# Approximation byq-Bernstein Polynomials in the Case q

Authors: Xuezhi Wu
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

LetB
n
,
q
(
f
;
x
), q
∈
(
0
,
∞
) be the q-Bernstein polynomials of a function f
∈
C
[
0,1
]. It has been known that, in general, the sequence B
n
,
q
n
(
f
) with q
n
→
1
+ is not an approximating sequence for f
∈
C
[
0,1
], in contrast to the standard case q
n
→
1
-. In this paper, we give the sufficient and necessary condition under which the sequence B
n
,
q
n
(
f
) approximates  f for any f
∈
C
[
0,1
] in the case q
n
>
1. Based on this condition, we get that if 1
<
q
n
<
1
+
ln
⁡
2
/
n for sufficiently large n, then B
n
,
q
n
(
f
) approximates f for any f
∈
C
[
0,1
]. On the other hand, if B
n
,
q
n
(
f
) can approximate f for any f
∈
C
[
0,1
] in the case q
n
>
1, then the sequence (
q
n
) satisfies lim
¯
n
→
∞
n
(
q
n
-
1
)
≤
ln
2.


---

## Body

## 1. Introduction

Letq
>
0. For any nonnegative integer k, the q
-integer  [
k
]
q is defined by
(1)
[
k
]
q
:
=
1
+
q
+
⋯
+
q
k
-
1
,
(
k
=
1,2
,
…
)
,
[
0
]
q
:
=
0
,
and the q
-factorial  [
k
]
q
! by
(2)
[
k
]
q
!
≔
[
1
]
q
[
2
]
q
⋯
[
k
]
q
,
(
k
=
1,2
,
…
)
,
[
0
]
q
!
:
=
1
.
For integers k, n with 0
≤
k
≤
n, the q
-binomial coefficient is defined by
(3)
[
n
k
]
q
:
=
[
n
]
q
!
[
k
]
q
!
[
n
-
k
]
q
!
.
In [1], Phillips proposed the q-Bernstein polynomials: for each positive integer n and f
∈
C
[
0,1
], the q-Bernstein polynomial of f is
(4)
B
n
,
q
(
f
;
x
)
:
=
∑
k
=
0
n
‍
f
(
[
k
]
q
[
n
]
q
)
p
n
k
(
q
;
x
)
,
where
(5)
p
n
k
(
q
;
x
)
=
[
n
k
]
q
x
k
∏
s
=
0
n
-
k
-
1
(
1
-
q
s
x
)
.
Note that, for q
=
1, B
n
,
q
(
f
;
x
) is the classical Bernstein polynomial B
n
(
f
;
x
):
(6)
B
n
(
f
;
x
)
:
=
∑
k
=
0
n
‍
f
(
k
n
)
(
n
k
)
x
k
(
1
-
x
)
n
-
k
.In recent years, theq-Bernstein polynomials have been investigated intensively and a great number of interesting results related to the q-Bernstein polynomials have been obtained. Reviews of the results on q-Bernstein polynomials are given in [2, Chapter 7] and [3, 4].Theq-Bernstein polynomials inherit some of the properties of the classical Bernstein polynomials, for example, theend-point interpolation property and theshape-preserving properties in the case 0
<
q
<
1, representation via divided differences. We can also define the generalized Bézier curve and de Casteljau algorithm, which can be used for evaluating q-Bernstein polynomials iteratively. These properties stipulate the importance of q-Bernstein polynomials for the computer-aided geometric design. Like the classical Bernstein polynomials, the q-Bernstein polynomials reproduce linear functions and are degree reducing on the set of polynomials. Apart from that, the basic q-Bernstein polynomials p
n
k
(
q
;
x
) admit a probabilistic interpretation via the stochastic process and the q-binomial distribution in the case 0
<
q
<
1; see [5].On the other hand, when passing fromq
=
1 to q
≠
1 convergence properties of the q-Bernstein polynomials dramatically change. More specially, in the case 0
<
q
<
1, B
n
,
q are positive linear operators on C
[
0,1
], and the convergence properties of the q-Bernstein polynomials have been investigated intensively (see, e.g., [6–11]). In the case q
