Volume 10 (2009), Issue 4, Article 105, 10 pp.
PROPERTIES OF q−MEYER-KÖNIG-ZELLER DURRMEYER OPERATORS
HONEY SHARMA
R AYAT & BAHRA I NSTITUTE OF P HARMACY
V ILLAGE S AHAURAN K HARAR D ISTT. M OHALI P UNJAB , I NDIA
pro.sharma.h@gmail.com
Received 09 April, 2009; accepted 26 June, 2009
Communicated by I. Gavrea
A BSTRACT. We introduce a q analogue of the Meyer-König-Zeller Durrmeyer type operators
and investigate their rate of convergence.
Key words and phrases: q−integers, q−Meyer-König-Zeller Durrmeyer type operators, A-Statistical convergence, Weighted
space, Weighted modulus of smoothness, Lipschitz class.
2000 Mathematics Subject Classification. 41A25, 41A35.
1. I NTRODUCTION
Abel et al. [5] introduced the Meyer-Konig-Zeller Durrmeyer operators as
Z 1
∞
X
(1.1)
Mn (f ; x) =
mn,k (x)
bn,k (t)f (t)dt,
0 ≤ x < 1,
0
k=0
where
n+k−1
mn,k (x) =
xk (1 − x)n
k
and
n+k k
bn,k (t) = n
t (1 − t)n−1 .
k
Very recently H. Wang [6], O. Dogru and V. Gupta [2], A. Altin, O. Dogru and M.A. Ozarslan
[7] and T. Trif [3] studied the q-Meyer-Konig-Zeller operators. This motivated us to introduce
the q analogue of the Meyer-Konig-Zeller Durrmeyer operators.
Before introducing the operators, we mention certain definitions based on q−integers; details
can be found in [10] and [12].
For each non-negative integer k, the q-integer [k] and the q-factorial [k]! are respectively
defined by
(
(1 − q k ) (1 − q), q 6= 1
[k] :=
,
k,
q=1
The author would like to thank Dr Vijay Gupta, NSIT, New Delhi for his valuable suggestions and remarks during the preparation of this
work.
094-09
2
H ONEY S HARMA
and
(
[k] [k − 1] · · · [1],
k≥1
[k]! :=
.
1,
k=0
For the integers n, k satisfying n ≥ k ≥ 0, the q-binomial coefficients are defined by
hni
[n]!
:=
.
[k]![n − k]!
k
We use the following notations
(a +
b)nq
=
n−1
Y
(a + q j b) = (a + b)(a + qb) · · · (a + q n−1 b)
j=0
and
(t; q)0 = 1,
(t; q)n =
n−1
Y
j
(1 − q t),
(t; q)∞
∞
Y
=
(1 − q j t).
j=0
j=0
Also it can be seen that
(a; q)∞
.
(aq n ; q)∞
(a; q)n =
The q−Beta function is defined as
Z
1
tm−1 (1 − qt)n−1
dq t
q
Bq (m, n) =
0
for m, n ∈ N and we have
Bq (m, n) =
(1.2)
[m − 1]![n − 1]!
.
[m + n − 1]!
It can be easily checked that
(1.3)
n−1
Y
j
(1 − q x)
j=0
∞ X
n+k−1
k
k=0
xk = 1.
Now we introduce the q-Meyer-Konig-Zeller Durrmeyer operator as follows
Z 1
∞
X
(1.4)
Mn,q (f ; x) =
mn,k,q (x)
bn,k,q (t)f (qt)dq t, 0 ≤ x < 1
(1.5)
:=
k=0
∞
X
0
mn,k,q (x)An,k,q (f ),
k=0
where 0 < q < 1 and
(1.6)
n+k−1 k
mn,k,q (x) = Pn−1 (x)
x ,
k
(1.7)
bn,k,q (t) =
[n + k]! k
t (1 − qt)n−1
.
q
[k]![n − 1]!
Here
Pn−1 (x) =
n−1
Y
(1 − q j x).
j=0
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 105, 10 pp.
