CREATIVE MATH. & INF.
19 (2010), No. 1,
45 - 52
Online version at http://creative-mathematics.ubm.ro/
Print Edition: ISSN 1584 - 286X Online Edition: ISSN 1843 - 441X
Statistical approximation by q-integrated
Meyer-König-Zeller-Kantorovich operators
V IJAY G UPTA and H ONEY S HARMA
A BSTRACT. In the present paper we introduce a q-analogue of the integrated Meyer-König-ZellerKantorovich type operators and investigate their statistical approximation properties.
1. I NTRODUCTION
In the last decade some new generalizations of well known positive linear
operators, based on the q-integers were introduced and studied by several authors. For instance q-Meyer-König and Zeller operators were studied by Trif [11],
Doğru and Gupta [6], Dogru et al. [4] and Dogru and Duman [5] etc. In 2008
M. Ali Özarslan and Oktay Duman [2] proposed an approximation theorem by
Meyer-König and Zeller type operators. C. Radu [10] in 2008 proposed statistical approximation by some linear operators of discrete type. In what follows we
mention some basic definitions and notations used in q-calculus, details can be
found in [9] and [7].
For any fixed real number q > 0, we denote q-integers by [k], k ∈ N
1 + q + q 2 + . . . + q k−1 if q 6= 1,
[k] =
k
if q = 1.
We set [0]q = 0. In general, for a real number k ∈ R, we denote the q-number k by

 1 − qk
if q =
6 1,
[k] =
 1−q
k
if q = 1.
The q-factorial is defined as follows
[1] · [2] · . . . · [k] if k = 1, 2, . . .
[k]! =
1
if k = 0,
and the q-binomial coefficients are given by
[n]!
n
,
=
k
[k]! [n − k]!
0 ≤ k ≤ n.
Received: 19.03.2009. In revised form: 27.01.2010. Accepted: 08.02.2010.
2000 Mathematics Subject Classification. 41A25, 41A35.
Key words and phrases. q-integers, q-integrated Meyer-König-Zeller-Kantorovich operators, statistical
approximation.
45
46
V. Gupta and H. Sharma
Also
∞ X
1
n+k−1
xk = n−1
k
Q
k=0
(1 − q j x)
(1.1)
j=0
and
[k + 1]
[k]
q k [n]
−
=
.
[n + k + 1] [n + k]
[n + k][n + k + 1]
The q-analogue of integration (see [3]) is defined as
Z a
∞
X
f (t)dq t = (1 − q)a
f (q j a)q j .
0
(1.2)
(1.3)
j=0
For q ∈ (0, 1), x ∈ [0, 1] and n ∈ N, we propose the q-Meyer-König-ZellerKantorovich operators as
Z [k+1]/[n+k+1]
∞ X
n+k+1
xk q −k Pn−1 (x)
Mn,q (f ; x) = [n+1]
f (t)dq t, (1.4)
k
[k]/[n+k]
k=0
where Pn−1 (x) =
n−1
Q
(1 − q j x).
j=0
Remark 1.1. It can be seen that for q → 1− the q-Meyer-König-Zeller and Kantorovich operator becomes the operator studied in [1].
2. M OMENTS
Lemma 2.1. For r = 0, 1, 2 . . . and n > r, we have
Pn−1 (x)
∞ X
k=0
r
Q
n+k−1
k
xk
=
[n + k − 1]r
(1 − q n−j x)
j=1
[n − 1]r
(2.5)
where [n − 1]r = [n − 1][n − 2] . . . [n − r].
Proof. Clearly for r = 0, relation holds.
For r = 1, using identity (1.1), we get
∞ ∞ X
X
xk
(1−q n−1 x)
n+k−1
n+k−2
Pn−1 (x)
=
Pn−2 (x)
xk
k
k
[n+k−1]
[n − 1]
k=0
k=0
(1 − q n−1 x)
.
=
[n − 1]
By method of induction the above lemma can be proved easily.
Lemma 2.2. The following inequality holds true
1
1
≤ r+1
, r ≥ 0.
[n + k + r]
q [n + k − 1]
Proof of lemma is very technical and omitted here.
