) is an at most 1-order algebraic polynomial with respect to x; that is, there exists a positive absolute constant C depending on α and q, such that ∑
j
=
0
q
+
1
‍
(
P
j
,
2
q
+
2
,
α
x
/
(
1
+
x
2
q
+
1
)
)
≤
C
(
1
+
α
x
).Remark 7.
The result ofV
n
,
α
,
2
q
+
2
(
f
;
x
) can be easily obtained by imitating Theorem 6; here we omit it because it will be mentioned in the proof of next theorem.Theorem6 is to focus on the pointwise approximation of the operators V
n
,
α
,
q
(
f
;
x
); now we will study their uniform approximation.Theorem 8.
Fixq
∈
N
0; for every f
∈
S
q
q, one has(33)
V
n
,
α
,
q
(
f
;
·
)
-
f
(
·
)
q
,
∞
=
O
1
q
!
n
q
/
2
.Proof.
From the proof of Theorem6, for f
∈
S
2
q
+
1
2
q
+
1, we can get (34)
V
n
,
α
,
2
q
+
1
f
;
x
-
f
x
≤
2
f
(
2
q
+
1
)
∞
∑
k
=
0
∞
b
n
,
k
,
α
(
x
)
x
-
k
/
n
2
q
+
1
2
q
+
1
!
=
2
f
(
2
q
+
1
)
∞
(
2
q
+
1
)
!
V
n
,
α
(
x
-
t
2
q
+
1
;
x
)
.Using the Hölder inequality with Lemma 3, we obtain (35)
w
2
q
+
1
x
V
n
,
α
,
2
q
+
1
f
;
x
-
f
x
≤
2
f
(
2
q
+
1
)
∞
(
2
q
+
1
)
!
1
n
q
+
1
/
2
∑
j
=
0
2
q
+
1
‍
P
j
,
4
q
+
2
,
α
(
x
)
1
+
x
2
q
+
1
2
1
/
2
≤
2
C
f
(
2
q
+
1
)
∞
(
2
q
+
1
)
!
1
n
q
+
1
/
2
.For all x
∈
R
0, we have(36)
V
n
,
α
,
2
q
+
1
(
f
;
·
)
-
f
(
·
)
q
,
∞
≤
2
C
f
(
2
q
+
1
)
∞
2
q
+
1
!
1
n
q
+
1
/
2
.On the other hand, for f
∈
S
2
q
+
2
2
q
+
2, similar to the proof of Theorem 6, we get (37)
V
n
,
α
,
2
q
+
2
f
;
x
-
f
x
≤
2
f
2
q
+
2
∞
2
q
+
2
!
V
n
,
α
x
-
t
2
q
+
2
;
x
.By Lemma 3, we obtain (38)
w
2
q
+
2
x
V
n
,
α
,
2
q
+
2
f
;
x
-
f
x
≤
2
f
(
2
q
+
2
)
∞
(
2
q
+
2
)
!
∑
j
=
0
q
+
1
‍
1
n
2
q
+
2
-
j
P
j
,
2
q
+
2
,
α
(
x
)
1
+
x
2
q
+
2
≤
2
C
f
(
2
q
+
2
)
∞
(
2
q
+
2
)
!
1
n
q
+
1
.For all x
∈
R
0, we further have(39)
V
n
,
α
,
2
q
+
2
(
f
;
·
)
-
f
(
·
)
q
,
∞
≤
2
C
f
(
2
q
+
2
)
∞
(
2
q
+
2
)
!
1
n
q
+
1
.Combining the above two inequalities (36) and (39), for all f
∈
S
q
q and fixed q
∈
N
0, the desired equality (33) is obtained.Remark 9.
Theorem8 indicates that the rate of convergence by the operator V
n
,
α
,
q
(
f
;
x
) is 1
/
n
q
/
2.Corollary 10.
Letf
∈
S
q
q with some q
∈
N
0, for all x
∈
R
+; then(40)
lim
n
→
∞
V
n
,
α
,
q
(
f
;
x
)
=
f
(
x
)
.Finally, we will discuss the convergence of the truncated sumB
n
,
α
,
q
(
f
;
γ
n
,
x
).Theorem 11.
Letf
∈
S
q
q with some q
∈
N
0, for fixed x
∈
R
+; then(41)
lim
n
→
∞
B
n
,
α
,
q
(
f
;
γ
n
,
x
)
=
f
(
x
)
.Moreover, the assertion (41) holds uniformly on every rectangle x
∈
[
a
,
b
] with 0
<
a
<
b.Proof.
Notice that(42)
B
n
,
α
,
q
f
;
γ
n
,
x
-
f
x
=
V
n
,
α
,
q
(
f
;
x
)
-
f
(
x
)
-
K
n
,
α
,
q
(
f
;
γ
n
,
x
)
.Using Corollary 10 and Lemma 5, we easily get the assertion (41).Remark 12.
Theorem11 demonstrates that the generalized Baskakov operators V
n
,
α
,
q
(
f
;
x
) can be replaced by the truncated operators B
n
,
α
,
q
(
f
;
γ
n
,
x
) in a certain sense from the computational point of view.



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*Source: 564854-2015-03-16.xml*