# L

Authors: Zoltán Finta
Journal: Journal of Operators
 (2014)

---

## Abstract

For a newq-Kantorovich operator we establish direct approximation theorems in the space L
p
[
0,1
]
,
1
≤
p
≤
∞
, via Ditzian-Totik modulus of smoothness of second order.


---

## Body

## 1. Introduction

Letq
>
0. Then for each non-negative integer n, the q-integer [
n
]
≡
[
n
]
q and the q-factorial [
n
]
!
≡
[
n
]
q
! are defined by [
n
]
=
1
+
q
+
⋯
+
q
n
-
1
(
n
=
1,2
,
…
),[
0
]
=
0, and [
n
]
!
=
[
1
]
[
2
]
⋯
[
n
]
(
n
=
1,2
,
…
),[
0
]
!
=
1. For integers 0
≤
k
≤
n, the q-binomial coefficient is defined by
(1)
[
n
k
]
=
[
n
]
!
[
k
]
!
[
n
-
k
]
!
.
Further, we set
(2)
p
n
,
k
(
q
;
x
)
=
[
n
k
]
x
k
(
1
-
x
)
(
1
-
q
x
)
⋯
(
1
-
q
n
-
k
-
1
x
)
for k
=
0,1
,
…
,
n
-
1 and p
n
,
n
(
q
;
x
)
=
x
n. Following Phillips [1], the operators B
n
,
q
:
C
[
0,1
]
→
C
[
0,1
] defined by
(3)
(
B
n
,
q
f
)
(
x
)
≡
B
n
,
q
(
f
,
x
)
=
∑
k
=
0
n
‍
p
n
,
k
(
q
;
x
)
f
(
[
k
]
[
n
]
)
are called q-Bernstein operators. They quickly gained the popularity and were studied widely by a number of authors. A survey of the obtained results and references on the subject can be found in [2]. The q-Bernstein operators are not defined for f
∈
L
p
[
0,1
]. For this reason we introduce the following operators based on q-integers:
(4)
(
K
n
,
q
f
)
(
x
)
≡
K
n
,
q
(
f
,
x
)
=
[
n
+
1
]
∑
k
=
0
n
‍
p
n
,
k
(
q
;
x
)
q
-
k
∫
[
k
]
/
[
n
+
1
]
[
k
+
1
]
/
[
n
+
1
]
‍
f
(
u
)
d
u
.
For q
=
1, we recover the well-known Kantorovich operators [3]. In [4, 5] other Kantorovich type operators involving q-integers are considered, which are defined with the aid of q-Riemann integral (see [6]), and their rate of convergence are studied for only continuous functions on [
0,1
].The aim of the paper is to establish direct approximation theorems for (4) in the space L
p
[
0,1
],1
≤
p
≤
∞. To describe our results, we will give the definitions of Ditzian-Totik modulus of smoothness of second order and the correspondingK-functionals (cf. [7]). For f
∈
L
p
[
0,1
], 1
≤
p
≤
∞, and φ
(
x
)
=
x
(
1
-
x
), x
∈
[
0,1
], we set
(5)
ω
φ
2
(
f
,
δ
)
p
=
sup
⁡
0
<
h
≤
δ
∥
f
(
x
+
h
φ
(
x
)
)
-
2
f
(
x
)
+
f
(
x
-
h
φ
(
x
)
)
∥
p
,
K
2
,
φ
(
f
,
δ
)
p
=
inf
⁡
g
∈
W
p
2
{
∥
f
-
g
∥
p
+
δ
∥
φ
2
g
′′
∥
p
}
,
K
¯
2
,
φ
(
f
,
δ
)
p
=
inf
⁡
g
∈
W
p
2
{
∥
f
-
g
∥
p
+
δ
∥
φ
2
g
′′
∥
p
+
δ
2
∥
g
′′
∥
p
}
,
where W
p
2
=
{
g
∈
L
p
[
0,1
]
:
g
′
∈
A
C
loc
⁡
[
0,1
]
,
∥
φ
2
g
′′
∥
p
<
∞
} and g
′
∈
A
C
loc
⁡
[
0,1
] means that g is differentiable such that g
′ is absolutely continuous in every interval [
a
,
b
]
⊆
[
0,1
]. It is known from [7] that ω
φ
2
(
f
,
δ
)
p, K
2
,
φ
(
f
,
δ
)
p and K
¯
2
,
φ
(
f
,
δ
)
p are equivalent; that is, there exists C
>
0 such that
(6)
C
-
1
ω
φ
2
(
f
,
δ
)
p
≤
K
2
,
φ
(
f
,
δ
)
p
≤
C
ω
φ
2
(
f
,
δ
)
p
,
(7)
C
-
1
ω
φ
2
(
f
,
δ
)
p
≤
K
¯
2
,
φ
(
f
,
δ
)
p
≤
C
ω
φ
2
(
f
,
δ
)
p
.
Here C denotes a positive constant independent of n and x, but it is not necessarily the same in different cases. Finally, we define the modulus of continuity of f
∈
C
[
0,1
] by
(8)
ω
(
f
,
δ
)
=
sup
⁡
{
|
f
(
x
)
-
f
(
y
)
|
:
x
,
y
∈
[
0,1
]
,
|
x
-
y
|
≤
δ
}
.The paper is organized as follows. In Section2 some auxiliary results are established and in Section 3 we give the main results, establishing direct approximation theorems for (4) in L
p
[
0,1
], 1
≤
p
≤
∞.