>
1, B
n
,
q are not positive linear operators on C
[
0,1
], and the lack of positivity makes the investigation of convergence in the case q
>
1 essentially more difficult. There are many unexpected results concerning convergence of q-Bernstein polynomials in the case q
>
1 (see [2, 12–17]). For example, the rate of approximation by q-Bernstein polynomials (
q
>
1
) in C
[
0,1
] for functions analytic in {
z
:
|
z
|
<
q
+
ɛ
} is q
-
n versus 1
/
n for the classical Bernstein polynomials, while, for some infinitely differentiable functions on [
0,1
], their sequences of q-Bernstein polynomials (
q
>
1
) may be divergent (see [12]). In [2, 15], strong asymptotic estimates for the norm ∥
B
n
,
q
∥ as n
→
∞ for fixed q
>
1 and as q
→
∞ are obtained. It was shown in [2] that ∥
B
n
,
q
∥
→
+
∞ faster than any geometric progression n
→
∞ for fixed q
>
1. This fact provides an explanation for the unpredictable behavior of q-Bernstein polynomials (q
>
1) with respect to convergence.This paper is devoted to studying approximation properties ofq-Bernstein polynomials for q takingvarying values that tend to 1. We note that, from the very first papers (see [1]), there was interest in such approximation properties. In the case 0
<
q
n
<
1, many interesting results including the convergence, the rate of convergence, Voronvskaya-type theorems, and the direct and converse theorem are obtained (see [1, 6, 8–11]). It was shown in [1, 8] that, in the case q
n
≤
1, the condition q
n
→
1 is necessary and sufficient for the sequence (
B
n
,
q
n
(
f
)
) to be approximating for any f
∈
C
[
0,1
].Naturally, the question arises as to whether the sequence(
B
n
,
q
n
(
f
)
) to be approximating for any f
∈
C
[
0,1
] as q
n tends to 1 from above. It turns out that, in general, the answer is negative. Indeed, Ostrovska showed in [13] that if q
n
-
1
↓
0 slower than (
ln
⁡
n
)
/
n, then the sequence (
B
n
,
q
n
(
f
)
) may not be approximating for some f
∈
C
[
0,1
] (e.g., f
(
x
)
=
x). However, in [14] Ostrovska showed that if q
n
→
1
+ fast enough, the sequence (
B
n
,
q
n
(
f
)
) is approximating for any f
∈
C
[
0,1
]: a sufficient condition is q
n
=
1
+
o
(
n
-
1
3
-
n
).In this paper, we continue to study the convergence of the sequence(
B
n
,
q
n
) as q
n tends to 1 from above. Clearly, the convergence of the sequence (
B
n
,
q
n
) depends heavily on the operator norms ∥
B
n
,
q
∥. We remark that for ∥
B
n
,
q
n
∥
=
1 for all 0
<
q
n
<
1. In contrast to this, ∥
B
n
,
q
n
∥ vary with q
n
>
1. By the delicate analysis of ∥
B
n
,
q
n
∥, we obtain the sufficient and necessary condition under which (
B
n
,
q
n
(
f
;
·
)
)
(
q
n
>
1
) approximates f for any f
∈
C
[
0,1
]. Based on this condition we get that if (
B
n
,
q
n
(
f
;
·
)
) can approximate f for any f
∈
C
[
0,1
], then the sequence (
q
n
) satisfies lim
⁡
¯
n
→
∞
n
(
q
n
-
1
)
≤
ln
⁡
2. On the other hand, if 1
<
q
n
≤
1
+
ln
⁡
2
/
n for sufficient large n, then (
B
n
,
q
n
(
f
;
·
)
) approximates f for any f
∈
C
[
0,1
].


## 2. Statement of Results

From here on we assume thatq
n
>
1. The following theorem gives the sufficient and necessary condition for convergence of the sequence (
B
n
,
q
n
(
f
)
) for any f
∈
C
[
0,1
].Theorem 1.
