Hacettepe Journal of Mathematics and Statistics
Volume 42 (2) (2013), 139 – 148
APPROXIMATION PROPERTIES OF STANCU
TYPE MEYER- KÖNIG AND
ZELLER OPERATORS
Mediha Örkcü
∗
Received 02 : 03 : 2011 : Accepted 26 : 04 : 2012
Abstract
In this paper, we introduce a Stancu type modification of the q- MeyerKönig and Zeller operators and investigate the Korovkin type statistical
approximation properties of this modification via A−statistical convergence. We also compute rate of convergence of the defined operators
by means of modulus of continuity. Furthermore, we give an rth order generalization of our operators and obtain approximation results of
them.
Keywords: q− Meyer-König and Zeller operators, Stancu type operators, rth order
generalization, statistical convergence, modulus of continuity
2000 AMS Classification: 41A25, 41A36
1. Introduction
The classical Meyer-König and Zeller (MKZ) operators defined on C [0, 1] were introduced in 1960 (see [17]). In order to give the monotonicity properties, Cheney and
Sharma [5] modified these operators as follows:

∞
k
 (1 − x)n+1 P
n+k
k
x , x ∈ [0, 1)
f n+k
k
Mn (f ; x) =
k=0

f (1) ,
x = 1.
During the last decade, q−Calculus was intensively used for the construction of generalizations of many classical approximation processes of positive type. The first researches
have been achieved by Lupaş [15] and Phillips [20]. Phillips introduced the q−type generalization of the classical Bernstein operator and obtained the rate of convergence and
the Voronoskaja type asymptotic formula for these operators. Later, Trif [22] defined the
∗Department of Mathematics, Gazi University, Faculty of Sciences, Teknik-okullar, 06500
Ankara, Turkey
Email: (M. Örkcü) medihaakcay@gazi.edu.tr
140
M. Örkcü
MKZ operators based on the q−integers. Also, in order to give some explicit formulae
for second moment of the MKZ operators based on the q−integers, Doğru and Duman
[6] have presented the following q−MKZ operators for q ∈ (0, 1]:
 n
qn [k] ∞
k
 Q (1 − q s x) P
n+k
f [n+k]q
x , x ∈ [0, 1)
k
q
q
Mn (f ; x) =
k=0
 s=0
f (1) ,
x = 1.
Now we recall some definitions about q−Calculus. Let q > 0. For any n ∈ N0 = N∪{0},
the q−integer of the number n and the q−factorial are respectively defined by
[n]q =
1−q n
1−q
n
if
if
q=
6 1
, [n]q ! =
q=1
The q−binomial coefficients are defined as
n
k q
[1]q [2]q ...[n]q
1
if
if
n = 1, 2, ...
n = 0.
[n]q !
, k = 0, 1, ..., n. It is obvious
[k]q ![n−k]q !
n
and k 1 = nk , the ordinary binomial
=
that for q = 1 one has [n]1 = n, [n]1 ! = n!
coefficients. Details of q−Calculus can be found in [4].
Recently, Nowak [18] introduced a q−type generalization of Stancu’s operators [21].
Beside, an other two q−analogues of Stancu operators earlier introduced by Lupaş [16].
Agratini [1] presented approximation properties of the mentioned class of operators. In
[23], a Stancu type generalization of Baskakov-Durrmeyer operators was constructed
and some approximation properties were obtained by Verma, Gupta and Agrawal. In
this paper, we introduce a Stancu type modification of the q−MKZ operators and
investigate the Korovkin type statistical approximation properties of this modification
via A−statistical convergence.