## 2. Auxiliary Results

In the sequel we need some lemmas.Lemma 1.
LetK
n
,
q, 0
<
q
<
1, be defined by (4) and δ
q
,
n
(
x
)
=
φ
(
x
)
+
[
n
+
1
]
-
1
/
2. Then K
n
,
q
(
1
,
x
)
=
1,|
K
n
,
q
(
t
-
x
,
x
)
|
≤
2
/
[
n
+
1
] and K
n
,
q
(
(
t
-
x
)
2
,
x
)
≤
(
8
/
[
n
+
1
]
)
δ
q
,
n
2
(
x
).Proof.
By (4), we obtain
(9)
K
n
,
q
(
f
,
x
)
=
∑
k
=
0
n
‍
p
n
,
k
(
q
;
x
)
∫
0
1
‍
f
(
[
k
]
+
q
k
u
[
n
+
1
]
)
d
u
.
Recall some properties of the q-Bernstein operators (see [1]):
(10)
B
n
,
q
(
1
,
x
)
=
1
,
B
n
,
q
(
t
-
x
,
x
)
=
0
,
B
n
,
q
(
(
t
-
x
)
2
,
x
)
=
φ
2
(
x
)
[
n
]
.
Then, by (9) and (10), we find K
n
,
q
(
1
,
x
)
=
B
n
,
q
(
1
,
x
)
=
1 and
(11)
K
n
,
q
(
t
-
x
,
x
)
=
∑
k
=
0
n
‍
p
n
,
k
(
q
;
x
)
∫
0
1
‍
(
[
k
]
+
q
k
u
[
n
+
1
]
-
x
)
d
u
=
∑
k
=
0
n
‍
p
n
,
k
(
q
;
x
)
∫
0
1
‍
(
[
k
]
+
q
k
u
[
n
+
1
]
-
[
k
]
[
n
]
)
d
u
+
B
n
,
q
(
t
-
x
,
x
)
=
∑
k
=
0
n
‍
p
n
,
k
(
q
;
x
)
∫
0
1
‍
(
[
k
]
+
q
k
u
[
n
+
1
]
-
[
k
]
[
n
]
)
d
u
.
But [
n
+
1
]
=
[
n
]
+
q
n implies that
(12)
|
[
k
]
+
q
k
u
[
n
+
1
]
-
[
k
]
[
n
]
|
=
|
q
k
[
n
]
u
-
q
n
[
k
]
|
[
n
]
[
n
+
1
]
≤
2
[
n
+
1
]
for k
=
0,1
,
…
,
n. Hence, by (10),
(13)
|
K
n
,
q
(
t
-
x
,
x
)
|
≤
B
n
,
q
(
1
,
x
)
2
[
n
+
1
]
≤
2
[
n
+
1
]
.
Further, by (9), (10) and (12), we have
(14)
K
n
,
q
(
(
t
-
x
)
2
,
x
)
=
∑
k
=
0
n
‍
p
n
,
k
(
q
;
x
)
∫
0
1
‍
(
[
k
]
+
q
k
u
[
n
+
1
]
-
x
)
2
d
u
≤
2
∑
k
=
0
n
‍
p
n
,
k
(
q
;
x
)
×
∫
0
1
‍
(
[
k
]
+
q
k
u
[
n
+
1
]
-
[
k
]
[
n
]
)
2
d
u
+
2
B
n
,
q
(
(
t
-
x
)
2
,
x
)
≤
8
[
n
+
1
]
2
+
2
[
n
+
1
]
[
n
]
φ
2
(
x
)
[
n
+
1
]
≤
8
[
n
+
1
]
δ
n
,
q
2
(
x
)
,
because
(15)
[
n
+
1
]
=
1
+
q
[
n
]
≤
2
[
n
]
.Lemma 2.
Letf
∈
L
p
[
0,1
] and let K
n
,
q
f be given by (4). Then (i)
∥
K
n
,
q
f
∥
p
≤
C
∥
f
∥
p, where 1
≤
p
<
∞, q
=
q
n
∈
(
0,1
) and there exists C
0
>
0 such that n
3
(
1
-
q
n
)
≤
C
0
<
1 for all n
=
1,2
,
…;
(ii)
∥
K
n
,
q
f
∥
∞
≤
C
∥
f
∥
∞, where 0
<
q
<
1.Proof.
By Riesz-Thorin theorem (see, e.g., [8]), we separate the proof for p
=
1 and p
=
∞.
Case
1 (
p
=
1
). Because the q-binomial coefficients are increasing functions of q, we get for the polynomials p
n
,
k
(
q
;
x
) and p
n
,
k
(
1
;
x
) of degree n that
(16)
p
n
,
k
(
q
;
x
)
p
n
,
k
(
1
;
x
)
=
[
n
k
]
x
k
(
1
-
x
)
(
1
-
q
x
)
⋯
(
1
-
q
n
-
k
-
1
x
)
(
n
k
)
x
k
(
1
-
x
)
n
-
k
≤
1
-
q
x
1
-
x
⋯
1
-
q
n
-
k
-
1
x
1
-
x
≤
(
1
-
q
n
x
1
-
x
)
n
-
k
-
1
≤
(
1
+
(
1
-
q