Letq
n
>
1. Then the sequence (
B
n
,
q
n
(
f
)
) converges to f in C
[
0,1
] for any f
∈
C
[
0,1
] if and only if
(7)
sup
⁡
n
∈
ℕ
sup
⁡
x
∈
[
q
n
-
1
,
1
]
∑
k
=
2
n
‍
|
p
n
n
-
k
(
q
n
;
x
)
|
<
∞
.Based on Theorem1, we obtain the following necessary condition for convergence of the sequence (
B
n
,
q
n
(
f
)
). Indeed, we show that if lim
⁡
¯
n
→
∞
n
(
q
n
-
1
)
>
ln
⁡
2, then sup
⁡
n
∈
ℕ
|
p
n
n
-
[
ln
⁡
n
]
(
q
n
;
x
0
)
|
=
∞ with x
0
=
(
1
+
q
n
)
/
2
q
n.Theorem 2.
Letq
n
>
1. If the sequence (
B
n
,
q
n
(
f
)
) converges to f in C
[
0,1
], for any f
∈
C
[
0,1
], then
(8)
lim
⁡
n
→
∞
¯
n
(
q
n
-
1
)
≤
ln
⁡
2
.Finally, we give the sufficient condition for convergence of the sequence(
B
n
,
q
n
(
f
)
).Theorem 3.
Letq
n
>
1. If the sequence (
q
n
) satisfies q
n
≤
1
+
ln
⁡
2
/
n for sufficiently large n, then, for any f
∈
C
[
0,1
], (
B
n
,
q
n
(
f
;
x
)
) converges to f
(
x
) uniformly on [
0,1
].The following corollary follows immediately for Theorem3.Corollary 4.
Letq
n
>
1. If the sequence (
q
n
) satisfies
(9)
lim
⁡
n
→
∞
¯
n
(
q
n
-
1
)
<
ln
⁡
2
,
then, for any f
∈
C
[
0,1
], (
B
n
,
q
n
(
f
;
x
)
) converges to f
(
x
) uniformly on [
0,1
].Remark 5.
Using the same technique as in the proof of Theorem3, we can prove a slightly stronger conclusion: if
(10)
1
<
q
n
≤
1
+
ln
⁡
2
n
+
C
n
2
for some positive constant C and sufficiently large n, then, for any f
∈
C
[
0,1
], (
B
n
,
q
n
(
f
;
x
)
) converges to f
(
x
) uniformly on [
0,1
].


## 3. Proofs of Theorems1–3

Forf
∈
C
[
0,1
], we set
(11)
∥
f
∥
:
=
max
⁡
x
∈
[
0,1
]
|
f
(
x
)
|
,
∥
f
∥
s
:
=
∥
f
∥
C
[
q
n
-
s
-
1
,
q
n
-
s
]
:
=
max
⁡
x
∈
[
q
n
-
s
-
1
,
q
n
-
s
]
|
f
(
x
)
|
.
Let F
n
(
x
)
:
=
∑
k
=
0
n
|
p
n
k
(
q
n
;
x
)
|, x
∈
[
0,1
]. Clearly,
(12)
∥
B
n
,
q
n
∥
=
∥
F
n
∥
=
max
⁡
x
∈
[
0,1
]
(
∑
k
=
0
n
‍
|
p
n
n
-
k
(
q
n
;
x
)
|
)
.
Note that ∑
k
=
0
n
p
n
k
(
q
n
;
x
)
=
1 for x
∈
[
0,1
] and p
n
k
(
q
n
;
x
)
≥
0 for x
∈
[
0
,
q
n
-
n
+
1
] and k
=
0,1
,
…
,
n. This means that
(13)
F
n
(
x
)
=
1
,
x
∈
[
0
,
q
n
-
n
+
1
]
,
F
n
(
x
)
≥
1
,
x
∈
[
q
n
-
n
+
1
,
1
]
.
It follows that
(14)
∥
B
n
,
q
n
∥
=
∥
F
n
∥
=
max
⁡
0
≤
s
≤
n
-
2
∥
F
n
∥
s
.Proof of Theorem1.