Now we recall the concepts of regularity of a summability matrix and A−statistical
convergence. Let A := (ank ) be an infinite summability matrix. For a given sequence x :=
∞
P
(xk ), the A−transform of x, denoted by Ax := ((Ax)n ), is defined as (Ax)n :=
ank xk
k=1
provided the series converges for each n. A is said to be regular if lim(Ax)n = L whenever
n
lim x = L [11]. Suppose that A is non-negative regular summability
matrix. Then x is
P
A−statistically convergent to L if for every ε > 0, lim
ank = 0 and we write
n k:|x −L|≥ε
k
stA − lim x = L [9, 12]. Actually, x is A−statistically convergent to L if and only if, for
every ε > 0, δA (k ∈ N : |xk − L| ≥ ε) = 0, where δA (K) denotes
P the A− density of subset
K of the natural numbers and is given by δA (K) := limn ∞
k=1 ank χK (k) provided the
limit exists, where χK is the charateristic function of K. If A = C1 , the Cesáro matrix
of order one, then A−statistically convergence reduces to the statistical convergence [8].
Also, taking A = I, the identity matrix, A−statistical convergence coincides with the
ordinary convergence.
The rest of this paper is organized as follows. In Section 2, we present a Stancu type
generalization of q−MKZ operators. Furthermore, we obtain statistical Korovkin-type
approximation result of the defined operators and compute their rate of convergence by
means of modulus of continuity. In Section 3, we give an rth order generalization of our
operators and get approximation results of them.
2. Construction of the Operators
In this section, we present a Stancu type generalization of q−MKZ operators and
obtain statistical Korovkin-type approximation result.
Approximation properties of Stancu type Meyer-König and Zeller operators
For f ∈ C [0, 1] , n ∈ N and q ∈ (0, 1]
 ∞
 P mq,α (x) f qn [k]q ,
q,α
n,k
[n+k]q
Mn (f ; x) =
 k=0
f (1),
141
x ∈ [0, 1)
x = 1,
where
"
(2.1)
mq,α
n,k (x) =
n+k
k
n k−1 # Q x + α [i] Q 1 − q s x + α [s]
q
q
i=0
q
s=0
n+k
Q .
1 + α [i]q
i=0
As usual, we use the test functions ei (x) = xi for i = 0, 1, 2.
2.1. Lemma. For all x ∈ [0, 1) and n ∈ N, we have
(2.2)
Mnq,α (e0 ; x) = 1,
(2.3)
Mnq,α (e1 ; x) = q n x,
(2.4)
q 2n
q 2n+1
q 2n
x (x + α) ≤ Mnq,α (e2 ; x) ≤
x (x + α) +
x.
1+α
1+α
[n]q
Proof. Item (2.2) easily follows by
∞
X
mq,α
n,k (x) = 1.
k=0
A direct computation yields
Mnq,α (e1 ; x) = q n
∞
X
[n + k − 1]q !
k=1
[k − 1]q ! [n]q !
k−1
Q i=0
x + α [i]q
Q
n 1 − q s x + α [s]q
s=0
n+k
Q 1 + α [i]q
i=0
k Q
=q
x
x + α [i]q
∞
X
[n + k]q ! i=1
n
k=0
Q
n 1 − q s x + α [s]q
s=0
n+k+1
Q [k]q ! [n]q !
i=1
1 + α [i]q
k−1
Q Q
n x
x + α [i + 1]q
1 − q s x + α [s]q
∞
X
[n
+
k]
!
q
s=0
i=0
= qn
n+k
Q [k]q ! [n]q !
k=0
1 + α [i + 1]q
i=0
= q n x,
which guarantees (2.3). Now we will prove (2.4). We immediately see
Mnq,α (e2 ; x) = q
∞
X
[n + k − 1]q !
2n
k=1
[k − 1]q ! [n]q !
k−1
Q i=0
x + α [i]q
Q
n 1 − q s x + α [s]q
s=0
n+k
Q i=0
1 + α [i]q
[k]q
[n + k]q
.
142
M. Örkcü
Since [k]q = q [k − 1]q + 1 for k ≥ 1, we get
Q
Qk−1 n
1 − q s x + α [s]q
x + α [i]q
s=0
i=0
Mnq,α (e2 ; x) = q
Qn+k [k − 2]q ! [n]q !
1 + α [i]q
k=2
i=0
Q
Qk−1 n
s
∞
[n + k − 1]q ! 2n X
[n + k − 1]q ! i=0 x + α [i]q
s=0 1 − q x + α [s]q
+q
Qn+k [n + k]q
[k − 1]q ! [n]q !