(2.6)
Statistical approximation by q-integrated Meyer-König-Zeller-Kantorovich operators
47
Lemma 2.3. For all x ∈ [0, 1], n ∈ N and q ∈ (0, 1), we have
Mn,q (e0 ; x) = 1,
1
(1 − q n−1 x)
Mn,q (e1 ; x) ≤
2x +
,
([3] − 1)
q[n − 1]
2
3x
1 (1 − q n−1 x)
1 3x
+
1
+
Mn,q (e2 ; x) ≤
[3] q 2
q3
q
[n − 1]1
n−1
n−2
x)(1 − q
x)
1 (1 − q
+ 5
.
q
[n − 1]2
(2.7)
(2.8)
(2.9)
Proof. In (1.4), by using (1.1), (1.2) and (1.3), we have
Z [k+1]/[n+k+1]
∞ X
n+k+1
xk q −k Pn−1 (x)
dq t
k
[k]/[n+k]
k=0
∞ X
[k]
[k + 1]
n+k+1
k −k
−
= [n + 1]
x q Pn−1 (x)
k
[n + k + 1] [n + k]
k=0
∞ X
q k [n]
n+k+1
xk q −k Pn−1 (x)
= [n + 1]
k
[n + k][n + k + 1]
k=0
∞
X n+k−1
=
xk Pn−1 (x)
k
Mn,q (e0 ; x) = [n + 1]
k=0
= 1.
Using Lemma 2.1, Lemma 2.2 and the identities (1.1) and (1.2), we obtain the
relation (2.8) as follows
!
∞
∞
[k]2 X 2j
[k + 1]2 X 2j
q −
q
tdq t = (1 − q)
[n + k + 1]2
[n + k]2
[k]/[n+k]
k=0
k=0
1
[k + 1]2
[k]2
=
−
(1 + q) [n + k + 1]2
[n + k]2
k
1
q [n]
q[k] + 1
[k]
=
+
(1 + q) [n + k][n + k + 1] [n + k + 1] [n + k]
1
q k [n]
q
1
1
=
[k]
+
+
(1 + q) [n + k][n + k + 1]
[n + k + 1] [n + k]
[n + k + 1]
k
1
q [n]
[k]
2
1
≤
+ 2
.
(1 + q) [n + k][n + k + 1] q [n + k − 1] q [n + k − 1]
Z
[k+1]/[n+k+1]
48
V. Gupta and H. Sharma
∞ X
[n + 1]
n+k+1
Mn,q (e1 , x) ≤
xk q −k
k
(1 + q)
k=0
[k]
2
1
1
q k [n]
+ 2
=
[n + k][n + k + 1] q [n + k − 1] q [n + k − 1]
([3] − 1)
!
∞ ∞ k
X
X
[k]
1
x
n+k−1
n+k−1
2
xk Pn−1 (x)+
Pn−1 (x)
k
k
[n + k − 1]
q
[n+k−1]
k=0
k=0
!
∞ ∞ X
1
xk
1 X n+k−1
n+k−2
k
=
2
Pn−1 (x)
x Pn−1 (x)+
k
k−1
([3]−1)
q
[n+k−1]
k=0
k=1
!
∞ ∞ X
1
1 X n+k−1
xk
n+k−1
k
=
2x
Pn−1 (x)
x Pn−1 (x)+
k
k
([3]−1)
q
[n+k−1]
k=0
k=0
(1 − q n−1 x)
1
2x +
.
=
([3] − 1)
q[n − 1]
A similar calculation reveals relation (2.9) as follows:
Z
[k+1]/[n+k+1]
(1 − q)
t dq t =
(1 − q 3 )
2
[k]/[n+k]
=
[k + 1]3
[k]3
−
[n + k + 1]3
[n + k]3
1
q k [n]
[k]2 b2 (n, k)+[k]b1 (n, k)+b0 (n, k)
[3] [n+k][n+k+1]
where
q2
1
q
+
+
[n + k + 1]2
[n + k]2
[n + k + 1][n + k]
2q
1
b1 (n, k) =
+
2
[n + k + 1]
[n + k + 1][n + k]
1
,
b0 (n, k) =
[n + k + 1]2
b2 (n, k) =
∞
Mn,q (e2 , x) =
1 X
[3]
k=0
n+k−1
k
xk [k]2 b2 (n, k) + [k]b1 (n, k) + b0 (n, k) .