From Corollary 7 in [12] we know that, for any polynomial P
(
x
), we have
(15)
B
n
,
q
n
(
P
;
x
)
⟶
P
(
x
)
uniformly in [
0,1
] as n
→
∞. It follows from the well-known Banach-Steinhaus theorem that (
B
n
,
q
n
(
f
)
)
(
q
n
>
1
) approximates f for any f
∈
C
[
0,1
] if and only if
(16)
sup
⁡
n
∈
ℕ
∥
B
n
,
q
n
∥
=
sup
⁡
n
∈
ℕ
sup
⁡
x
∈
[
0,1
]
(
∑
k
=
0
n
‍
|
p
n
n
-
k
(
q
n
;
x
)
|
)
<
+
∞
.
We set
(17)
G
s
,
n
(
x
)
=
∑
k
=
s
+
2
n
‍
|
p
n
n
-
k
(
q
n
;
x
)
|
,
s
=
0,1
,
…
,
n
-
2
.
Since p
n
n
-
k
(
q
n
;
x
)
≥
0 for x
∈
[
q
n
-
s
-
1
,
q
n
-
s
] and k
=
0,1
,
…
,
s
+
1, we get, for x
∈
[
q
n
-
s
-
1
,
q
n
-
s
],
(18)
∑
k
=
0
s
+
1
‍
|
p
n
n
-
k
(
q
n
;
x
)
|
=
∑
k
=
0
s
+
1
‍
p
n
n
-
k
(
q
n
;
x
)
=
1
-
∑
k
=
s
+
2
n
‍
p
n
n
-
k
(
q
n
;
x
)
≤
1
+
G
s
,
n
(
x
)
,
and, therefore,
(19)
∥
F
n
∥
s
≤
∥
1
+
2
G
s
,
n
∥
s
=
1
+
2
∥
G
s
,
n
∥
s
,
s
=
0,1
,
…
,
n
-
2
.
Next we will show that
(20)
∥
G
s
,
n
∥
s
≤
∥
G
s
-
1
,
n
∥
s
-
1
,
s
=
1,2
,
…
,
n
-
2
.
Note that, for x
∈
[
q
n
-
s
-
1
,
q
n
-
s
],
(21)
G
s
,
n
(
x
)
=
∑
k
=
s
+
2
n
‍
|
p
n
n
-
k
(
q
n
;
x
)
|
,
G
s
-
1
,
n
(
q
n
x
)
=
∑
k
=
s
+
1
n
‍
|
p
n
n
-
k
(
q
n
;
q
n
x
)
|
.
If we show that, for x
∈
[
q
n
-
s
-
1
,
q
n
-
s
] and k
=
s
+
1
,
…
,
n
-
1,
(22)
|
p
n
n
-
k
-
1
(
q
n
;
x
)
|
≤
|
p
n
n
-
k
(
q
n
;
q
n
x
)
|
,
then
(23)
G
s
,
n
(
x
)
≤
G
s
-
1
,
n
(
q
n
x
)
,
x
∈
[
q
n
-
s
-
1
,
q
n
-
s
]
,
and (20) follows. Indeed, for x
∈
(
q
n
-
s
-
1
,
q
n
-
s
) and k
=
s
+
1
,
…
,
n
-
1,
(24)
|
p
n
n
-
k
(
q
n
;
q
n
x
)
|
|
p
n
n
-
k
-
1
(
q
n
;
x
)
|
=
[
n
k
]
q
n
(
q
n
x
)
n
-
k
∏
j
=
0
s
-
1
(
1
-
q
n
j
+
1
x
)
∏
j
=
s
k
-
1
(
q
n
j
+
1
x
+
1
)
[
n
k
+
1
]
q
n
x
n
-
k
-
1
∏
j
=
0
s
(
1
-
q
n
j
x
)
∏
j
=
s
+
1
k
(
q
n
j
x
-
1
)
=
[
k
+
1
]
q
n
q
n
n
-
k
x
[
n
-
k
]
q
n
(
1
-
x
)
.
