MATEMATIQKI VESNIK
originalni nauqni rad
research paper
65, 2 (2013), 187–196
June 2013
ON CONVERGENCE OF q-CHLODOVSKY-TYPE
MKZD OPERATORS
Harun Karsli and Vijay Gupta
Abstract. In the present paper, we define a new kind of MKZD operators for functions
defined on [0, bn ], named q-Chlodovsky-type MKZD operators, and give some approximation
properties.
1. Introduction
For a function defined on the interval [0, 1], the Meyer-König and Zeller operators Mn (f, x) [10] are defined as
µ
¶
∞
X
k
Mn (f ; x) =
mn,k (x) f
(1.1)
n+k
k=0
¡n+k−1¢
where mn,k =
xk (1 − x)n . In 1989 Guo [2] introduced the integrated
k
fn by the means of the operators (1.1), to
Meyer-König and Zeller operators M
approximate Lebesgue integrable functions on the interval [0, 1]. Such operators
have been defined as
Z
∞
X
fn (f ; x) =
M
m
e n,k (x)
f (t) dt
(1.2)
Ik
k=0
¡
¢ k
k+1
k
, n+k+1
] and m
e n,k (x) = (n + 1) n+k+1
x (1 − x)n . Similar results
where Ik = [ n+k
k
may be also found in the papers [3, 4].
Recently, Karsli [8] defined the following MKZD operators for functions defined
on [0, bn ], named Chlodovsky-type MKZD operators as
µ ¶ Z bn
µ ¶
∞
X
n+k
x
t
Ln (f ; x) =
mn,k
f (t)bn,k
dt, 0 ≤ x, t ≤ bn ,
(1.3)
bn
bn
bn
0
k=0
2010 AMS Subject Classification: 41A25, 41A36
Keywords and phrases: q-Chlodovsky-type MKZD operators; modulus of continuity; PeetreK functional; Lipschitz space.
187
188
H. Karsli, V. Gupta
where (bn ) is a positive increasing sequence with the properties
lim bn = ∞ and lim
n→∞
n→∞
bn
=0
n
¡
¢k
n−1
and bn,k (t) = n n+k
. We now deal with the q-analogue of Chlodovskyk t (1 − t)
type MKZD operators Ln,q , defined as
µ ¶ Z bn
µ ¶
∞
X
[n + k]q
x
qt
Ln,q (f ; x) =
mn,k,q
q −k f (t)bn,k,q
dq t, 0 ≤ x ≤ bn ,
bn
bn
bn
0
k=0
(1.4)
where
¸
·
n−1
Y
n+k−1
xk
(1 − q s x)
mk,n,q (x) =
k
q
s=0
and
·
bn,k,q (t) = [n]q
n+k
k
¸
tk
q
n−2
Y
(1 − q s t) (0 ≤ t, x ≤ 1),
s=0
provided the q-integral and the infinite series on the r.h.s. of (1.4) are well-defined.
It can be easily verified that in the case q = 1 the operators defined by (1.4) reduce
to the Chlodovsky-type MKZD operators defined by (1.3).
Actually the q-analogue of the linear positive operators was started in the last
decade when Phillips [11] first introduced q-Bernstein polynomials, and later their
Durrmeyer variants were studied and discussed in [5, 6]. Very recently Govil and
Gupta [1] studied the approximation properties of q-MKZD operators. Here our
aim is to study the q-analogue of summation-integral-type CMKZD operators. We
shall prove that the operators Ln,q f being defined in (1.4) converge to the limit f .
Before getting onto the main subject, we first give definitions of q-integer,
q-binomial coefficient and q-integral, which are required in this paper. For any
fixed real number q > 0 and non-negative integer r the q-integer of the number r
is defined by
½
(1 − q r )/(1 − q), q 6= 1
[r]q =
r,
q = 1.
The q-f actorial is defined by
½
[r]q [r − 1]q · · · [1]q , r = 1, 2, 3, . . .