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P ROPERTIES OF q−M EYER -KÖNIG -Z ELLER D URRMEYER O PERATORS
3
Remark 1. It can be seen that for q → 1− , the q−Meyer-Konig-Zeller Durrmeyer operator
becomes the operator studied in [4] for α = 1.
2. M OMENTS
Lemma 2.1. For gs (t) = ts , s = 0, 1, 2, . . ., we have
Z 1
[n + k]![k + s]!
(2.1)
bn,k,q (t)gs (qt)dq t = q s
.
[k]![k + s + n]!
0
Proof. By using the q−Beta function (1.2), the above lemma can be proved easily.
Here, we introduce two lemmas proved in [8], as follows:
Lemma 2.2. For r = 0, 1, 2, . . . and n > r, we have
Qr
∞ n−j
X
x)
n+k−1
xk
j=1 (1 − q
(2.2)
Pn−1 (x)
=
,
r
r
[n
+
k
−
1]
[n
−
1]
k
k=0
where [n − 1]r = [n − 1][n − 2] · · · [n − r].
Lemma 2.3. The identity
(2.3)
1
1
≤ r+1
,
[n + k + r]
q [n + k − 1]
r≥0
holds.
Theorem 2.4. For all x ∈ [0, 1], n ∈ N and q ∈ (0, 1), we have
(2.4)
(2.5)
(2.6)
(2.7)
Mn,q (e0 ; x) = 1,
(1 − q n−1 x)
,
Mn,q (e1 ; x) ≤ x +
q[n − 1]
(1 + q n−2 )
Mn,q (e1 ; x) ≥ 1 −
x + q n−2 (1 − q)x2 ,
[n + 1]
Mn,q (e2 ; x) ≤ x2 +
(1 + q)2 (1 − q n−1 x)
(1 + q) (1 − q n−1 x)(1 − q n−2 x)
x
+
.
q3
[n − 1]
q4
[n − 1][n − 2]
Proof. We have to estimate Mn,q (es ; x) for s = 0, 1, 2. The result can be easily verified for
s = 0. Using the above lemmas and equation (1.3), we obtain relations (2.5) and (2.6) as
follows
∞ X
n+k−1
[k + 1]
Mn,q (e1 , x) = qPn−1 (x)
xk
k
[n
+
k
+
1]
k=0
∞ X
n+k−1
q[k] + 1
xk
≤ qPn−1 (x)
2
k
q
[n
+
k
−
1]
k=0
∞ X
n+k−1 k
= xPn−1 (x)
x
k
k=0
∞ Pn−1 (x) X n + k − 1
xk
+
q
k
[n + k − 1]
k=0
=x+
(1 − q n−1 x)
.
q[n − 1]
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 105, 10 pp.
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4
H ONEY S HARMA
Also,
∞ X
n + k − 2 [k + 1] [n + k − 1]
xk
[n
+
k
+
1]
k=1
∞
X n + k − 1 [n + k + 1] − 1 xk+1
≥ Pn−1 (x)
[n + k + 2]
k
k=0
∞
X
n+k−1
[n + k + 1]
1
−
xk+1
≥ Pn−1 (x)
k
[n + k + 2] [n + 1]
k=0
∞
X
1
n+k−1
q n+k+1
xk+1 −
x
≥ Pn−1 (x)
1−
[n
+
k
+
2]
[n
+
1]
k
k=0
∞ X
n+k−1
q n−2 (1 − (1 − q)[k])
1
≥ Pn−1 (x)
1−
xk+1 −
x
k
[n
+
k
−
1]
[n
+
1]
k=0
∞ n−2
X
q x
n+k−1 k
1
n−2
2
=x−
+ q (1 − q)x Pn−1 (x)
x −
x
[n + 1]
k
[n
+
1]
k=0
(1 + q n−2 )
= 1−
x + q n−2 (1 − q)x2 .