n
)
x
1
-
x
)
n
.
Now let x
∈
[
1
/
(
2
n
2
)
,
1
-
1
/
(
2
n
2
)
] and k
∈
{
0,1
,
…
,
n
-
2
}. Then, in view of n
3
(
1
-
q
n
)
≤
C
0, n
=
1,2
,
…, we obtain
(17)
p
n
,
k
(
q
;
x
)
p
n
,
k
(
1
;
x
)
≤
(
1
+
2
n
2
(
1
-
q
n
)
)
n
=
{
(
1
+
2
n
2
(
1
-
q
n
)
)
1
/
(
2
n
2
(
1
-
q
n
)
)
}
2
n
3
(
1
-
q
n
)
≤
C
.
Using [7, Theorem 8.4.8, page 108] translated from [
-
1,1
] to [
0,1
], we find that
(18)
∥
p
n
,
k
(
q
;
x
)
∥
1
≤
C
∥
p
n
,
k
(
q
;
x
)
∥
L
1
[
1
/
(
2
n
2
)
,
1
-
1
/
(
2
n
2
)
]
≤
C
∥
p
n
,
k
(
1
;
x
)
∥
L
1
[
1
/
(
2
n
2
)
,
1
-
1
/
(
2
n
2
)
]
≤
C
∥
p
n
,
k
(
1
;
x
)
∥
1
=
C
n
+
1
.
On the other handn
3
(
1
-
q
n
)
≤
C
0
<
1, n
=
1,2
,
…, implies that q
n
≥
1
-
C
0
>
0, n
=
1,2
,
…. Hence, by (4) and (18), we get
(19)
∥
K
n
,
q
f
∥
1
≤
[
n
+
1
]
∑
k
=
0
n
‍
q
-
k
∥
p
n
,
k
(
q
;
x
)
∥
1
×
∫
[
k
]
/
[
n
+
1
]
[
k
+
1
]
/
[
n
+
1
]
‍
|
f
(
u
)
|
d
u
≤
C
[
n
+
1
]
q
n
(
n
+
1
)
∥
f
∥
1
≤
C
∥
f
∥
1
.
Case
2 (
p
=
∞
). Due to (9) and (10), we have
(20)
|
K
n
,
q
(
f
,
x
)
|
≤
∑
k
=
0
n
‍
p
n
,
k
(
q
;
x
)
∫
0
1
‍
|
f
(
[
k
]
+
q
k
u
[
n
+
1
]
)
|
d
u
≤
B
n
,
q
(
1
,
x
)
∥
f
∥
∞
=
∥
f
∥
∞
.Lemma 3.
IfK
n
,
q
f is defined by (4), E
[
n
+
1
]
=
[
1
/
[
n
+
1
]
,
1
-
1
/
[
n
+
1
]
],g
∈
W
p
2 and 1
≤
p
<
∞, then
(21)
∥
K
n
,
q
(
∫
x
t
‍
(
t
-
v
)
g
′′
(
v
)
d
v
,
x
)
∥
L
p
(
E
[
n
+
1
]
)
≤
C
[
n
+
1
]
∥
φ
2
g
′′
∥
p
,
where q
=
q
n
∈
(
0,1
) satisfies that there exists C
0
>
0 such that n
3
(
1
-
q
n
)
≤
C
0
<
1 for all n
=
1,2
,
….Proof.
Consider the following cases.
Case
1 (
p
=
1
). Because of [7, Lemma 9.6.1, page 140], we have |
u
-
v
|
/
φ
2
(
v
)
≤
|
u
-
x
|
/
φ
2
(
x
). Then, by (4),
(22)
∫
E
[
n
+
1
]
‍
|
K
n
,
q
(
∫
x
t
‍
(
t
-
v
)
g
′′
(
v
)
d
v
,
x
)
|
d
x
≤
∫
E
[
n
+
1
]
‍
∑
k
=
0
n
‍
p
n
,
k
(
q
;
x
)
[
n
+
1
]
q
-
k
×
∫
[
k
]
/
[
n
+
1
]
[
k
+
1
]
/
[
n
+
1
]
‍
|
u
-
x
|
φ
2
(
x
)
×
|
∫
x
u
‍
φ
2
(
v
)
|
g
′′
(
v
)
|
d
v
|
d
u
d
x
.
But
(23)
|
[
k
]
[
n
+
1
]
-
[
k
]
[
n
]
|
=
q
n
[
k
]
[
n
]
[
n
+
1
]
≤
1
[
n
+
1
]
,
|
[
k
+
1
]
[
n
+
1
]
-
[
k
]
[
n
]
|
=
|
1
+
q
[
k
]
1
+
q
[
n
]
-
[
k
]
[
n
]
|
=
[
n
]
-
[
k
]
[
n
]
[
n
+
1
]
=
q
k
[
n
-
k
]
[
n
]
[
n
+
1
]
≤
1
[
n
+
1
]
imply
(24)
[
n
+
1
]
q
-
k
∫
[
k
]
/
[
n
+
1
]
[
k
+
1
]
/
[
n
+
1
]
‍
|
u
-
x
|
d
u
≤
[
n
+
1
]
q
-
k
∫
[
k
]
/
[
n
+
1
]
[
k
+
1
]
/
[
n
+
1
]
‍
|
u
-
[
k
]
[
n
]
|
d
u
+
|
[
k
]
[
n
]
-
x
|
≤
1
[
n
+
1
]
+
|
[
k
]
[
n
]
-
x
|
.
Hence, by (22),
(25)
∫