[r]q ! =
1,
r = 0.
and q-binomial coefficient is defined as
· ¸
[n]q !
n
=
,
r q
[r]q ![n − r]q !
for integers n ≥ r ≥ 0. The q-integral is defined as (see [9])
Z a
∞
X
f (x) dq x = (1 − q) a
f (aq n ) q n
0
n=0
189
q-Chlodovsky-type MKZD operators
provided the sum converges absolutely. Note that the series on the right-hand side
is guaranteed to be absolutely convergent as the function f is such that, for some
M > 0, α > −1, |f (x)| < M xα in a right neighbourhood of x = 0.
Definition 1.1. A function f is q-integrable on [0, ∞) if the series
Z ∞
X
f (x) dq x = (1 − q)
f (q n ) q n
0
n∈Z
converges absolutely. We use the notation
(a − b)nq =
n−1
Y
(a − q j b).
j=0
The q-analogue of Beta function (see [7]) is defined as
Z 1
Bq (m, n) =
tm−1 (1 − qt)qn−1 dq t, m, n > 0.
0
Also
Bq (m, n) =
[m − 1]![n − 1]!
.
[m + n − 1]!
2. Auxiliary results
In this section we give certain results, which are necessary to prove our main
theorem.
Lemma 2.1. For s ∈ N,
s
(Ln,q t ) (x) =
bsn
∞
X
µ
mn,k,q
k=0
x
bn
¶
[n + k]q !
[k + s]q !
[k]q !
[k + s + n]q !
.
(2.1)
Proof. We have
µ
¶
qt
mn,k,q
q t bn,k,q
dq t
(Ln,q t ) (x) =
bn
bn
0
k=0
µ ¶ Z bn ·
¸ µ ¶kµ
¶n−1
∞
X
[n + k]q
x
t
qt
n+k−1
=
mn,k,q
ts
1−
dq t.
k
bn
bn
bn q
0
q bn
∞
X
[n + k]q
s
x
bn
¶Z
bn
µ
−k s
k=0
Setting u = t/bn , we get
s
(Ln,q t ) (x) =
=
∞
X
[n + k]q
k=0
∞
X
k=0
bn
[n + k]q
bn
µ
mn,k,q
µ
mn,k,q
x
bn
x
bn
¶
·
bs+1
n
¶
·
bs+1
n
n+k−1
k
n+k−1
k
¸ Z
¸
q
0
1
uk+s (1 − qu)n−1
dq u
q
Bq (k + s + 1, n)
q
190
H. Karsli, V. Gupta
= bsn
bsn
=
∞
X
k=0
∞
X
µ
[n + k]q mn,k,q
µ
mn,k,q
k=0
x
bn
¶
x
bn
¶
[n + k − 1]q ! Γq (k + s + 1) Γq (n)
[n − 1]q ! [k]q ! Γq (k + s + n + 1)
[n + k]q !
[k + s]q !
[k]q !
[k + s + n]q !
.
For s = 0, 1 and 2 in (2.1), we get respectively
∞
X
(Ln,q 1) (x) =
µ
mn,k,q
k=0
x
bn
¶
¸ µ ¶k n−1
¶
∞ ·
X
Yµ
x
n+k−1
s x
=
1−q
= 1,
k
bn
bn
q
s=0
k=0
(2.2)
since
Qn−1 ³
s=0
(Ln,q t)(x) = bn
= bn
∞
X
1 − qs
1 − q s bxn
µ
mn,k,q
k=0
n−1
Yµ
1
x
bn
¶X
∞
x
bn
¸ µ ¶k
∞ ·
X
x
n+k−1
´=
.
k
bn
q
k=0
¶
[n + k]q !
[k]q !
[n + k − 1]q !
[k + 1]q !
[n + k + 1]q !