Using Lemma 2.2, we get
b2 (n, k) ≤
b1 (n, k) ≤
b0 (n, k) ≤
3
+ k − 1]2
3
q 4 [n + k − 1]2
1
q 5 [n + k − 1]2
q 3 [n
∞ X
k=0
Statistical approximation by q-integrated Meyer-König-Zeller-Kantorovich operators
49
∞
2
3 X n+k−1
[k]
n+k−1
Pn−1 (x)
xk [k]2 b2 (n, k)Pn−1 (x) ≤ 3
xk
k
k
q
[n+k−1]2
k=1
∞ 3 X n+k−2
[k]
= 3
xk
Pn−1 (x)
q
=
3
q3
k=1
∞ X
k=0
k−1
n+k−1
k
[n + k − 2]
xk+1
[k + 1]
Pn−1 (x)
[n + k − 1]
∞ 1
3 X n+k−1
xk+1
Pn−1 (x)
= 3
k
q
[n + k − 1]
k=0
∞ 3 X n+k−1
+ 2
xk+2 Pn−1 (x)
k
q
k=0
=
3x (1 − q n−1 x)
3x2
+ 3
2
q
q
[n − 1]
and
∞ X
n+k−1
xk [k]b1 (n, k)Pn−1 (x) ≤
k
k=0
∞ 3 X n+k−1
[k]
xk
Pn−1 (x)
4
k
q
[n + k − 1][n + k − 2]
k=0
∞ 3 X n+k−2
3x (1 − q n−1 x)
1
≤ 4
Pn−1 (x) = 4
.
xk
k−1
q
[n + k − 2]
q
[n − 1]
k=1
Also
∞ X
1 (1 − q n−1 x)(1 − q n−2 x)
n+k−1
xk b0 (n, k)Pn−1 (x) ≤ 5
,
k
q
[n − 1]2
k=0
therefore
Mn,q (e2 , x) ≤
1
[3]
2
n−1
3x
3x (1 − q
x) 3x (1 − q n−1 x)
1 (1 − q n−1 x)(1 − q n−2 x)
+ 3
+ 4
+ 5
2
q
q
[n − 1]
q
[n − 1]
q
[n − 1]2
.
3. STATISTICAL APPROXIMATION PROPERTIES
In this section, by using a Bohman-Korovkin type theorem proved in [8], we
present the statistical approximation properties of the operator Mn,q given by
(1.4).
At this moment, we recall the concept of statistical convergence.
A sequence (xn )n is said to be statistically convergent to a number L, denoted
by st − lim xn = L if, for every ε > 0,
n
δ{n ∈ N : |xn − L| ≥ ε} = 0,
(3.10)
50
V. Gupta and H. Sharma
where
δ(S) :=
N
1 X
χS(j)
N
k=1
is the natural density of set S ⊆ N and χS is the characteristic function of S.
Let CB (D) represent the space of all continuous functions on D and bounded
on entire real line, where D is any interval on real line. It can be easily shown that
CB (D) is a Banach space with supreme norm. Also Mn.q (f, x), n ∈ N are well
defined for any f ∈ CB ([0, 1]).
Theorem A. [5]. Let (Ln )n be a sequence of positive linear operators from CB ([a, b])
into B([a, b]) and satisfies the condition that
st − lim kLn ei − ei k = 0 for all i = 0, 1, 2.
n
Then
st − lim kLn f − f k = 0 for all f ∈ CB ([a, b]).
n
We consider a sequence (qn )n , qn ∈ (0, 1), such that
st − lim qn = 1.
n
(3.11)
As an application of Theorem A, we have the following result for our operators.
Theorem 3.1. Let (qn )n be a sequence satisfying (3.11). Then for the operators Mn,q f
satisfying the condition
st − lim kMn,q ei − ei k = 0 for all i = 0, 1, 2
n
we have
st − lim kMn,q f − f k = 0 for all f ∈ CB ([a, b]).
n
Proof. It is clear that
st − lim kMn,qn (e0 ; ·) − e0 k = 0.
n
(3.12)
Based on Lemma 2.3, we have
(1 − qnn−1 x)
2
−1 x+
|Mn,qn (e1 ; ·) − e1 | ≤ [3]qn − 1
([3]qn − 1)qn [n − 1]qn 3−[3]qn 1
qnn−2
+
≤
+
|
[3]qn −1 |([3]qn −1)qn [n−1]qn |
([3]qn −1)[n−1]qn |
≤ |3 − [3]qn | +
1
qnn−2
+
.
|qn [n − 1]qn | |[n − 1]qn |
Since st − lim qn = 1, we get
n
st − lim
n
1
=0
[n − 1]qn
(3.13)
and
st − lim(|3 − [3]qn |) = 0.
n
(3.14)
Statistical approximation by q-integrated Meyer-König-Zeller-Kantorovich operators
51
Hence, we get
kMn,qn (e1 ; ·) − e1 k < .