Hence, (22) is equivalent to the following inequality:
(25)
(
q
n
k
+
1
-
1
)
q
n
n
-
k
x
≥
(
q
n
n
-
k
-
1
)
(
1
-
x
)
,
which is also equivalent to the inequality
(26)
x
≥
q
n
n
-
k
-
1
q
n
n
+
1
-
1
.
For x
∈
(
q
n
-
s
-
1
,
q
n
-
s
) and k
=
s
+
1
,
…
,
n
-
1, we have
(27)
x
>
q
n
-
s
-
1
≥
q
n
-
s
-
1
(
q
n
n
-
q
n
s
+
1
)
q
n
n
+
1
-
1
=
q
n
n
-
s
-
1
-
1
q
n
n
+
1
-
1
≥
q
n
n
-
k
-
1
q
n
n
+
1
-
1
.
This proves (26). On the other hand, p
n
n
-
k
-
1
(
q
n
;
x
)
=
0
=
p
n
n
-
k
(
q
n
;
q
n
x
) for x
∈
{
q
n
-
s
-
1
,
q
n
-
s
}, which completes the proof of (20). From (14), (19), and (20), we get
(28)
∥
G
0
,
n
∥
0
≤
∥
F
n
∥
=
∥
B
n
,
q
n
∥
≤
1
+
2
∥
G
0
,
n
∥
0
.
This implies that (16) is equivalent to
(29)
sup
⁡
n
∈
ℕ
∥
G
0
,
n
∥
0
=
sup
⁡
n
∈
ℕ
sup
⁡
x
∈
[
q
n
-
1
,
1
]
∑
k
=
2
n
‍
|
p
n
n
-
k
(
q
n
;
x
)
|
<
∞
.
Theorem 1 is proved.Proof of Theorem2.
First we show that(30)
q
n
-
1
=
O
(
1
n
)
.
Otherwise, we may assume that
(31)
lim
⁡
n
→
∞
n
(
q
n
-
1
)
=
+
∞
,
which implies
(32)
lim
⁡
n
→
∞
q
n
n
-
1
=
lim
⁡
n
→
∞
exp
⁡
(
(
n
-
1
)
ln
⁡
q
n
)
≥
lim
⁡
n
→
∞
exp
⁡
(
(
n
-
1
)
min
⁡
{
(
q
n
-
1
)
2
,
ln
⁡
2
}
)
=
+
∞
.
We have
(33)
∥
G
0
,
n
∥
0
≥
∥
p
n
n
-
2
(
q
n
;
·
)
∥
0
≥
|
p
n
n
-
2
(
q
n
;
q
n
+
1
2
q
n
)
|
=
(
q
n
n
-
1
)
(
q
n
n
-
1
-
1
)
(
q
n
2
-
1
)
(
q
n
-
1
)
(
1
+
q
n
2
q
n
)
n
-
2
×
(
1
-
1
+
q
n
2
q
n
)
(
q
n
1
+
q
n
2
q
n
-
1
)
=
(
1
-
q
n
-
n
+
1
)
(
q
n
n
-
1
)
8
(
1
+
q
n
2
)
n
-
3
≥
(
1
-
q
n
-
n
+
1
)
(
q
n
n
-
1
)
8
⟶
+
∞
,
(
as
n
⟶
∞
)
.
This leads to a contradiction by Theorem 1. Hence, (30) holds.
Next, we show Theorem2. Assume that lim
⁡
¯
n
→
∞
n
(
q
n
-
1
)
>
ln
⁡
2. Then by (30) we may suppose that, for some A, B, ln
⁡
2
<
A
<
B
<
+
∞,
(34)
1
+
A
n
≤
q
n
≤
1
+
B
n
.
For0
<
a
<
b, we set h
(
x
)
=
(
x
a
-
1
)
/
(
x
b
-
1
), x
>
1. Direct computation gives that
(35)
h
′
(
x
)
=
b
x
a
-
1
(
x
b
-
a
-
(
(
b
-
a
)
/
b
)
x
b
-
a
/
b
)
(
x
b
-
1
)
2
.