µ ¶k
[k + 1]q
x
[n − 1]q ! [k]q ! [n + k + 1]q bn
¶ ∞
µ ¶k
n−1
Yµ
[k + 1]q [n + k − 1]q
x X [n + k − 2]q !
x
= bn
1 − qs
bn
[n − 1]q ! [k − 1]q ! bn
[n + k + 1]q
[k]q
s=0
k=1
µ
¶
µ
¶
n−1
∞
k
Y
[k + 1]q [n + k − 1]q
x X [n + k − 2]q !
x
= bn
1 − qs
bn
[n − 1]q ! [k − 1]q ! bn
[k]q [n + k + 1]q
s=0
k=1
µ
¶
µ
¶
∞
n−1
k
Y
[k + 1]q [n − 1]q
x X [n + k − 2]q !
x
≥ bn
1 − qs
b
[n
−
1]
!
[k
−
1]
!
b
[k]q [n + 1]q
n
n
q
q
s=0
k=1
¶ ∞
µ ¶k
Yµ
[n − 1]q n−1
x X [n + k − 2]q !
x
=
bn
1 − qs
[n + 1]q s=0
bn
[n − 1]q ! [k − 1]q ! bn
k=1
¶
µ ¶k+1
µ
n−1
∞
[n − 1]q Y
x X [n + k − 1]q ! x
=
bn
1 − qs
[n + 1]q s=0
bn
[n − 1]q ! [k]q ! bn
k=0
µ
¶ Yµ
¶
∞
[n − 1]q x X
[n + k − 1]q ! x k n−1
s x
=
1−q
bn
[n + 1]q bn
[n − 1]q ! [k]q ! bn
bn
s=0
k=0
µ
·
¸
µ
¶
¶
∞
n−1
k Y
[n − 1]q X n + k − 1
x
x
x
1 − qs
=
k
[n + 1]q
b
b
n
n
q
s=0
s=0
k=0
k=0
=
[n − 1]q
[n + 1]q
x,
(2.3)
q-Chlodovsky-type MKZD operators
191
and
(Ln,q t2 )(x) = b2n
∞
X
µ
mn,k,q
x
bn
¶
[n + k]q !
[k + 2]q !
[k + 2 + n]q !
¶ ∞
µ ¶k
n−1
Yµ
[k + 2]q [k + 1]q
x X [n + k − 1]q ! x
= b2n
1 − qs
bn
[n − 1]q ! [k]q ! bn
[k + 2 + n]q [k + 1 + n]q
s=0
k=0
µ
¶
µ
¶
2
n−1
∞
k
Y
1 + q + q [k]q + 2q 2 [k]q + q 3 [k]q
x X [n + k − 1]q ! x
= b2n
1 − qs
bn
[n − 1]q ! [k]q ! bn
[k + 2 + n]q [k + 1 + n]q
s=0
k=0
µ
¶
n−1
Y
x
1
1 − qs
×
≤ b2n
b
[n
−
1]q !
n
s=0
µ ¶
∞
´
X
[n + k − 3]q ! x k ³
2
×
1 + q + q [k]q + 2q 2 [k]q + q 3 [k]q
[k]q !
bn
k=0
µ
¶
µ ¶
∞
n−1
X
Y
[n + k − 3]q ! x k
1
2
s x
= (1 + q) bn
1−q
bn [n − 1]q [n − 2]q
[n − 3]q ! [k]q ! bn
s=0
k=0
¶
µ ¶
∞
X
Yµ
[n + k − 2]q ! x k+1
¡
¢ n−1
1
x
+ q + 2q 2 b2n
1 − qs
bn [n − 1]q
[n − 2]q ! [k]q ! bn
s=0
k=0
¶
µ ¶
n−1
∞
Yµ
X
[n + k − 3]q ! x k
x
1
1 − qs
+ q 3 b2n
[k]q
bn [n − 1]q !
[k − 1]q !
bn
s=0
k=0
k!
k=1
¡
¢
1
x
= (1 + q) b2n
+ q + 2q 2 b2n
[n − 1]q [n − 2]q
[n − 1]q
¶
µ ¶
n−1
∞
´
Yµ
X
[n + k − 3]q ! x k ³
1
s x
3 2
1−q
1 + q [k − 1]q
+ q bn
bn [n − 1]q !
