1. Introduction
Let
q
>
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, the
q
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].
The
q
Bernstein polynomials inherit some of the properties of the classical Bernstein polynomials, for example, the endpoint interpolation property and the shapepreserving 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 computeraided 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 from
q
=
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 of
q
Bernstein polynomials for
q
taking varying 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, Voronvskayatype 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 that
q
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.
Let
q
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 Theorem 1, 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.
Let
q
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.
Let
q
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 Theorem 3.
Corollary 4.
Let
q
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 Theorem 3, 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 Theorems <xref reftype="statement" rid="thm1">1</xref>–<xref reftype="statement" rid="thm3">3</xref>
For
f
∈
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 Theorem <xref reftype="statement" rid="thm1">1</xref>.
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 wellknown BanachSteinhaus 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 Theorem <xref reftype="statement" rid="thm2">2</xref>.
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 Theorem 2. 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
.
For
0
<
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 Theorem <xref reftype="statement" rid="thm3">3</xref>.
From Theorem 1, 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.