Since the function g
(
y
)
=
x
y is convex on (
-
∞
,
+
∞
) for a fixed x
>
0, we get that
(36)
x
b
-
a
=
x
(
(
b
-
a
)
/
b
)
·
b
+
(
a
/
b
)
·
0
≤
b
-
a
b
x
b
+
a
b
.
This means that h
′
(
x
)
≤
0 and h
(
x
) is nonincreasing on (
1
,
+
∞
). Hence, for x
∈
(
1
,
ξ
0
), ξ
0
>
1, we have
(37)
h
(
ξ
0
)
≤
h
(
x
)
≤
lim
⁡
x
→
1
+
h
(
x
)
=
a
b
.
Put x
0
=
(
1
+
q
n
)
/
2
q
n
∈
(
q
n
-
1
,
1
). Then, for k
0
=
[
ln
⁡
n
], we have
(38)
∥
G
0
,
n
∥
0
≥
∥
p
n
n
-
k
0
(
q
n
;
·
)
∥
0
≥
|
p
n
n
-
k
0
(
q
n
;
x
0
)
|
=
(
q
n
n
-
1
)
⋯
(
q
n
n
-
k
0
+
1
-
1
)
(
q
n
k
0
-
1
)
⋯
(
q
n
-
1
)
x
0
n
-
k
0
×
(
1
-
x
0
)
∏
s
=
1
k
0
-
1
‍
(
q
n
s
x
0
-
1
)
≥
(
q
n
n
-
k
0
-
1
)
k
0
x
0
n
-
k
0
(
1
-
x
0
)
×
(
q
n
k
0
-
1
x
0
-
1
)
⋯
(
q
n
x
0
-
1
)
(
q
n
k
0
-
1
)
⋯
(
q
n
-
1
)
.
Using (34), the inequalities
(39)
q
n
s
+
1
x
0
-
1
q
n
s
-
1
≥
1
,
s
=
1
,
…
,
k
0
-
2
,
x
0
n
-
k
0
(
1
-
x
0
)
(
q
n
x
0
-
1
)
≥
q
n
-
n
+
k
0
-
1
(
q
n
-
1
)
2
4
≥
(
1
+
B
n
)
-
n
(
q
n
-
1
)
2
4
≥
(
q
n
-
1
)
2
exp
⁡
(
-
B
)
4
,
and the nonincreasing property of h
(
x
), we continue to obtain that
(40)
∥
G
0
,
n
∥
0
≥
(
(
1
+
A
n
)
n
-
ln
⁡
n
-
1
)
k
0
exp
⁡
(
-
B
)
4
(
q
n
-
1
)
2
(
q
n
k
0
-
1
)
(
q
n
k
0
-
1
-
1
)
≥
(
(
1
+
A
n
)
n
-
ln
⁡
n
-
1
)
k
0
×
exp
⁡
(
-
B
)
4
(
A
/
n
)
2
(
(
1
+
B
/
n
)
k
0
-
1
)
(
(
1
+
B
/
n
)
k
0
-
1
-
1
)
.
We observe that
(41)
lim
⁡
n
→
∞
(
1
+
A
n
)
n
-
ln
⁡
n
=
exp
⁡
(
lim
⁡
n
→
∞
(
n
-
ln
⁡
n
)
ln
⁡
(
1
+
A
n
)
)
=
exp
⁡
(
lim
⁡
n
→
∞
A
(
n
-
ln
⁡
n
)
n
)
=
exp
⁡
(
A
)
>
2
,
and, for s
=
k
0, k
0
-
1,
(42)
lim
⁡
n
→
∞
(
1
+
B
/
n
)
s
-
1
B
ln
⁡
n
/
n
=
lim
⁡
n
→
∞
exp
⁡
(
s
ln
⁡
(
1
+
B
/
n
)
)
-
1
B
ln
⁡
n
/
n
=
lim
⁡
n
→
∞
s
ln
⁡
(
1
+
B
/
n
)
B
ln
⁡
n
/
n
=
1
.