E
[
n
+
1
]
‍
|
K
n
,
q
(
∫
x
t
‍
(
t
-
v
)
g
′′
(
v
)
d
v
,
x
)
|
d
x
≤
∫
E
[
n
+
1
]
‍
∑
k
=
0
n
‍
p
n
,
k
(
q
;
x
)
φ
-
2
(
x
)
×
(
|
[
k
]
[
n
]
-
x
|
+
1
[
n
+
1
]
)
×
|
∫
x
[
k
*
]
/
[
n
+
1
]
‍
φ
2
(
v
)
|
g
′′
(
v
)
|
d
v
|
d
x
=
I
q
,
n
,
where k
* is either k or k
+
1 such that
(26)
|
∫
x
[
k
*
]
/
[
n
+
1
]
‍
φ
2
(
v
)
|
g
′′
(
v
)
|
d
v
|
=
max
⁡
j
=
k
,
k
+
1
|
∫
x
[
j
]
/
[
n
+
1
]
‍
φ
2
(
v
)
|
g
′′
(
v
)
|
d
v
|
.
Using the method in [7, pages 146-147], let
(27)
D
(
l
,
q
,
n
,
x
)
=
{
k
:
l
φ
(
x
)
[
n
+
1
]
-
1
/
2
|
k
n
-
x
|
≤
|
k
n
-
x
|
<
(
l
+
1
)
φ
(
x
)
[
n
+
1
]
-
1
/
2
}
.
Then
(28)
I
q
,
n
=
∫
E
[
n
+
1
]
φ
-
2
(
x
)
∑
l
=
0
∞
∑
k
∈
D
(
l
,
q
,
n
,
x
)
p
n
,
k
(
q
;
x
)
‍
‍
‍
×
(
|
[
k
]
[
n
]
-
x
|
+
1
[
n
+
1
]
)
×
|
∫
x
[
k
*
]
/
[
n
+
1
]
‍
φ
2
(
v
)
|
g
′′
(
v
)
|
d
v
|
d
x
.
Forx
∈
E
[
n
+
1
] we prove that
(29)
∑
k
∈
D
(
l
,
q
,
n
,
x
)
‍
p
n
,
k
(
q
;
x
)
(
|
[
k
]
[
n
]
-
x
|
+
1
[
n
+
1
]
)
≤
C
(
l
+
1
)
4
φ
(
x
)
[
n
+
1
]
-
1
/
2
.
To obtain (29), we need some useful estimations:
(30)
|
k
n
-
[
k
]
[
n
]
|
≤
4
[
n
+
1
]
,
k
=
0,1
,
…
,
n
,
(31)
[
n
+
1
]
-
1
/
2
≤
2
1
-
C
0
φ
(
x
)
,
x
∈
E
[
n
+
1
]
.
Indeed, for k
∈
{
1,2
,
…
,
n
} we have 0
≤
k
-
[
k
]
=
(
1
-
q
)
+
(
1
-
q
2
)
+
⋯
+
(
1
-
q
k
-
1
)
≤
(
k
-
1
)
(
1
-
q
k
-
1
)
≤
n
3
(
1
-
q
n
)
≤
C
0
<
1. Hence, by (15),
(32)
|
k
n
-
[
k
]
[
n
]
|
≤
|
k
n
-
k
[
n
]
|
+
|
k
[
n
]
-
[
k
]
[
n
]
|
=
k
n
n
-
[
n
]
[
n
]
+
k
-
[
k
]
[
n
]
≤
2
[
n
]
≤
4
[
n
+
1
]
.
Further, for x
∈
E
[
n
+
1
] we have
(33)
φ
2
(
x
)
=
x
(
1
-
x
)
≥
1
[
n
+
1
]
(
1
-
1
[
n
+
1
]
)
=
1
[
n
+
1
]
q
[
n
]
[
n
+
1
]
≥
1
[
n
+
1
]
1
-
C
0
2
,
taking into account (15) and q
≥
q
n
>
1
-
C
0
>
0, because of n
3
(
1
-
q
n
)
≤
C
0
<
1.
Now letl
≥
1 and x
∈
E
[
n
+
1
]. By (15), we have E
[
n
+
1
]
⊆
[
1
/
(
2
n
)
,
1
-
1
/
(
2
n
)
]
⊆
[
1
/
(
2
n
2
)
,
1
-
1
/
(
2
n
2
)
]. Then, in view of (30), (17), [7, Lemma 9.4.4, page 128], (15) and (31), we obtain
(34)
∑
k
∈
D
(
l
,
q
,
n
,
x
)
‍
p
n
,
k
(
q
;
x
)
(
|
[
k
]
[
n
]
-
x
|
+
1
[
n
+
1
]
)
≤
C
∑
k
∈
D
(
l
,
q
,
n
,
x
)
‍
p
n
,
k
(
1
;
x
)
(
|
k
n
-
x
|
+
1
[
n
+
1
]
)
≤
C
[
n
+
1
]
2
l
4
φ
4
(
x
)
∑
k
=
0
n
‍
p
n
,
k
(
1
;
x
)
(
|
k
n
-
x
|
5
+
1
[
n
+
1
]
|
k
n
-
x
|
4
)
≤
C
[
n
+
1
]
2
l
4
φ
4
(
x
)
(
φ
5
(
x
)
n
5
/
2
+
1
[
n
+
1
]
φ
4
(
x
)
n
2
)
≤
C
l
4
(
[
n
+
1
]
2
[
n
]
5
/
2
φ
(
x
)
+
[
n
+
1
]
2
[
n
]
2
1
[
n
+
1
]
)
≤
C
l
4