1
+
α
[i]
k=1
i=0
q [n + k]q
∞
X
[n + k − 2]q !
2n+1
and hence
(2.5)
Mnq,α (e2 ; x) =
q
Q
n
s
x + α [i]q
s=0 1 − q x + α [s]q [n + k + 1]q
Qn+k+2 [k]q ! [n]q !
[n + k + 2]q
1 + α [i]q
i=0
Qk
Qn
s
∞
x
+
α
[i]
1
−
q
x
+
α
[s]
X
[n
+
k]
!
i=0
q
s=0
q
q
+ q 2n
Qn+k+1 [k]q ! [n]q !
1 + α [i]q [n + k + 1]q
k=0
i=0
∞
X
[n + k]q !
2n+1
k=0
Qk+1 i=0
[n+k+2]q −1
q
Since q ∈ (0, 1) , x ∈ [0, 1) and by using [n + k + 1]q =
[n + k + 2]q , we obtain
Mnq,α (e2 ; x) = q 2n
∞
X
[n + k]q !
k=0
[k]q ! [n]q !
k+1
Q x + α [i]q
Q
n 1 − q s x + α [s]q
s=0
n+k+2
Q i=0
1 + α [i]q
i=0
− q 2n
∞
X
[n + k]q !
k=0
[k]q ! [n]q !
k+1
Q i=0
x + α [i]q
+q
k=0
Q
n 1 − q s x + α [s]q
s=0
n+k+2
Q i=0
∞
X
[n + k]q !
2n
and [n + k + 1]q <
k Q
i=0
[k]q ! [n]q !
1 + α [i]q [n + k + 2]q
x + α [i]q
n+k+1
Q i=0
Q
n s=0
1 − q s x + α [s]q
1 + α [i]q [n + k + 1]q
q 2n
≥
x (x + α)
1+α
2n
−
q
x (x + α)
1+α
∞
X
[n + k]q !
k=0
k+1
Q i=2
[k]q ! [n]q !
x + α [i]q
n+k+2
Q i=2
+
q 2n
x (x + α)
1+α
∞
X
[n + k]q !
k=0
[k]q ! [n]q !
k Q
i=1
i=1
(2.6)
q 2n
=
x (x + α) .
1+α
s=0
1 − q s x + α [s]q
1 + α [i]q [n + k + 2]q
x + α [i]q
n+k+1
Q Q
n Q
n 1 − q s x + α [s]q
s=0
1 + α [i]q [n + k + 2]q
Approximation properties of Stancu type Meyer-König and Zeller operators
143
On the other hand, using the inequalities [n + k + 1]q < [n + k + 2]q , [n + k + 1]q > [n]q
for all k = 0, 1, 2, ..., n ∈ N and from (2.5) it follows
Q
k+1
n Q 1 − q s x + α [s]q
x + α [i]q
∞
2n+1
X
[n + k]q ! i=2
q
s=0
x (x + α)
Mnq,α (e2 ; x) ≤
+
n+k+2
Q 1+α
[k]q ! [n]q !
k=0
1 + α [i]q
i=2
k
Q
+
∞
q 2n X [n + k]q ! i=1
x
[n]q
[k]q ! [n]q !
Q
n x + α [i]q
1 − q s x + α [s]q
s=0
n+k+1
Q k=0
i=1
1 + α [i]q
,
which guarantees
Mnq,α (e2 ; x) ≤
(2.7)
q 2n+1
q 2n
x (x + α) +
x.
1+α
[n]q
Then, by combining (2.6) and (2.7), the proof is completed.
The well-known Korovkin theorem (see [3, 13]) was improved via the concept of statistical convergence by Gadjiev and Orhan in [10]. This theorem can be stated as the
following:
2.2. Theorem. If the sequence of positive linear operators Ln : C [a, b] → B [a, b] satisfies
the conditions
st − lim kLn ej − ej k = 0,
n
where j ∈ {0, 1, 2} , then, for any function f ∈ C [a, b] , we have
st − lim kLn f − f k = 0.
n
In other words, the sequence of functions (Ln f )n≥1 is statistically uniform convergent
to f on C [a, b] . Here B [a, b] stands for the space of all real valued bounded functions
defined on [a, b] , endowed with the sup-norm.