[k − 1]q !
bn
s=0
k=1
¡
¢
1
1
x
= (1 +
+ q + 2q 2 b2n
[n − 1]q [n − 2]q
[n − 1]q bn
¶
µ ¶
n−1
∞
Yµ
X
[n + k − 2]q ! x k+1
1
s x
3 2
1−q
+ q bn
bn [n − 1]q !
[k]q !
bn
s=0
k=0
¶
µ ¶
n−1
∞
Yµ
X
[n + k − 1]q ! x k+2
x
1
1 − qs
+ q 4 b2n
bn [n − 1]q !
[k]q !
bn
s=0
q) b2n
k=0
=
q) b2n
(1 +
[n − 1]q [n − 2]q
¡
¢
+ q + 2q 2 + q 3
bn
x + q 4 x2 .
[n − 1]q
From (2.2), (2.3) and (2.4), an easy computation gives
¡
¢
¢
¡
q + 2q 2 + q 3 bn
(1 + q) b2n
2
Ln,q (t − x) (x) ≤
+
x
[n − 1]q [n − 2]q
[n − 1]q
(2.4)
192
H. Karsli, V. Gupta
"
+ q4 − 2
#
[n − 1]q
+ 1 x2 := An,q (x).
[n + 1]q
(2.5)
1
It is observed here that for 0 < q < 1, one has [n]q → 1−q
as n → ∞. This implies
¢
¡
2
2
that (Ln,q t )(x) and Ln,q (t − x) (x) does not converge to x2 and 0 respectively,
as n → ∞. To obtain some convergence results for q-CMKZD operators defined
in (1.4), we will consider a sequence (qn ) of real numbers such that 0 < qn < 1,
limn→∞ qn = 1, and
bn
= 0.
lim
(2.6)
n→∞ [n]qn
3. Main results
Now we are ready to obtain some convergence results on q-CMKZD operators.
Theorem 3.1. Let (qn ) be a sequence of real numbers such that 0 < qn < 1
and limn→∞ qn = 1. If f ∈ C[0, ∞), we have
q
|(Ln,qn f )(x) − f (x)| ≤ 2ω(f, An,qn (x)),
(3.1)
where ω(f, ·) is the usual modulus of continuity of f in the space of continuous
functions.
Proof. Using (1.4) for q = qn , we have
|(Ln,qn f )(x) − f (x)|
¯∞
¯
µ ¶ Z bn
µ
¶
¯X [n + k]qn
¯
x
qn t
= ¯¯
mn,k,qn
qn−k f (t)bn,k,qn
dqn t − f (x)¯¯
bn
bn
bn
0
k=0
µ
¶
¶
µ
Z
∞
bn
X
[n + k]qn
x
qn t
−k
mn,k,qn
dqn t
qn |f (t) − f (x)| bn,k,qn
≤
bn
bn
bn
0
k=0
¶
¶
µ ¶ Z bn
µ
µ
∞
X
[n + k]qn
x
|t − x|
qn t
≤
mn,k,qn
+ 1 ω(f, δ)bn,k,qn
dqn t
qn−k
bn
bn
δ
bn
0
k=0
µ ¶ Z bn
µ
¶
∞
X
[n + k]qn
x
qn t
= ω(f, δ)
mn,k,qn
qn−k bn,k,qn
d qn t
bn
bn
bn
0
k=0
µ
¶
µ ¶ Z bn
∞
qn t
ω(f, δ) X [n + k]qn
x
−k
+
mn,k,qn
qn |t − x| bn,k,qn
d qn t
δ
bn
bn
bn
0
k=0
¢
ª1/2
ω(f, δ) ©¡
Ln,qn (t − x)2 (x)
δ
ω(f, δ)
1/2
≤ ω(f, δ) +
{An,qn (x)}
δ
Now, if we choose δ 2 = An,qn (x), we get
≤ ω(f, δ) +
|(Ln,qn f )(x) − f (x)| ≤ 2ω(f,
and the proof of Theorem 3.1 is thus complete.
q
An,qn (x)),
193
q-Chlodovsky-type MKZD operators
It is easy to see that, the right-hand side of formula (3.1) can diverge. Indeed,
for x = b2n we cannot guarantee δ → 0 as n → ∞ .
