Filomat 27:4 (2013), 721–730
DOI 10.2298/FIL1304721M
Published by Faculty of Sciences and Mathematics,
University of Ni
ˇ
s, Serbia
Available at: http://www.pmf.ni.ac.rs/filomat
Approximation Theorems for q-Bernstein-Kantorovich Operators
N. I. Mahmudov
a
, P. Sabancigil
a
a
Eastern Mediterranean University, Gazimagusa, TRNC, Mersin 10, Turkey
Abstract. In the present paper we introduce a q-analogue of the Bernstein-Kantorovich operators and
investigate their approximation properties. We study local and global approximation properties and
Voronovskaja type theorem for the q-Bernstein-Kantorovich operators in case 0 < q < 1.
1. Introduction
In the last two decades interesting generalizations of Bernstein polynomials were proposed by Lupas¸
[15] and by Phillips [20]. Generalizations of the Bernstein polynomials based on the q-integers attracted
a lot of interest and was studied widely by a number of authors. A survey of the obtained results and
references on the subject can be found in [19]. Recently some new generalizations of well known positive
linear operators, based on q-integers were introduced and studied by several authors, see [23], [5], [6], [8],
[21], [22], [16].
The classical Kantorovich operator B
∗
n
, n = 1, 2, ... is defined by (cf. [14])
B
∗
n
f ; x
:=
(
n + 1
)
n
k=0
n
k
x
k
(
1 − x
)
n−k
k+1/n+1
k/n+1
f
(
t
)
dt
=
n
k=0
n
k
x
k
(
1 − x
)
n−k
1
0
f
k + t
n + 1
dt, f : [0, 1] → R. (1)
These operators have been extensively considered in the mathematical literature. Also, a number of
generalizations have been introduced by different authors (see, for instance [24], [25], [26]).
In this paper, inspired by (1), we introduce a q-type generalization of Bernstein-Kantorovich polynomial
operators as follows.
B
∗
n,q
f, x
:=
n
k=0
p
n,k
q; x
1
0
f
[k] + q
k
t
[
n + 1]
d
q
t,
where f ∈ C [0, 1] , 0 < q < 1.
2010 Mathematics Subject Classification. Primary 41A46 ; Secondary 33D99, 41A25)
Keywords. q-integers, positive operator, q-Bernstein-Kantorovich operator
Received: 26 November 2011; Accepted: 8 December 2012
Communicated by Gradimir Milovanovic
Email addresses: nazim.mahmudov@emu.edu.tr (N. I. Mahmudov), pembe.sabancigil@emu.edu.tr (P. Sabancigil)
N. I. Mahmudov, P .Sabancigil / Filomat 27:4 (2013), 721–730 722
The paper is organized as follows. In Section 2, we give standard notations that will be used throughout
the paper, introduce q-Bernstein-Kantorovich operators and evaluate the moments of B
∗
n,q
. In Section 3
we study local and global convergence properties of the q-Bernstein-Kantorovich operators and prove
Voronovskaja-type asymptotic formula. In the final section we give statistical approximation result for the
q-Bernstein-Kantorovich operators.
2. q-Bernstein-Kantorovich operators
Let q > 0. For any n ∈ N ∪
{
0
}
, the q-integer [n] = [n]
q
is defined by
[n] := 1 + q + ... + q
n−1
, [0] := 0;
and the q-factorial [n]! = [n]
q
! by
[n]! := [1] [2] ... [n] , [0]! := 1.
For integers 0 ≤ k ≤ n, the q-binomial coefficient is defined by
n
k
:=
[
n
]!
[k]! [n −k]!
.
The q-analogue of integration in the interval [0, A] (see [13]) is defined by
A
0
f
(
t
)
d
q
t := A
1 − q
∞
n=0
f
Aq
n
q
n
, 0 < q < 1.
Let 0 < q < 1. Based on the q-integration we propose the Kantorovich type q-Bernstein polynomial as
follows.