[n + 1]
Mn,q (e1 , x) = qPn−1 (x)
k−1
[k]
Similar calculations reveal the relation (2.7) as follows
∞ X
[k + 1][k + 2]
n+k−1
2
Mn,q (e2 , x) = q Pn−1 (x)
xk
k
[n + k + 1][n + k + 2]
k=0
∞
X
n + k − 1 q 3 [k]2 + (2q + 1)q[k] + (q + 1) k
1
≤ 4 Pn−1 (x)
x
q
k
[n
+
k
−
1][n
+
k
−
2]
k=0
∞
Pn−1 (x) X [n + k − 2]!
=
(q[k] + 1)xk+1
q
[k]![n
−
1]!
k=0
∞ Pn−1 (x)(2q + 1)x X n + k − 1
xk
+
q3
k
[n + k − 1]
k=0
∞ Pn−1 (x)(1 + q) X n + k − 1
xk
+
q4
k
[n + k − 1]2
k=0
∞ X
n+k−1 k
2
= x Pn−1 (x)
x
k
k=0
∞ Pn−1 (x) X n + k − 1
xk
(2q + 1) (1 − q n−1 x)
+x
+x
q
k
[n + k − 1]
q3
[n − 1]
k=0
(1 + q) (1 − q n−1 x)(1 − q n−2 x)
q4
[n − 1][n − 2]
2
(1 + q) (1 − q n−1 x)
(1 + q) (1 − q n−1 x)(1 − q n−2 x)
= x2 +
x
+
.
q3
[n − 1]
q4
[n − 1][n − 2]
+
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 105, 10 pp.
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P ROPERTIES OF q−M EYER -KÖNIG -Z ELLER D URRMEYER O PERATORS
5
Remark 2. From Lemma 2.3, it is observed that for q → 1− , we obtain
Mn (e0 ; x) = 1,
(1 − x)
,
Mn (e1 ; x) ≤ x +
(n − 1)
2
Mn (e1 ; x) ≥ 1 −
x,
(n + 1)
4x(1 − x)
2(1 − x)2
Mn (e2 ; x) ≤ x2 +
+
,
(n − 1)
(n − 1)(n − 2)
which are moments for a new generalization of the Meyer-Konig-Zeller operators for α = 1 in
[4].
Corollary 2.5. The central moments of Mn,q are
Mn,q (ψ0 ; x) = 1,
(1 − q n−1 x)
Mn,q (ψ1 ; x) ≤
,
q[n − 1]
(1 + q)2 (1 − q n−1 x)
(1 + q) (1 − q n−1 x)(1 − q n−2 x)
Mn,q (ψ2 ; x) ≤
x
+
q3
[n − 1]
q4
[n − 1][n − 2]
n−2
(1 + q ) 2
+2
x,
[n + 1]
where ψi (x) = (t − x)i for i = 0, 1, 2.
Proof. By the linearity of Mn,q and Theorem 2.4, we directly get the first two central moments.
Using simple computations, the third moment can be easily verified as follows
Mn,q (ψ2 ; x) = Mn,q (e2 ; x) + x2 Mn,q (e0 ; x) − 2xMn,q (e1 ; x)
(1 + q)2 (1 − q n−1 x)
(1 + q) (1 − q n−1 x)(1 − q n−2 x)
x
+
q3
[n − 1]
q4
[n − 1][n − 2]
(1 + q n−2 )
+ 1−
x − q n−2 (1 − q)x2
[n + 1]
2
n−1
(1 + q) (1 − q x)
(1 + q) (1 − q n−1 x)(1 − q n−2 x)
≤
x
+
q3
[n − 1]
q4
[n − 1][n − 2]
n−2
(1 + q ) 2
+2
x.
[n + 1]
≤
Remark 3. For q → 1− , we get
Mn (ψ2 ; x) ≤
4x
2(1 − x)2
+
n − 1 (n − 1)(n − 2)
which is similar to the result in [4].
Theorem 2.6. The sequence Mn,qn (f ) converges to f uniformly on C[0, 1] for each f ∈ C[0, 1]
iff qn → 1 as n → ∞.
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 105, 10 pp.