From Lemma 2.1 and Theorem 3.1, we can immediately give the following
Bohman-Korovkin-type theorem.
Theorem 3.2. Let (qn ) be a sequence of real numbers such that 0 < qn < 1
and limn→∞ qn = 1. Then, for f ∈ C[0, ∞), the sequence Ln,qn (f, x) converges
uniformly to f (x) on any closed finite subinterval [0, A], where A > 0 being a
constant.
Definition 3.3. For f ∈ C[a, b] and t > 0, the Peetre-K Functional are
defined by
n
o
K(f, δ) := inf
kf
−
gk
+
t
kgk
.
2 [a,b]
C[a,b]
C
2
g∈C [a,b]
Theorem 3.4. If g ∈ C 2 [0, A], then
|(Ln,q g)(x) − g(x)| ≤ An,q (x) kgkC 2 [0,A] ,
where A > 0 is a constant.
Proof. By Taylor formula with integral reminder term, we write
Z t−x
0
g(t) = g(x) + (t − x)g (x) +
(t − x − u)2 g 00 (x + u) du.
(3.2)
0
If we apply the operator (1.4) to (3.2), we get
|(Ln,q g)(x) − g(x)|
¯
µ
µZ
¯ 0
¯
= ¯g (x)(Ln,q (t − x))(x) + Ln,q
t−x
0
¶¶ ¯
¯
(t − x − u) g (x + u) du (x)¯¯
2 00
≤ kg 0 kC[0,A] |(Ln,q (t − x))(x)|
¯µ
µZ t−x
¶¶ ¯
¯
¯
00
2
¯
+ kg kC[0,A] ¯ Ln,q
(t − x − u) du (x)¯¯.
0
Since
Z
0
t−x
(t − x − u)2 du =
(t − x)2
,
2
one gets from (2.5)
1/2
|(Ln,q g)(x) − g(x)| ≤ kg 0 kC[0,A] {An,q (x)}
+ kg 00 kC[0,A] An,q (x).
Now noting that
kgkC 2 [a,b] = kgkC[a,b] + kg 0 kC[a,b] + kg 00 kC[a,b] ,
we get
|(Ln,q g)(x) − g(x)| ≤ An,q (x) kgkC 2 [0,A] ,
194
H. Karsli, V. Gupta
and this completes the proof of Theorem 3.4.
Now, we are ready to prove the following theorem.
Theorem 3.5. Let (qn ) be a sequence of real numbers such that 0 < qn < 1
and limn→∞ qn = 1. If f ∈ C[0, ∞), then
k(Ln,qn f ) − f kC[0,A] ≤ 2K(f, Bn,qn ),
where Bn,qn is the maximum value of An,qn (x) on [0, A], A > 0 is a constant;
namely,
"
#
¡
¢
[n − 1]q
q + 2q 2 + q 3 bn
(1 + q) b2n
4
Bn,q =
+
A+ q −2
+ 1 A2 .
[n − 1]q [n − 2]q
[n − 1]q
[n + 1]q
Proof. By the linearity property of (Ln,qn ), we get
|(Ln,qn f )(x) − f (x)|
≤ |(Ln,qn f )(x) − (Ln,qn g)(x)| + |(Ln,qn g)(x) − g(x)| + |g(x) − f (x)|
≤ kf − gkC[0,A] |(Ln,qn 1)(x)| + kf − gkC[0,A] + |(Ln,qn g)(x) − g(x)| .