(

4
2
[
n
+
1
]
-
1
/
2
φ
(
x
)

+

4
[
n
+
1
]
-
1
/
2
2
1
-
C
0
φ
(
x
)

)

≤
C
(
l
+
1
)
4
φ
(
x
)
[
n
+
1
]
-
1
/
2
.
For x
∈
E
[
n
+
1
], by (10), (15) and (31), we have
(35)
∑
k
∈
D
(
0
,
q
,
n
,
x
)
p
n
,
k
(
q
;
x
)
(
|
[
k
]
[
n
]
-
x
|
+
1
[
n
+
1
]
)
‍
≤
{
B
n
,
q
(
(
t
-
x
)
2
,
x
)
}
1
/
2
+
1
[
n
+
1
]
B
n
,
q
(
1
,
x
)
≤
2
φ
(
x
)
[
n
+
1
]
-
1
/
2
+
2
1
-
C
0
φ
(
x
)
[
n
+
1
]
-
1
/
2
≤
C
φ
(
x
)
[
n
+
1
]
-
1
/
2
.
We now define(36)
F
(
l
,
x
)
=
{
v
:
v
∈
[
0,1
]
,
[
n

+

1
]
-
1
/
2
|
v
-
x
|
≤
5
(
l
+
1
)
φ
(
x
)
[
n

+

1
]
-
1
/
2

+

5
[
n
+
1
]
-
1
}
,
G
(
l
,
v
)
=
{
v
:
x
∈
E
[
n
+
1
]
,
v
∈
F
(
l
,
x
)
}
.
Then [
k
*
]
/
[
n
+
1
]
∈
F
(
l
,
x
), where k
∈
D
(
l
,
q
,
n
,
x
), because (23), (30) and the definition of D
(
l
,
q
,
n
,
x
) imply that
(37)
|
[
k
*
]
[
n
+
1
]
-
x
|
≤
|
[
k
*
]
[
n
+
1
]
-
[
k
]
[
n
]
|
+
|
[
k
]
[
n
]
-
k
n
|
+
|
k
n
-
x
|
≤
1
[
n
+
1
]
+
4
[
n
+
1
]
+
(
l
+
1
)
φ
(
x
)
[
n
+
1
]
-
1
/
2
≤
5
(
l
+
1
)
φ
(
x
)
[
n
+
1
]
-
1
/
2
+
5
[
n
+
1
]
-
1
.
Hence, by (29), (28) and (15), we get
(38)
I
q
,
n
≤
C
[
n
+
1
]
∑
l
=
0
∞
‍
1
(
l
+
1
)
4
×
∫
E
[
n
+
1
]
‍
[
n
+
1
]
1
/
2
φ
(
x
)
∫
F
(
l
,
x
)
‍
φ
2
(
v
)
|
g
′′
(
v
)
|
d
v
d
x
≤
C
[
n
+
1
]
∑
l
≤
n
1
/
4
‍
1
(
l
+
1
)
4
×
∫
0
1
‍
φ
2
(
v
)
|
g
′′
(
v
)
|
[
n
+
1
]
1
/
2
{
∫
G
(
l
,
v
)
‍
φ
-
1
(
x
)
d
x
}
d
v
+
C
[
n
+
1
]
∑
l
≥
n
1
/
4
‍
1
(
l
+
1
)
4
×
∫
0
1
‍
φ
2
(
v
)
|
g
′′
(
v
)
|
n
1
/
2
×
{
∫
G
(
l
,
v
)
‍
φ
-
1
(
x
)
d
x
}
d
v
.
Applying the estimate (see [7, (9.6.14), page 147])
(39)
∫
x
:
|
v
-
x
|
≤
h
φ
(
x
)
+
h
2
‍
φ
-
1
(
x
)
d
x
≤
M
h
for h
=
5
(
l
+
1
)
[
n
+
1
]
-
1
/
2 and recalling that h
2
≥
5
[
n
+
1
]
-
1, we have
(40)
∫
G
(
l
,
v
)
‍
φ
-
1
(
x
)
d
x
≤
5
M
(
l
+
1
)
[
n
+
1
]
-
1
/
2
.
Substituting this estimate in (38) for l
≤
n
1
/
4 and following the procedure given in [7, page 147], we obtain the statement of our lemma, because of (25).
Case
2 (
1
<
p
<
∞
). Let x
∈
E
[
n
+
1
]. The function δ
q
,
n
2 is concave on [
0,1
]; therefore
(41)
|
t
-
v
|
δ
q
,
n
2
(
v
)
≤
|
t
-
x
|
δ
q
,
n
2
(
x
)
(see [9, (4)]). Then, by Lemma 1,
(42)
|
K
n
,
q
(
∫
x
t
‍
(
t
-
v
)
g
′′
(
v
)
d
v
,
x
)
|
≤
K
n
,
q
(
|
t
-
x
|
δ
q
,
n
2
(
x
)
|
∫
x
t
‍
δ
q
,
n
2
(
v
)
|
g
′′
(
v
)
|
d
v
|
,
x
)
=
K
n
,
q
(
(
t
-
x
)
2
δ
q
,
n
2
(
x
)
|
1
t
-
x
∫
x
t
‍
δ
q
,
n
2
(
v
)
|
g
′′
(
v
)
|
d
v
|
,
x
)
≤
K
n
,
q
(
(
t
-
x
)
2
δ
q
,
n
2
(
x
)
,
x
)
M
(
δ
q
,
n
2
|
g
′′
|
,
x
)
≤
8
[
n
+
1
]
M
(
δ
q
,
n
2
|
g
′′
|
,
x
)
,
where M
(
δ
q
,
n
2
|
g
′′
|
,
x
)
=
sup
⁡
t
|
(
1
/
(
t
-
x
)
)
∫
x
t
‍
δ
q
,
n
2
(
v
)
|
g
′′
(
v
)
|
d
v
| is the Hardy-Littlewood maximal function. It is known that ∥
M
(
δ
q
,
n
2
|
g
′′
|
)
∥
p
≤
(
p
/
(
p
-
1
)
)
∥
δ
q
,
n
2
g
′′
∥
p,1
<
p
<
∞. Hence, by (42),
(43)
∥
K
n
,
q
(
∫
x
t
‍
(
t
-
v
)
g
′′
(
v
)
d
v
,
x
)
∥
L
p
(
E
[
n
+
1
]
)
≤
8
[
n
+
1
]
p
p
-
1
∥
δ
q
,
n
2
g
′′
∥
p
.
But δ
q
,
n
2
(
x
)
≤
2
(
φ
2
(
x
)
+
(
1
/
[
n
+
1
]
)
)
≤
C
φ
2
(
x
), because of (31). Thus
(44)
∥
K
n
,
q
(
∫
x
t
‍
(
t
-
v
)
g
′′
(
v
)
d
v
,
x
)
∥
L
p
(
E
[
n
+
1
]
)
≤
C
[
n
+
1
]
∥
φ
2
g
′′
∥
p
,
which completes the proof.Lemma 4.
IfP
l is a polynomial of degree at most l, where l
<
n, then K
n
,
q
P
l is also a polynomial of degree at most l.Proof.
It is sufficient to prove thatK
n
,
q
(
t
l
,
x
) is a polynomial of degree at most l. By (4), we have
(45)
K
n
,
q
(
t
l
,
x
)
=
[
n
+
1
]
∑
k
=
0
n
‍
p
n
,
k
(
q
;
x
)
q
-
k
∫
[
k
]
/
[
n
+
1
]
[
k
+
1
]
/
[
n
+
1
]
‍
u
l
d
u
=
1
l
+
1
∑
k
=
0
n
‍
p
n
,
k
(
q
;
x
)
×
{
(
[
k
+
1
]
[
n
+
1
]
)
l
+
(
[
k
+
1
]
[
n
+
1
]
)
l
-
1
[
k
]
[
n
+
1
]
+
⋯
+
(
[
k
]
[
n
+
1
]
)
l
}
=
1
l
+
1
(
[
n
]
[
n
+
1
]
)
l
∑
k
=
0
n
‍
p
n
,
k
(
q
;
x
)
×
{
(
1
+
q
[
k
]
[
n
]
)
l
+
(
1
+
q
[
k
]
[
n
]
)
l
-
1
[
k
]
[
n
]
+
⋯
+
(
[
k
]
[
n
]
)
l
}
=
1
l
+
1
(
[
n
]
[
n
+
1
]
)
l
∑
k
=
0
n
‍
p
n
,
k
(
q
;
x
)
×
{
a
0
(
l
,
q
,
n
)
(
[
k
]
[
n
]
)
l
+
a
1
(
l
,
q
,
n
)
[
k
]
[
n
]
+
⋯
+
a
l
(
l
,
q
,
n
)
(
[
k
]
[
n
]
)
l
}
=
1
l
+
1
(
[
n
]
[
n
+
1
]
)
l
{
a
0
(
l
,
q
,
n
)
B
n
,
q
(
1
,
x
)
+
a
1
(
l
,
q
,
n
)
B
n
,
q
(
t
,
x
)
+
⋯
+
a
l
(
l
,
q
,
n
)