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6
H ONEY S HARMA
Proof. By the Korovkin theorem (see [1]), Mn,qn (f ; x) converges to f uniformly on [0, 1] as
n → ∞ for f ∈ C[0, 1] iff Mn,qn (ti ; x) → xi for i = 1, 2 uniformly on [0, 1] as n → ∞.
From the definition of Mn,q and Theorem 2.4, Mn,qn is a linear operator and reproduces
constant functions.
Moreover, as qn → 1, then [n]qn → ∞, therefore by Theorem 2.4, we get
Mn,qn (ti ; x) → xi
for i = 0, 1, 2.
Hence, Mn,qn (f ) converges to f uniformly on C[0, 1].
Conversely, suppose that Mn,qn (f ) converges to f uniformly on C[0, 1] and qn does not tend
to 1 as n → ∞. Then there exists a subsequence (qnk ) of (qn ) s.t. qnk → q0 (q0 6= 1) as k → ∞.
Thus
1
1 − qnk
=
→ (1 − q0 ).
[n]qnk
1 − qnk n
Taking n = nk and q = qnk in Mn,q (e2 , x), we have
Mn,qnk
(1 − qnn−1
x)(1 − q0 )
k
(e2 ; x) ≤ x +
6 x
=
qnk
which is a contradiction. Hence qn → 1. This completes the proof.
Remark 4. Similar results are proved for the q−Bernstein-Durrmeyer operator in [11].
3. W EIGHTED S TATISTICAL A PPROXIMATION P ROPERTIES
In this section, we present the statistical approximation properties of the operator Mn,q by
using a Bohman-Korovkin type theorem [9].
Firstly, we recall the concepts of A-statistical convergence, weight functions and weighted
spaces as considered in [9].
Let A = (ajn )j,n be a non-negative regular summability matrix. A sequence
(xn )n is said
P
to be A-statistically convergent to a number L if, for every ε > 0, lim
ajn = 0. It is
j
n:|xn −L|≥ε
denoted by stA − lim xn = L. For A := C1 , the Cesàro matrix of order one is defined as
n
(
cjn :=
1
j
1≤n≤j
0
n > j.
A-statistical convergence coincides with statistical convergence.
A weight function is a real continuous function ρ on R s.t. lim ρ(x) = ∞, ρ(x) ≥ 1 for all
|x|→∞
x ∈ R.
The weighted space of real-valued functions f (denoted as Bρ (R)) is defined on R with the
property |f (x)| ≤ Mf ρ(x) for all x ∈ R, where Mf is a constant depending on the function f .
We also consider the weighted subspace Cρ (R) of Bρ (R) given by
Cρ (R) := {f ∈ Bρ (R) : f continuous on R} .
(x)|
Bρ (R) and Cρ (R) are Banach spaces with the norm k·kρ , where kf kρ := sup |fρ(x)
.
x∈R
We next present a Bohman-Korovkin type theorem ([9, Theorem 3]) as follows.
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 105, 10 pp.
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P ROPERTIES OF q−M EYER -KÖNIG -Z ELLER D URRMEYER O PERATORS
7
Theorem 3.1. Let A = (ajn )j,n be a non-negative regular summability matrix and let (Ln )n be
a sequence of positive linear operators from Cρ1 (R) into Bρ2 (R), where ρ1 and ρ2 satisfy
ρ1 (x)
= 0.
|x|→∞ ρ2 (x)
lim
Then
stA − lim kLn f − f kρ2 = 0
n
f ∈ Cρ1 (R)
for all
if and only if
stA − lim kLn Fv − Fv kρ1 = 0,
n
where Fv (x) =
xv ρ1 (x)
,
1+x2
v = 0, 1, 2,
v = 0, 1, 2.
We next consider a sequence (qn )n , qn ∈ (0, 1), such that
st − lim qn = 1.
(3.1)
n
Theorem 3.2. Let (qn )n be a sequence satisfying (3.1). Then for all f ∈ Cρ0 (R+ ), we have
st − lim kMn,q (f ; ·) − f kρα = 0,
n
α > 0.
Proof. It is clear that
st − lim kMn,qn (e0 ; ·) − e0 kρ0 = 0.