From Theorem 3.4, one has
|(Ln,qn f )(x) − f (x)| ≤ 2 kf − gkC[0,A] + An,qn (x) kgkC 2 [0,A] ,
and hence
k(Ln,qn f ) − f kC[0,A] ≤ 2 kf − gkC[0,A] + Bn,qn kgkC 2 [0,A] .
(3.3)
If we take the infimum on the right-hand side of (3.3) over all g ∈ C 2 [0, A], we get
k(Ln,qn f ) − f kC[0,A] ≤ 2K(f, Bn,qn ).
This completes the proof.
Theorem 3.6. Let (qn ) be a sequence of real numbers such that 0 < qn < 1
and limn→∞ qn = 1. If f ∈ Lipα
M [0, ∞), then for any A > 0 and x ∈ [0, A] the
inequality
α
|(Ln,qn f )(x) − f (x)| ≤ M {Bn,qn } 2
holds with the constant M , which is independent of n and Bn,qn is as defined in
Theorem 3.5.
Proof. For convenience we write Ln,qn (f ; x) instead of (Ln,qn f )(x). Note that
|Ln,qn (f ; x) − f (x)| ≤ Ln,qn (|f (t) − f (x)| ; x)
µ ¶ Z bn
µ
¶
∞
X
[n + k]qn
x
qn t
=
mn,k,qn
qn−k |f (t) − f (x)| bn,k,qn
dqn t
bn
bn
bn
0
k=0
195
q-Chlodovsky-type MKZD operators
Z
bn
≤M
0
If we choose p1 =
qn−k |t − x|
α
∞
X
[n + k]q
k=0
2
α
and p2 =
2
2−α ,
bn
then
1
p1
µ
n
mn,k,qn
+
1
p2
x
bn
¶
µ
bn,k,qn
qn t
bn
¶
dqn t.
= 1. Therefore
|Ln,qn (f ; x) − f (x)|
µ ¶
µ
¶¾ p1
Z bn ½
∞
X
[n + k]qn
1
x
qn t
2
|t − x| qn−k
bn,k,qn
≤M
mn,k,qn
×
bn
bn
bn
0
k=0
½
¶¾ p1
µ ¶
µ
∞
X
[n + k]qn
2
x
qn t
−k
× qn
mn,k,qn
bn,k,qn
dqn t.
bn
bn
bn
k=0
By Hölder inequality, we have
|Ln,qn (f ; x) − f (x)|
½Z bn
µ ¶
µ
¶
¾ p1
∞
X
[n + k]qn
1
x
qn t
2
≤M
qn−k |t − x|
mn,k,qn
bn,k,qn
dqn t
×
bn
bn
bn
0
k=0
½Z bn
µ ¶
µ
¶
¾ p1
∞
X
[n + k]qn
2
x
qn t
−k
mn,k,qn
bn,k,qn
dqn t
×
qn
bn
bn
bn
0
k=0
½Z bn
¶
¶
¾ α2
µ
µ
∞
X [n + k]q
x
qn t
2
n
=M
qn−k |t − x|
mn,k,qn
bn,k,qn
dqn t .
bn
bn
bn
0
k=0
From (2.5) we obtain
α
|Ln,qn (f ; x) − f (x)| ≤ M {An,qn (x)} 2 .
This implies that for x ∈ [0, A]
α
|(Ln,qn f )(x) − f (x)| ≤ M {Bn,qn } 2
which in view of (2.5) and (2.6) tends to zero as n → ∞.
Acknowledgement. The authors are thankful to the referees for their valuable remarks and suggestions.
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(received 16.06.2011; in revised form 06.02.2012; available online 15.03.2012)
Abant Izzet Baysal University, Faculty of Science and Arts, Department of Mathematics, 14280
Golkoy Bolu
E-mail: karsli h@ibu.edu.tr
School of Applied Sciences, Netaji Subhas Institute of Technology, Sector 3 Dwarka New Delhi110075, India
E-mail: vijaygupta2001@hotmail.com