B
∗
n,q
f, x
=
n
k=0
p
n,k
q; x
1
0
f
[
k
]
+ q
k
t
[
n + 1]
d
q
t, 0 ≤ x ≤ 1, n ∈ N
where
p
n,k
q; x
:=
n
k
x
k
(
1 − x
)
n−k
q
,
(
1 − x
)
n
q
:=
n−1
s=0
1 − q
s
x
.
It can be seen that for q → 1
−
the q-Bernstein-Kantorovich operator becomes the classical Bernstein-
Kantorovich operator.
Lemma 2.1. For all n ∈ N, x ∈ [0, 1] and 0 < q ≤ 1 we have
B
∗
n,q
(
t
m
, x
)
=
m
j=0
m
j
[
n
]
j
[n + 1]
m
m − j + 1
m−j
i=0
m − j
i
q
n
− 1
i
B
n,q
t
j+i
, x
. (2)
N. I. Mahmudov, P .Sabancigil / Filomat 27:4 (2013), 721–730 723
Proof. The recurrence formula can be derived by direct computation.
B
∗
n,q
(
t
m
, x
)
=
n
k=0
p
n,k
q; x
m
j=0
1
0
m
j
[
k
]
j
q
k
(
m−j
)
t
m−j
[n + 1]
m
d
q
t
=
n
k=0
p
n,k
q; x
m
j=0
m
j
q
k
(
m−j
)
[
k
]
j
[n + 1]
m
m − j + 1
=
m
j=0
m
j
[n]
j
[
n + 1]
m
m − j + 1
n
k=0
q
k
− 1 + 1
m−j
[k]
j
[
n
]
j
p
n,k
q; x
=
m
j=0
m
j
[
n
]
j
[n + 1]
m
m − j + 1
n
k=0
m−j
i=0
m − j
i
q
k
− 1
i
[
k
]
j
[n]
j
p
n,k
q; x
=
m
j=0
m
j
[n]
j
[n + 1]
m
m − j + 1
m−j
i=0
m − j
i
q
n
− 1
i
n
k=0
[k]
j+i
[n]
j+i
p
n,k
q; x
=
m
j=0
m
j
[n]
j
[
n + 1]
m
m − j + 1
m−j
i=0
m − j
i
q
n
− 1
i
B
n,q
t
j+i
, x
.
Lemma 2.2. For all n ∈ N, x ∈ [0, 1] and 0 < q ≤ 1 we have
B
∗
n,q
(
1, x
)
= 1, B
∗
n,q
(
t, x
)
=
2q
[2]
[
n
]
[n + 1]
x +
1
[2]
1
[n + 1]
,
B
∗
n,q
t
2
, x
=
q
q + 2
[3]
q [n] [n −1]
[
n + 1]
2
x
2
+
4q + 7q
2
+ q
3
[2] [3]
[n]
[
n + 1]
2
x +
1
[3]
1
[
n + 1]
2
.
Proof. Taking into account (2), by direct computation, we obtain explicit formulas for B
∗
n,q
(
t, x
)
and
B
∗
n,q
t
2
, x
as follows.
B
∗
n,q
(
t, x
)
=
1
[
n + 1] [2]
B
n,q
(
1, x
)
+
q
n
− 1
B
n,q
(
t, x
)
+
[
n
]
[
n + 1]
B
n,q
(
t, x
)
=
q
n
− 1
[2] [
n + 1]
+
[
n
]
[
n + 1]
x +
1
[2] [
n + 1]
=
2q
[2]
[
n
]
[
n + 1]
x +
1
[2] [
n + 1]
and
B
∗
n,q
t
2
, x
=
1
[3] [
n + 1]
2
B
n,q
(
1, x
)
+ 2
q
n
− 1
B
n,q
(
t, x
)
+
q
n
− 1
2
B
n,q
t
2
, x
+
2 [n]
[2] [n + 1]
2
B
n,q
(
t, x
)
+
q
n
− 1
B
n,q
t
2
, x
+
[n]
2
[n + 1]
2
B
n,q
t
2
, x
=
1
[3] [n + 1]
2
+
[
n
]
2
[n + 1]
2
+
2 [n]
q
n
− 1
[2] [n + 1]
2
+
q
n
− 1
2
[3] [n + 1]
2
1 −
1
[
n
]
x
2
+
[n]
2
[
n
] [
n + 1]
2
+
2 [n]
q
n
− 1
[2] [
n
] [
n + 1]
2
+
q
n
− 1
2
[3] [
n
] [
n + 1]
2
+
2 [n]
[2] [
n + 1]
2
+
2
q
n
− 1
[3] [
n + 1]
2
x
=
2q + 3q
2
+ q
3
[2] [3]
q
[
n
] [
n − 1]
[n + 1]
2
x
2
+
4q + 7q
2
+ q
3
[2] [3]
[
n
]
[n + 1]
2
x +
1
[3] [n + 1]
2
.