(3.2)
n
Based on equation (2.5), we have
|Mn,qn (e1 , x) − e1 (x)|
1
≤k e0 k 2
2
1+x
qn [n − 1]qn
1
.
≤
[n − 1]qn
1
Since st − lim qn = 1, we get st − lim [n−1]
= 0 and thus
q
n
n
n
st − lim kMn,qn (e1 ; ·) − e1 kρ0 = 0.
(3.3)
n
By using (2.7), we have
|Mn,qn (e2 , x) − e2 (x)|
1
1
≤k e0 k
+
1 + x2
[n − 1]qn [n − 1]qn [n − 2]qn
1
1
≤
+
.
[n − 1]qn [n − 2]2qn
Consequently,
(3.4)
st − lim kKn,qn (e2 ; ·) − e2 kρ0 = 0.
n
Finally, using (3.2), (3.3) and (3.4), the proof follows from Theorem 3.1 by choosing A = C1 ,
the Cesàro matrix of order one and ρ1 (x) = 1 + x2 , ρ2 (x) = 1 + x2+α , x ∈ R+ , α > 0.
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 105, 10 pp.
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8
H ONEY S HARMA
4. O RDER
A PPROXIMATION
OF
We now recall the concept of modulus of continuity. The modulus of continuity of f (x) ∈
C[0, a], denoted by ω(f, δ), is defined by
ω(f, δ) =
(4.1)
|f (x) − f (y)|.
sup
|x−y|≤δ;x,y∈[0,a]
The modulus of continuity possesses the following properties (see [9]):
ω(f, λδ) ≤ (1 + λ)ω(f, δ)
(4.2)
and
ω(f, nδ) ≤ nω(f, δ),
n ∈ N.
Theorem 4.1. Let (qn )n be a sequence satisfying (3.1). Then
√
(4.3)
|Mn,q (f ; x) − f | ≤ 2ω(f, δn )
for all f ∈ C[0, 1], where
δn = Mn,q (qt − x)2 ; x .
(4.4)
Proof. By the linearity and monotonicity of Mn,q , we get
|Mn,q (f ; x) − f | ≤ Mn,q (|f (t) − f (x)|; x)
Z 1
∞
X
=
mn,k,q (x)
bn,k,q (t)|f (qt) − f (x)|dq t.
0
k=0
Also
(4.5)
|f (qt) − f (x)| ≤
(qt − x)2
1+
δ2
ω(f, δ).
By using (4.5), we obtain
∞
X
1
(qt − x)2
|Mn,q (f ; x) − f | ≤
mn,k,q (x)
bn,k,q (t) 1 +
ω(f, δ)dq t
2
δ
0
k=0
1
2
= Mn,q (e0 ; x) + 2 Mn,q (qt − x) ; x ω(f, δ)
δ
Z
and
Mn,q (qt − x)2 ; x = q 2 Mn,q (e2 ; x) + x2 Mn,q (e0 ; x) − 2qxMn,q (e1 ; x)
(1 + q)2 (1 − q n−1 x)
x
q
[n − 1]
(1 + q) (1 − q n−1 x)(1 − q n−2 x)
+
q2
[n − 1][n − 2]
(1 + q n−2 )
2
+ 2xq
− 2q n−1 (1 − q)x3 .
[n + 1]
By (3.1) and the above equation, we get
(4.6)
lim Mn,q (qt − x)2 ; x = 0.
n→∞,qn →1
√
So, letting δn = Mn,q ((qt − x)2 ; x) and taking δ = δn , we finally obtain
√
|Mn,q (f ; x) − f | ≤ 2ω(f, δn ).
≤ (1 − q)2 x2 +
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 105, 10 pp.
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P ROPERTIES OF q−M EYER -KÖNIG -Z ELLER D URRMEYER O PERATORS
9
As usual, a function f ∈ LipM (α), (M > 0 and 0 < α ≤ 1), if the inequality
|f (t) − f (x)| ≤ M |t − x|α
(4.7)
for all t, x ∈ [0, 1].