B
n
,
q
(
t
l
,
x
)
}

.
Since B
n
,
q
(
t
m
,
x
) is a polynomial of degree min
⁡
{
m
,
n
} (see [1]), we obtain the assertion of the lemma.


## 3. Main Results

The direct approximation results are included in the following theorem.Theorem 5.
Letf
∈
L
p
[
0,1
] be arbitrary and let K
n
,
q
f be given by (4). (i)
If1
≤
p
<
∞, q
=
q
n
∈
(
0,1
), and C
0
>
0 satisfies that n
3
(
1
-
q
n
)
≤
C
0
<
1 for all n
=
1,2
,
…, then
(46)
∥
K
n
,
q
f
-
f
∥
p
≤
C
{
ω
φ
2
(
f
,
[
n
+
1
]
-
1
/
2
)
p
+
[
n
+
1
]
-
1
∥
f
∥
p
}
.
(ii)
Ifq
=
q
n
∈
(
0,1
) satisfies that q
n
→
1 as n
→
∞, then
(47)
∥
K
n
,
q
f
-
f
∥
∞
≤
C
{
ω
φ
2
(
f
,
[
n
+
1
]
-
1
/
2
)
∞
+
ω
(
f
,
1
2
[
n
+
1
]
-
1
)
}
.Proof.
(i) Let g
∈
W
p
2. By Taylor’s formula,
(48)
g
(
t
)
=
g
(
x
)
+
g
′
(
x
)
(
t
-
x
)
+
∫
x
t
‍
(
t
-
v
)
g
′′
(
v
)
d
v
and Lemma 1, we obtain
(49)
|
K
n
,
q
(
g
,
x
)
-
g
(
x
)
|
≤
|
g
′
(
x
)
|
|
K
n
,
q
(
t
-
x
,
x
)
|
+
|
K
n
,
q
(
∫
x
t
‍
(
t
-
v
)
g
′′
(
v
)
d
v
,
x
)
|
≤
2
[
n
+
1
]
|
g
′
(
x
)
|
+
|
K
n
,
q
(
∫
x
t
‍
(
t
-
v
)
g
′′
(
v
)
d
v
,
x
)
|
.
Hence, for x
∈
E
[
n
+
1
], we have
(50)
∥
K
n
,
q
g
-
g
∥
L
p
(
E
[
n
+
1
]
)
≤
2
[
n
+
1
]
∥
g
′
∥
p
+
∥
K
n
,
q
(
∫
x
t
‍
(
t
-
v
)
g
′′
(
v
)
d
v
,
x
)
∥
L
p
(
E
[
n
+
1
]
)
.
Using the estimate ∥
g
′
∥
p
≤
C
(
∥
φ
2
g
′′
∥
p
+
∥
g
∥
p
),1
≤
p
<
∞ (see [7, (a), page 135]), and Lemma 3, we get, in view of (50), that
(51)
∥
K
n
,
q
g
-
g
∥
L
p
(
E
[
n
+
1
]
)
≤
C
[
n
+
1
]
-
1
(
∥
φ
2
g
′′
∥
p
+
∥
g
∥
p
)
.
On the other hand, the conditionn
3
(
1
-
q
n
)
≤
C
0
<
1, n
=
1,2
,
…, implies that 0
≤
n
-
[
n
]
=
(
1
-
q
)
+
(
1
-
q
2
)
+
⋯
+
(
1
-
q
n
-
1
)
≤
(
n
-
1
)
(
1
-
q
n
-
1
)
≤
n
3
(
1
-
q
n
)
≤
C
0
<
1 and q
≥
q
n
≥
1
-
C
0
n
-
3
≥
1
-
C
0
>
0. Then n
<
1
+
[
n
]
=
(
1
/
q
)
(
q
+
q
[
n
]
)
<
(
1
/
q
)
[
n
+
1
]
≤
(
1
/
(
1
-
C
0
)
)
[
n
+
1
]. In conclusion
(52)
(
1
-
C
0
)
n
<
[
n
+
1
]
.
Finally, we follow the procedure given in [7, pages 118-119]. The definition of K
2
,
φ
(
f
,
[
n
+
1
]
-
1
)
p implies the existence of g
n
∈
W
p
2 such that
(53)
∥
f
-
g
n
∥
p
≤
2
K
2
,
φ
(
f
,
[
n
+
1
]
-
1
)
p
,
(54)
∥
φ
2
g
n
′′
∥
p
≤
2
[
n
+
1
]
K
2
,
φ
(
f
,
[
n
+
1
]
-
1
)
p
.
Then, by (51), (53), (54) and (6), we get