N. I. Mahmudov, P .Sabancigil / Filomat 27:4 (2013), 721–730 724
Remark 2.3. It is observed from the above lemma that for q = 1, we get the moments of the Bernstein-Kantorovich
operators.
Lemma 2.4. For all n ∈ N, x ∈ [0, 1] and 0 < q ≤ 1 we have
B
∗
n,q
(
t − x
)
2
, x
≤
4
[n]
x
(
1 − x
)
+
1
[n]
, B
∗
n,q
(
t − x
)
4
, x
≤
C
[n]
2
x
(
1 − x
)
+
1
[n]
2
,
where C is a positive absolute constant.
Proof. Note that estimation of the moments for the q-Bernstein operators is given in [17]. The proof is
based on the estimations of the second and fourth order central moments of the q-Bersntein polynomials.
B
n,q
(
t − x
)
2
, x
=
1
[n]
x
(
1 − x
)
, B
n,q
(
t − x
)
4
, x
≤
C
[
n
]
2
x
(
1 − x
)
.
Indeed
B
∗
n,q
(
t − x
)
2
, x
=
n
k=0
p
n,k
q; x
1
0
[k] + q
k
t
[n + 1]
− x
2
d
q
t =
n
k=0
p
n,k
q; x
1
0
q
k
t
[n + 1]
−
q
n
[k]
[n] [n + 1]
+
[
k
]
[n]
− x
2
d
q
t
≤ 2
n
k=0
p
n,k
q; x
1
0
q
k
t
[
n + 1]
−
q
n
[k]
[
n
] [
n + 1]
2
d
q
t + 2
n
k=0
p
n,k
q; x
1
0
[
k
]
[
n
]
− x
2
d
q
t
≤
4
[3] [n + 1]
2
+
4
[n + 1]
2
+
2
[n]
x
(
1 − x
)
≤
4
[n]
x
(
1 − x
)
+
1
[n]
.
A similar calculus reveals:
B
∗
n,q
(
t − x
)
4
, x
=
n
k=0
p
n,k
q; x
1
0
[
k
]
+ q
k
t
[
n + 1]
− x
4
d
q
t =
n
k=0
p
n,k
q; x
1
0
q
k
t
[
n + 1]
−
q
n
[
k
]
[
n
] [
n + 1]
+
[k]
[
n
]
− x
4
d
q
t
≤ 4
n
k=0
p
n,k
q; x
1
0
q
k
t
[n + 1]
−
q
n
[
k
]
[n] [n + 1]
4
d
q
t + 4
n
k=0
p
n,k
q; x
1
0
[k]
[n]
− x
4
d
q
t
≤
32
[5] [n + 1]
4
+
32
[n + 1]
4
+
4
[n]
2
Cx
(
1 − x
)
≤
C
[n]
2
x
(
1 − x
)
+
1
[n]
2
.
Lemma 2.5. Assume that 0 < q
n
< 1, q
n
→ 1 and q
n
n
→ a as n → ∞. Then we have
lim
n→∞
[n]
q
n
B
∗
n,q
n
(
t − x; x
)
= −
1 + a
2
x +
1
2
,
lim
n→∞
[n]
q
n
B
∗
n,q
n
(
t − x
)
2
; x
= −
1
3
x
2
−
2
3
ax
2
+ x.