Theorem 2.2 is true for A−statistical convergence, where A is non-negative regular
summability matrix [7].
Now, we replace q and α in the definition of Mnq,α , by sequences (qn ) , 0 < qn ≤ 1,
and (αn ) , αn ≥ 0, respectively, so that
stA − lim qnn = 1, stA − lim
(2.8)
n
n
1
= 0 and stA − lim αn = 0.
n
[n]qn
For example, take A = C1 , the Cesáro matrix of order one, and define (qn ) and (αn )
sequences by
1
,
if n = m2 (m = 1, 2, 3...)
2
qn =
−n
1 − e n , if n 6= m2 .
n
e ,
if n = m2 (m = 1, 2, 3...)
αn =
.
0, if n 6= m2 .
Since the C1 −density (or natural density) of the set of all squares is zero, stA −lim qnn = 1,
n
stA − lim
n
1
[n]qn
= 0 and stA − lim αn = 0. It is observed that (qn ) and (αn ) satisfy the
n
equalities in (2.8) but they do not converge in the ordinary case.
144
M. Örkcü
2.3. Theorem. Let A = (ank ) be a non-negative regular summability matrix and let
(qn ) and (αn ) be two sequences satisfying (2.8). Then, for f ∈ C [0, 1] , the sequence
{Mnqn ,αn (f ; .)} is A−statistically uniform convergent to f on the interval [0, 1] .
Proof. By (2.2) and the definition of the operators Mnqn ,αn in the case of x = 1, it is
clear that
stA − lim kMnqn ,αn e0 − e0 k = 0.
n
Taking into account the case x = 1 and from (2.3),
(2.9)
kMnqn ,αn e1 − e1 k ≤ 1 − qnn .
For a given ε > 0, we define the following sets;
U := {n : kMnqn ,αn e1 − e1 k ≥ ε} , U 0 := {n : 1 − qnn ≥ ε} .
Then by (2.9), we can see U ⊆ U 0 . So, for all n ∈ N,
X
X
0≤
ank ≤
ank .
k∈U
k∈U 0
Letting n → ∞ and using (2.8), we conclude that lim
P
n k∈U
ank = 0, which gives
stA − lim kMnqn ,αn e1 − e1 k = 0.
n
Finally, by (2.4) and the definition of the operators Mnqn ,αn in the case of x = 1, we get
2n+1
q 2n
q
qn ,α
2
|Mn
(e2 ; x) − e2 | ≤ x (x + α) +
x−x 1 + α
[n]q
!
q 2n+1
αq 2n+1
q 2n
≤ 1−
x2 +
+
x
1+α
1+α
[n]q
≤1+
1 − α 2n+1
q 2n
−
q
[n]q
1+α
So, we can write
(2.10)
kMnqn ,αn e2 − e2 k ≤ 1 −
1 − αn 2n+1
q 2n
qn
+ n .
1 + αn
[n]qn
Since, for all n ∈ N, 0 < qnn ≤ qn ≤ 1, one can get stA − lim qn = 1. So, by (2.8) observe
n
that
q 2n
1 − αn 2n+1
stA − lim 1 −
qn
= stA − lim n = 0.
n [n]
n
1 + αn
qn
We define the following sets;
D := {n : kMnqn ,αn e2 − e2 k ≥ ε} ,
(
)
1 − αn 2n+1
qn2n
qn
≥ ε and D2 := n :
≥ε .
D1 := n : 1 −
1 + αn
[n]qn
Then, by (2.10), we get D ⊆ D1 ∪ D2 . Hence, for all n ∈ N
X
X
X
ank ≤
ank +
ank .
k∈D
k∈D1
k∈D2
Taking limit as j → ∞, we get
stA − lim kMnqn ,αn e2 − e2 k = 0.
n
Approximation properties of Stancu type Meyer-König and Zeller operators
145
So, from Theorem 2.2, the proof is completed.