Thus, for some a
∈
(
1
,
e
A
-
1
) and sufficiently large n, we have
(43)
∥
G
0
,
n
∥
0
≥
a
ln
⁡
n
-
1
(
ln
⁡
n
)
2
exp
⁡
(
-
B
)
A
2
4
B
2
⟶
+
∞
.
By Theorem 1, we know that there exists a function f
∈
C
[
0,1
] such that the sequence (
B
n
,
q
n
(
f
)
) does not converge to f in C
[
0,1
]. This leads to a contradiction. Hence, lim
⁡
¯
n
→
∞
n
(
q
n
-
1
)
≤
ln
⁡
2. Theorem 2 is proved.Proof of Theorem3.
From Theorem1, we know that it is sufficient to show that if q
n
≤
1
+
ln
⁡
2
/
n for sufficiently large n, then
(44)
sup
⁡
n
∈
ℕ
∥
G
0
,
n
∥
0
<
∞
.
For x
∈
(
q
n
-
1
,
1
), we set α
=
-
log
⁡
q
n
x. Then α
∈
(
0,1
) and x
=
q
n
-
α. Since, for k
=
2
,
…
,
n
-
1,
(45)
q
n
α
(
q
n
n
-
k
-
1
)
≤
q
n
n
-
k
+
α
-
1
≤
q
n
n
-
1
≤
(
1
+
ln
⁡
2
n
)
n
-
1
≤
1
,
by (37) we get that
(46)
|
p
n
n
-
k
-
1
(
q
n
;
x
)
|
|
p
n
n
-
k
(
q
n
;
x
)
|
=
(
q
n
n
-
k
-
1
)
(
q
n
k
-
α
-
1
)
(
q
n
k
+
1
-
1
)
q
n
-
α
≤
q
n
k
-
α
-
1
q
n
k
+
1
-
1
≤
k
-
α
k
+
1
.
On the other hand, by (37) we have
(47)
|
p
n
n
-
2
(
q
n
;
x
)
|
=
[
n
2
]
q
n
x
n
-
1
(
1
x
-
1
)
(
q
n
x
-
1
)
≤
(
q
n
n
-
1
)
(
q
n
n
-
1
-
1
)
(
q
n
2
-
1
)
(
q
n
-
1
)
(
q
n
α
-
1
)
(
q
n
1
-
α
-
1
)
≤
(
q
n
α
-
1
)
(
q
n
1
-
α
-
1
)
2
(
q
n
-
1
)
2
≤
α
(
1
-
α
)
2
.
It follows from (46) and (47) that
(48)
|
p
n
n
-
k
(
q
n
;
x
)
|
≤
α
(
1
-
α
)
⋯
(
k
-
1
-
α
)
k
!
.
Hence, for x
=
q
n
-
α, α
∈
(
0,1
),
(49)
G
0
,
n
(
x
)
=
∑
k
=
2
n
‍
|
p
n
n
-
k
(
q
n
;
x
)
|
≤
∑
k
=
2
∞
‍
α
(
1
-
α
)
⋯
(
k
-
1
-
α
)
k
!
.
Obviously (49) is satisfied for x
∈
{
0,1
}. We note that, for x
∈
[
0,1
],
(50)
(
1
-
x
)
α
=
1
-
α
x
-
∑
k
=
2
∞
‍
α
(
1
-
α
)
⋯
(
k
-
1
-
α
)
k
!
x
k
.
The above formula with x
=
1 means that
(51)
∑
k
=
2
∞
‍
α
(
1
-
α
)
⋯
(
k
-
1
-
α
)
k
!
=
1
-
α
.
Thus, by (49),
(52)
∥
G
0
,
n
∥
0
≤
sup
⁡
α
∈
[
0,1
]
∑
k
=
2
∞
‍
α
(
1
-
α
)
⋯
(
k
-
1
-
α
)
k
!
=
1
.
This completes the proof of Theorem 3.



---

*Source: 259491-2014-03-12.xml*