Proof. To prove the lemma we use formulas for B
∗
n,q
n
(
t; x
)
and B
∗
n,q
n
t
2
; x
given in Lemma 2.2.
lim
n→∞
[n]
q
n
B
∗
n,q
n
(
t − x; x
)
= lim
n→∞
[n]
q
n
2q
n
[2]
q
n
[
n
]
q
n
[
n + 1]
q
n
− 1
x +
1
[2]
q
n
[
n
]
q
n
[
n + 1]
q
n
= lim
n→∞
−
[n]
q
n
[
n + 1]
q
n
1 + q
n+1
n
[2]
q
n
x +
1
[2]
q
n
[n]
q
n
[
n + 1]
q
n
= −
1 + a
2
x +
1
2
.
N. I. Mahmudov, P .Sabancigil / Filomat 27:4 (2013), 721–730 725
lim
n→∞
[n]
q
n
B
∗
n,q
(
t − x
)
2
, x
= lim
n→∞
[n]
q
n
B
∗
n,q
t
2
, x
− x
2
− 2xB
∗
n,q
(
t − x, x
)
= lim
n→∞
[n]
q
n
q
n
q
n
+ 2
[3]
q
n
[
n
]
2
q
n
−
[
n
]
q
n
[n + 1]
2
q
n
− 1
x
2
+ lim
n→∞
[n]
q
n
4q
n
+ 7q
2
n
+ q
3
n
[2]
q
n
[3]
q
n
[
n
]
q
n
[n + 1]
2
q
n
x
− lim
n→∞
[n]
q
n
2xB
∗
n,q
n
(
t − x, x
)
= lim
n→∞
q
n
1 − q
n
n
2q
n
+ q
2
n
+ 2
x
2
− lim
n→∞
4q
n
+ 3q
2
n
+ 2q
3
n
x
2
+ lim
n→∞
4q
n
+ 7q
2
n
+ q
3
n
[2]
q
n
[3]
q
n
x
− lim
n→∞
[
n
]
q
n
2xB
∗
n,q
n
(
t − x, x
)
=
5
3
(
1 − a
)
x
2
− 3x
2
+ 2x +
(
1 + a
)
x
2
− x
= −
1
3
x
2
−
2
3
ax
2
+ x.
3. Local and global approximation
We begin by considering the following K-functional:
K
2
f, δ
2
:= inf
f − 1
+ δ
2
1
′′
: 1 ∈ C
2
[0
, 1]
, δ ≥ 0,
where
C
2
[0
, 1] :=
1 : 1, 1
′
, 1
′′
∈ C
[0
, 1]
.
Then, in view of a known result [7], there exists an absolute constant C
0
> 0 such that
K
2
f, δ
2
≤ C
0
ω
2
f, δ
(3)
where
ω
2
f, δ
:= sup
0<h≤δ
sup
x±h∈[0,1]
f
(
x − h
)
− 2 f
(
x
)
+ f
(
x + h
)
is the second modulus of smoothness of f ∈ C [0, 1].
Our first main result is stated below.
Theorem 3.1. There exists an absolute constant C > 0 such that
B
∗
n,q
f ; x
− f
(
x
)
≤ Cω
2
f,
δ
n
(
x
)
[n]
+ ω
f,
1 + q
n+1
x − 1
[2] [n + 1]
,
where f ∈ C [0, 1], δ
n
(
x
)
= φ
2
(
x
)
+
1
[n]
, 0 ≤ x ≤ 1 and 0 < q < 1.
Proof. Let
B
∗
n,q
f ; x
= B
∗
n,q
f ; x
+ f
(
x
)
− f
(
a
n
x + b
n
)
,
where f ∈ C [0, 1], a
n
=
2q
1+q
[
n
]
[
n+1]
and b
n
=
1
1+q
1
[
n+1]
. Using the Taylor formula
1
(
t
)
= 1
(
x
)
+ 1
′
(
x
) (
t − x
)
+
t
x
(
t − s
)
1
′′
(
s
)
ds, 1 ∈ C
2
[0
, 1] ,