Now we will give a theorem on a degree of approximation of continuous function f
by the sequence of Mnq,α (f ; x). For this aim, we will use the modulus of continuity of
function f ∈ C [0, 1] defined by
ω (f ; δ) = sup {|f (t) − f (x)| : t, x ∈ [0, 1] , |t − x| ≤ δ}
for any positive number δ.
2.4. Theorem. Let x ∈ [0, 1] and f be a continuous function defined on [0, 1] . Then,
for n ∈ N, the following inequalities holds:
|Mnq,α (f ; x) − f (x)| ≤ 2ω (f, δnq,α (x))
where
(2.11)
δnq,α
(
(x) =
q 2n+1
1 − 2q +
1+α
n
αq 2n+1
q 2n
+
1+α
[n]q
2
x +
! )1
2
x
.
Proof. Since the case of x = 1 is obvious, assume that x ∈ [0, 1) and f ∈ C [0, 1] . By the
known properties of modulus of continuity, we can write for any δ > 0, that
|t − x|
.
|f (t) − f (x)| ≤ ω (f, δ) 1 +
δ
Therefore, by the linearity and monotonicity of the operators Mnq,α , we obtain for any
δ > 0,
|Mnq,α (f ; x) − f (x)| ≤ Mnq,α (|f (t) − f (x)| ; x)
1
≤ ω (f, δ) 1 + Mnq,α (|t − x| ; x) .
δ
By the Cauchy-Schwarz inequality we have
q
1
(2.12) |Mnq,α (f ; x) − f (x)| ≤ ω (f, δ) 1 +
Mnq,α (t − x)2 ; x .
δ
Then, we can write from Lemma 2.1
q 2n
q 2n+1
x
x (x + α) + (1 − 2q n ) x2 +
Mnq,α (t − x)2 ; x ≤
1+α
[n]q
!
q 2n+1
αq 2n+1
q 2n
n
2
= 1 − 2q +
x +
+
x.
1+α
1+α
[n]q
In the inequality (2.12), taking
(
δ=
δnq,α
(x) =
the proof is completed.
q 2n+1
1 − 2q +
1+α
n
2
x +
αq 2n+1
q 2n
+
1+α
[n]q
! )1
2
x
,
2.5. Remark. Let (qn ) and (αn ) be two sequences satisfying (2.8). If we take q = qn
and α = αn in Theorem 2.4, then, we obtain the rate of A−statistical convergence of our
operators to the function f being approximated.
146
M. Örkcü
3. An rth order generalization of Mnq,α
Kirov and Popova [14] proposed a generalization of the rth order of positive linear
operators such that it keeps the linearity property but loose the positivity. Also, this
generalization is sensitive to the degree of smoothness of the function f as approximations
to f. In [19], using the similar method introduced by Kirov and Popova, Özarslan and
Duman consider a generalization of the MKZ-type operators on the q−integers. Recently,
Agratini [2] introduced a generalization of the rth order of Stancu type operators. For
every integer n ≥ 1, Ln : C [a, b] → C [a, b] be the operators defined by
X
(Ln f )(x) =
pn,k (x) f (xn,k ) , x ∈ [a, b]
k∈Jn
where Jn ⊂ N0 := {0} ∪ N and a net on the compact [a, b] namely (xn,k )k∈Jn and
P
pn,k (x) = 1. For r ∈ N0 by f ∈ C r [a, b] we mean the space of all functions f
k∈Jn
for which their rth derivative are continuous on the interval [a, b] and (Tr f ) (xn,k ; .) be
Taylor’s polynomial of r degree associated to the function f on the point xn,k , k ∈ Jn .
Agratini [2] considered the linear operators Ln,r : C r [a, b] → C [a, b],
X
(Ln,r f ) (x) =
(Tr f ) (xn,k ; .) pn,k (x)
k∈Jn
=
r
X X
f (i) (xn,k )
(x − xn,k )i pn,k (x) , x ∈ [a, b] .
i!
i=0
k∈Jn
Now, we introduce a generalization of the positive linear operators Mnq,α , by using the
Agratini’s method.