(55)
∥
K
n
,
q
g
n
-
g
n
∥
L
p
(
E
[
n
+
1
]
)
≤
C
[
n
+
1
]
-
1
(
∥
φ
2
g
n
′′
∥
p
+
∥
g
n
∥
p
)
≤
C
[
n
+
1
]
-
1
(
∥
φ
2
g
n
′′
∥
p
+
∥
f
-
g
n
∥
p
+
∥
f
∥
p
)
≤
C
{
K
2
,
φ
(
f
,
[
n
+
1
]
-
1
)
p
+
[
n
+
1
]
-
1
∥
f
∥
p
}
≤
C
{
ω
φ
2
(
f
,
[
n
+
1
]
-
1
/
2
)
p
+
[
n
+
1
]
-
1
∥
f
∥
p
}
.
Further, due to (54), (15) and (52), we have ∥
φ
2
g
n
′′
∥
p
≤
C
n
K
2
,
φ
(
f
,
n
-
1
)
p. Then, by [7, Theorem 7.3.1, page 84], we can choose g
n to be P
[
n
] the best [
n
]th degree polynomial approximation in L
p
[
0,1
]. Therefore, by (52) and (55), we obtain
(56)
∥
K
n
,
q
P
[
n
]
-
P
[
n
]
∥
L
p
[
1
/
(
(
1
-
C
0
)
n
)
,
1
-
1
/
(
(
1
-
C
0
)
n
)
]
≤
∥
K
n
,
q
P
[
n
]
-
P
[
n
]
∥
L
p
(
E
[
n
+
1
]
)
≤
C
{
ω
φ
2
(
f
,
[
n
+
1
]
-
1
/
2
)
p
+
[
n
+
1
]
-
1
∥
f
∥
p
}
.
Taking into account Lemma 4 and Theorem 8.4.8 of [7, page 108] translated from [
-
1,1
] to [
0,1
] and n to [
n
], we obtain, using (56), that
(57)
∥
K
n
,
q
P
[
n
]
-
P
[
n
]
∥
L
p
[
0,1
]
≤
C
∥
K
n
,
q
P
[
n
]
-
P
[
n
]
∥
L
p
[
1
/
(
(
1
-
C
0
)
n
)
,
1
-
1
/
(
(
1
-
C
0
)
n
)
]
≤
C
{
ω
φ
2
(
f
,
[
n
+
1
]
-
1
/
2
)
p
+
[
n
+
1
]
-
1
∥
f
∥
p
}
.
Hence, due to Lemma 2, (53) and (6), we find
(58)
∥
K
n
,
q
f
-
f
∥
p
≤
∥
K
n
,
q
(
f
-
P
[
n
]
)
-
(
f
-
P
[
n
]
)
∥
p
+
∥
K
n
,
q
P
[
n
]
-
P
[
n
]
∥
p
≤
C
∥
f
-
P
[
n
]
∥
p
+
∥
K
n
,
q
P
[
n
]
-
P
[
n
]
∥
p
≤
C
{
ω
φ
2
(
f
,
[
n
+
1
]
-
1
/
2
)
p
+
[
n
+
1
]
-
1
∥
f
∥
p
}
.
(ii) We introduce the operator
(59)
K
~
n
,
q
(
f
,
x
)
=
K
n
,
q
(
f
,
x
)
-
f
(
a
(
q
,
n
,
x
)
)
+
f
(
x
)
,
where a
(
q
,
n
,
x
)
=
(
(
1
+
q
)
/
2
)
(
[
n
]
/
[
n
+
1
]
)
x
+
(
1
/
2
[
n
+
1
]
). By Lemmas 1 and 2, we have K
~
n
,
q
(
1
,
x
)
=
1 and
(60)
|
K
~
n
,
q
(
f
,
x
)
|
≤
|
K
n
,
q
(
f
,
x
)
|
+
|
f
(
a
(
q
,
n
,
x
)
)
|
+
|
f
(
x
)
|
≤
3
∥
f
∥
∞
.
Further, by (4) and (10), we obtain
(61)
K
n
,
q
(
t
,
x
)
=
[
n
+
1
]
∑
k
=
0
n
‍
p
n
,
k
(
q
;
x
)
q
-
k
[
k
+
1
]
2
-
[
k
]
2
2
[
n
+
1
]
2
=
1
2
[
n
+
1
]
∑
k
=
0
n
‍
p
n
,
k
(
q
;
x
)
(
1
+
(
q
+
1
)
[
k
]
)
=
1
2
[
n
+
1
]
(
B
n
,
q
(
1
,
x
)
+
(
1
+
q
)
[
n
]
B
n
,
q
(
t
,
x
)
)
=
a
(
n
,
q
,
x
)
.
Thus K
~
n
,
q
(
t
,
x
)
=
x. Hence, in view of Taylor’s formula,
(62)
g
(
t
)