(3.15)
Define the following sets
A := {n ∈ N : kMn,qn (e1 ; ·) − e1 k ≥ },
A1 := {n ∈ N : (3 − [3]qn ) ≥ /3},
1
≥ /3}.
A2 := {n ∈ N :
[n − 1]qn
S
Thus we obtain A ⊆ A1 A2 i.e. δ(A) ≤ δ(A1 ) + δ(A2 ) = 0.
Hence
st − lim kMn,qn (e1 ; ·) − e1 k = 0.
n
(3.16)
A similar calculation reveals
6
q n−1
(1 − qnn−1 )2
1 |3−[3]qn |
+
+ n
+
|Mn,qn (e2 , ·) − e2 | ≤ 5
q
[3]qn
[n−1]qn
[n−1]qn
[n−2]2qn
≤
1
1
6
1 q n−1
1
1
q 2n−7 + 2qnn−6
|3−[3]qn | + 5
+ 5 n
+ 5
+ n
.
5
2
q
q [n−1]qn
q [n−1]qn
q [n−2]qn
[n−2]2qn
By using (3.13) and (3.14) we obtain kMn,qn (e2 , ·) − e2 k < .
Define the following sets
B = {n ∈ N : kMn,qn (e2 ; ·) − e2 k ≥ },
B1 := {n ∈ N : (3 − [3]qn ) ≥ /3},
1
B2 := n ∈ N :
≥ /36 ,
[n − 1]qn
1
B3 := n ∈ N :
≥ /18 .
[n − 2]qn
S
S
Thus we obtain B ⊆ B1 B2 B3 i.e. δ(B) ≤ δ(B1 ) + δ(B2 ) + δ(B3 ) = 0.
Hence
st − lim kMn,qn (e2 ; ·) − e2 k = 0.
(3.17)
n
Thus by using (3.11), (3.16), (3.17) and Theorem A, result holds. This completes
the proof of Theorem 3.1.
Corollary 3.1. Let (qn )n be a sequence satisfying lim qn = 1, then for all f ∈ CB ([0, 1])
n
we have
lim kMn,q (f ; ·) − f k = 0.
n
Follows by just replacing statistical convergence with uniform convergence.
R EFERENCES
[1] Abel, U., Gupta, V. and Ivan, M., Rate of convergence of Kantorovitch variant of Meyer-König Zeller
operators, Mathematical Inequality and Applications 8 (2005), No. 1, 135-146
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Chaos Solitons and Fractals (in press)
[3] Andrews, G. E., Askey, R. and Roy, R., Special Functions, Cambridge University Press, 1999
[4] Doğru, O., Duman, O. and Orhan, C., Statistical approximation by generalized Meyer-König and
Zeller type operators, Studia Sci. Math. Hungar. 40 (2003), No. 3, 359-371
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V. Gupta and H. Sharma
[5] Doğru, O. and Duman, O., Statistical approximation of Meyer-König and Zeller operators based on
q-integers, Publ. Math. Debrecen 68 (2006) No. 1-2, 199-214
[6] Doğru, O. and Gupta, V., Korovkin-type approximation properties of bivariate q-Meyer-König and Zeller
operators, Calcolo 43 (2006), No. 1, 51-63
[7] Ernst, T., The history of q-calculus and a new method, U.U.D.M. Report 16, Uppsala, Departament
of Mathematics, Uppsala University, 2000
[8] Gadjiev, A. D. and Orhan, C., Some approximation theorems via statistical convergence approximation,
Rocky Mountain J. Math. 32 (2002), 129-138
[9] Kac, V. and Cheung, P., Quantum calculus, Universitext, Springer-Verlag, New York, 2002
[10] Radu, C.,Statistical approximation by some linear operators of discrete type, Miskolc Math. Notes 9
(2008), No. 1, 61 -68
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29 (2000), No. 2, 221-229
S CHOOL OF A PPLIED S CIENCES
N ETAJI S UBHAS I NSTITUTE OF T ECHNOLOGY
S ECTOR 3 D WARKA
110078 N EW D ELHI , I NDIA
E-mail address: vijaygupta2001@hotmail.com
R AYAT AND B AHRA I NSTITUTE OF P HARMACY
V ILLAGE S AHAURAN K HARAR D ISTT.
M OHALI P UNJAB , I NDIA
D EPARTMENT OF M ATHEMATICS
D R . B.R. A MBEDKAR N ATIONAL I NSTITUTE OF T ECHNOLOGY
J ALANDHAR P UNJAB , I NDIA
E-mail address: honeycool 83@yahoo.co.in
E-mail address: pro.sharma.h@gmail.com