(3.1)
q,α
Mn,r
(f ; x) =
∞
X
(Tr f ) (xn,k ; .) mq,α
n,k (x)
k=0
=
∞ X
r
X
f (i) (xn,k )
(x − xn,k )i mq,α
n,k (x) ,
i!
i=0
k=0
where
mq,α
n,k
(x) is given by (2.1), xn,k =
q n [k]q
[n+k]q
, f ∈ C r [0, 1] and x ∈ [0, 1) . If x = 1 we
q,α
q,α
define that Mn,r
(f ; 1) = f (1) as stated before. Clearly, Mn,0
= Mnq,α , n ∈ N for every
f ∈ C [0, 1] , x ∈ [0, 1], q ∈ (0, 1] and n ∈ N.
q,α
We have the following approximation theorem for the operators Mn,r
.
3.1. Theorem. Let A = (ank ) be a non-negative regular summability matrix and let
(qn ) and (αn ) be two sequences satisfying (2.8). Let r ∈ N and x ∈ [0, 1] . Then, for all
f ∈ C r [0, 1] such that f (r) ∈ LipM (α) and for any n ∈ N, we have
qn ,αn
stA − Mn,r
f − f = 0.
Proof. For x ∈ [0, 1) , we obtain from (3.1) that
(3.2)
q,α
f (x) − Mn,r
(f ; x) =
∞
X
mq,α
n,k (x) {f (x) − (Tr f ) (xn,k ; x)} .
k=0
Using the Taylor’s formula, we can write
f (x) − (Tr f ) (xn,k ; x) =
Z
(x − xn,k )r 1
(1 − t)r−1
(r − 1)!
0
h
i
f (r) (xn,k + t (x − xn,k )) − f (r) (xn,k ) dt.
Approximation properties of Stancu type Meyer-König and Zeller operators
147
Since f (r) ∈ LipM (α) , we obtain
|x − xn,k |r+α
B (r, α + 1)
(r − 1)!
B (r, α + 1) r+α
=M
ϕx (xn,k ) ,
(r − 1)!
r
Q
where ϕx (t) = |x − t| and B (r, α + 1) =
(α + j)−1 , B being Beta function.
(3.3)
|f (x) − (Tr f ) (xn,k ; x)| ≤ M
j=1
In view of relations (3.2) and (3.3), we get
∞
B (r, α + 1) X q,α
q,α
f (x) − Mn,r
(f ; x) ≤ M
mn,k (x) ϕr+α
(xn,k ) .
x
(r − 1)!
k=0
Since the case of x = 1 is clear, we deduce
q,α
.
Mn,r f − f ≤ M B (r, α + 1) Mnq,α ϕr+α
(3.4)
x
(r − 1)!
Firstly, we replace q and α by two sequences (qn ) and (αn ) respectively, such that (2.8)
holds. Then, for ε > 0, we define the following sets,
qn ,αn
≥ε .
U := n ∈ N : Mn,r
f − f ≥ ε and U 0 := n ∈ N : Mnqn ,αn ϕr+α
x
From (3.4), we have U ⊆ U 0 . So, for all n ∈ N, that
X
X
ank ≤
ank .
k∈U 0
k∈U
Because ε is arbitrary and ϕr+α
∈ C [0, 1], lim
x
P
n→∞ k∈U 0
proof is completed.
Since
result.
ϕr+α
x
(t) = |x − t|
ank = 0 from Theorem 2.3. So, the
r+α
∈ C [0, 1] and
ϕr+α
x
(x) = 0, we can give the following
3.2. Corollary. Let x ∈ [0, 1] and r ∈ N. Then, for all f ∈ C r [0, 1] such that f (r) ∈
LipM (α) and for any n ∈ N, we have
q,α
Mn,r (f ; x) − f (x) ≤ 2 M B (r, α + 1) ω ϕr+α
, δnq,α (x)
x
(r − 1)!
where δnq,α (x) is given by (2.11).
4. Acknowledgements
The author would like to thank the referee for his/her valuable suggestions which
improved the paper considerably.
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