=

g
(
x
)
+
g
′
(
x
)
(
t
-
x
)

+

∫
x
t
‍
(
t
-
v
)
g
′′
(
v
)
d
v
,
g
∈
W
∞
2
,
and (59), we get
(63)
K
~
n
,
q
(
g
,
x
)
=
g
(
x
)
+
K
~
n
,
q
(
∫
x
t
‍
(
t
-
v
)
g
′′
(
v
)
d
v
,
x
)
=
g
(
x
)
+
K
n
,
q
(
∫
x
t
‍
(
t
-
v
)
g
′′
(
v
)
d
v
,
x
)
-
∫
x
a
(
q
,
n
,
x
)
‍
(
a
(
q
,
n
,
x
)
-
v
)
g
′′
(
v
)
d
v
.
But
(64)
|
a
(
q
,
n
,
x
)
-
x
|
=
|
(
1
+
q
)
[
n
]
-
2
[
n
+
1
]
2
[
n
+
1
]
x
+
1
2
[
n
+
1
]
|
=
|
(
1
-
q
)
[
n
]
-
2
2
[
n
+
1
]
x
+
1
2
[
n
+
1
]
|
=
|
1
-
(
1
+
q
n
)
x
2
[
n
+
1
]
|
≤
1
2
[
n
+
1
]
.
Hence, by (63), (41), Lemma 1 and δ
q
,
n
2
(
x
)
≤
2
(
φ
2
(
x
)
+
[
n
+
1
]
-
1
), we find
(65)
|
K
~
n
,
q
(
g
,
x
)
-
g
(
x
)
|
≤
K
n
,
q
(
|
t
-
x
|
δ
q
,
n
-
2
(
x
)
|
∫
x
t
‍
δ
q
,
n
2
(
v
)
|
g
′′
(
v
)
|
d
v
|
,
x
)
+
(
a
(
q
,
n
,
x
)
-
x
)
2
∥
g
′′
∥
∞
≤
δ
q
,
n
-
2
(
x
)
K
n
,
q
(
(
t
-
x
)
2
,
x
)
∥
δ
q
,
n
2
g
′′
∥
∞
+
1
4
[
n
+
1
]
2
∥
g
′′
∥
∞
≤
8
[
n
+
1
]
∥
δ
q
,
n
2
g
′′
∥
∞
+
1
4
[
n
+
1
]
2
∥
g
′′
∥
∞
≤
17
(
1
[
n
+
1
]
∥
φ
2
g
′′
∥
∞
+
1
[
n
+
1
]
2
∥
g
′′
∥
∞
)
.
Now, from (59), (60), (65) and (64), we have
(66)
|
K
n
,
q
(
f
,
x
)
-
f
(
x
)
|
≤
|
K
~
n
,
q
(
f
,
x
)
-
f
(
x
)
|
+
|
f
(
a
(
q
,
n
,
x
)
)
-
f
(
x
)
|
≤
|
K
~
n
,
q
(
f
-
g
,
x
)
|
+
|
K
~
n
,
q
(
g
,
x
)
-
g
(
x
)
|
+
|
g
(
x
)
-
f
(
x
)
|
+
|
f
(
a
(
q
,
n
,
x
)
)
-
f
(
x
)
|
≤
4
∥
f
-
g
∥
∞
+
17
(
1
[
n
+
1
]
∥
φ
2
g
′′
∥
∞
+
1
[
n
+
1
]
2
∥
g
′′
∥
∞
)
+
ω
(
f
,
|
a
(
q
,
n
,
x
)
-
x
|
)
≤
17
(
∥
f
-
g
∥
∞
+
1
[
n
+
1
]
∥
φ
2
g
′′
∥
∞
+
1
[
n
+
1
]
2
∥
g
′′
∥
∞
)
+
ω
(
f
,
1
2
[
n
+
1
]
-
1
)
.
Taking the infimum on the right-hand side over all g
∈
W
∞
2, and using the equivalence between K
¯
2
,
φ
(
f
,
[
n
+
1
]
-
1
)
∞ and ω
φ
2
(
f
,
[
n
+
1
]
-
1
/
2
)
∞ (see (7)), we obtain the statement of the theorem.



---

*Source: 958656-2014-05-15.xml*