Theorem 4.2. For all f ∈ LipM (α) and x ∈ [0, 1], we have
|Mn,q (f ; x) − f | ≤ M δnα/2 ,
(4.8)
where δn = Mn,q (ψ2 ; x).
Proof. Using inequality (4.7) and Hölder’s inequality with p = α2 , q =
2
,
2−α
we get
|Mn,q (f ; x) − f | ≤ Mn,q (|f (t) − f (x)|; x)
≤ M Mn,q (|t − x|α ; x)
≤ M Mn,q (|t − x|2 ; x)α/2 .
Taking δn = Mn,q (ψ2 ; x), we get
|Mn,q (f ; x) − f | ≤ M δnα/2 .
Theorem 4.3. For all f ∈ C[0, 1] and f (1) = 0, we have
|An,k,q (f )| ≤ An,k,q (|f |) ≤ ω(f, q n )(1 + q −n ),
(4.9)
(0 ≤ k ≤ n).
Proof. Clearly
|f (qt)| = |f (qt) − f (1)|
≤ ω(f, q n (1 − qt))
(1 − qt)
n
≤ ω(f, q ) 1 +
.
qn
Thus by using Lemma 2.1, we get
|An,k,q (f )| ≤ An,k,q (|f |)
Z 1
=
bn,k,q (t)|f (qt)|dq t
0
Z 1
(1 − qt)
n
≤ ω(f, q )
bn,k,q (t) 1 +
dq t
qn
0
Z 1
Z
1
1 1
n
= ω(f, q )
1+ n
bn,k,q (t)dq t − n
bn,k,q (t)(qt)dq t
q
q 0
0
1
1
[k + 1]
n
= ω(f, q )
1 + n − n−1
q
q
[k + n + 1]
n
−n
≤ ω(f, q )(1 + q ).
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 105, 10 pp.
http://jipam.vu.edu.au/
10
H ONEY S HARMA
R EFERENCES
[1] G.G. LORENTZ, Bernstein Polynomials, Mathematical Expositions, University of Toronto Press,
Toronto, 8 (1953).
[2] O. DOGRU AND V. GUPTA, Korovkin-type approximation properties of bivariate q−Meyer-Konig
and Zeller operators, Calcolo, 43(1) (2006), 51–63.
[3] T. TRIF, Meyer-Konig and Zeller operators based on q integers, Revue d’Analysis Numerique et de
Theorie de l’ Approximation, 29(2) (2000), 221–229.
[4] V. GUPTA, On new types of Meyer Konig and Zeller operators, J. Inequal. Pure and Appl.
Math., 3(4) (2002), Art. 57.
[5] U. ABEL, M. IVAN AND V. GUPTA, On the rate of convergence of a Durrmeyer variant of Meyer
Konig and Zeller operators, Archives Inequal. Appl., (2003), 1–9.
[6] H. WANG, Properties of convergence for q−Meyer Konig and Zeller operators, J. Math. Anal.
Appl., in press.
[7] A. ALTIN, O. DOGRU AND M.A. OZARSLAN, Rate of convergence of Meyer-Konig and Zeller
operators based on q integer, WSEAS Transactions on Mathematics, 4(4) (2005), 313–318.
[8] V. GUPTA AND H. SHARMA, Statistical approximation by q integrated Meyer-Konig-Zeller and
Kantrovich operators, Creative Mathematics and Informatics, in press.
[9] O. DUMAN AND C.O. RHAN, Statistical approximation by positve linear operators, Studia
Math., 161(2) (2006), 187–197.
[10] T. ERNST, The history of q-calculus and a new method, Department of Mathematics, Uppsala
University, 16 U.U.D.M. Report, (2000).
[11] V. GUPTA AND W. HEPING, The rate of convergence of q−Durrmeyer operators for 0 < q < 1,
Math. Meth. Appl. Sci., (2008).
[12] V. KAC AND P. CHEUNG, Quantum Calculus, Universitext, Springer-Verlag, NewYork, (2002).
J. Inequal. Pure and Appl. Math., 10(4) (2009), Art. 105, 10 pp.
http://jipam.vu.edu.au/