Bulletin of Mathematical Analysis and Applications
ISSN: 1821-1291, URL: http://www.bmathaa.org
Volume 4 Issue 3 (2012), Pages 87-98
INTEGRAL TYPE MODIFICATION FOR q−LAGUERRE
POLYNOMIALS
(COMMUNICATED BY PROFESSOR F. MARCELLAN)
GÜRHAN İÇÖZ, SERHAN VARMA AND FATMA TAŞDELEN
Abstract. Özarslan gave the approximation properties of linear positive operators including the q-Laguerre polynomials in [13]. In this paper, we will
give Kantorovich type generalization for this operator with the help of Riemann type q-integral. We also get approximation properties for the generalized
operator with modulus.
1. Introduction
In 1960, The Meyer-König and Zeller operators
∞
X
n+k k
k
n+1
x (1 − x)
Mn (f ; x) =
f
k+n+1
k
k=0
(0 ≤ x < 1) were introduced by Meyer-König and Zeller in [11].
In order to give the monotonicity properties, Cheney and Sharma [1] modified
these operators as:
∞
X
k
n+k k
n+1
∗
Mn (f ; x) =
f
x (1 − x)
k+n
k
k=0
(0 ≤ x < 1).
In [1], they also introduced the operators
X
∞
k
tx
(n)
Pn (f ; x) = exp
f
Lk (t) xk (1 − x)n+1
1−x
k+n
k=0
(n)
(0 ≤ x < 1 and − ∞<t≤
0) where Lk (t) denotes the Laguerre polynomials.
n+k
(n)
Since Lk (0) =
, then Mn∗ (f ; x) is the special case of the operators
k
Pn (f ; x).
02010 Mathematics Subject Classification: 41A25, 41A30, 41A35.
Keywords and phrases. q-Laguerre polynomials, generating functions, Riemann type q-integral,
second order modulus of smoothness.
c 2012 Universiteti i Prishtinës, Prishtinë, Kosovë.
Submitted May 9, 2012. Accepted August 16, 2012.
87
88
GÜRHAN İÇÖZ, SERHAN VARMA AND FATMA TAŞDELEN
The q-type generalization of the linear positive operators was initiated by Phillips
in [14]. He introduced the q-type generalization of the classical Bernstein operator
and obtained the rate of convergence and the Voronovskaja type asymptotic formula
for these operators.
q-Laguerre polynomials were defined by (Hahn [7, p. 29], Jackson [8, p. 57] and
Moak [12, p. 21, eq. 23])
n
k
k
k
q α+1 ; q n X
(q −n ; q)k q (2) (1 − q) q n+α+1 x
(α)
Ln (x; q) =
(q; q)n
(q α+1 ; q)k (q; q)k
k=0
where
(α; q)n =
1
(1 − α) (1 − αq) ... 1 − αq n−1
;n = 0
.
; n ∈ N, α ∈ C
Moak gave the following recurrence relation [12, p. 29, eq. 4.14] and generating
function [12, p. 29, eq. 4.17] for the q-Laguerre polynomials:
(α+1)
(α)
(α)
tLk−1 (t; q) = [k + α] q −α−k Lk−1 (t; q) − [k] q −α−k Lk (t; q)
(Re α > −1, k = 1, 2, ...) ,
Fα (x, t)
2
∞
xq α+1 ; q ∞ X
q m +αm [− (1 − q) xt]m
(x; q)∞ m=0 (q; q)m (xq α+1 ; q)m
=
∞
X
=
(α)
Lk (t; q) xk
(Re α > 1)
(1.1)
k=0
where
(a; q)∞ =
∞
Y
j=0
1 − aq j ,
(a ∈ C) .
Trif [16] defined the Meyer-König and Zeller operators based on q-integer as follows:
n
∞
Y
X
[k]
n+k k
Mn,q (f ; x) =
1 − qj x
f
x
k
[k + n]
j=0
(0 ≤ x < 1) where
1 − q k / (1 − q) ; q 6= 1
,
1
;q = 1
[1] [2] ... [k] ; k ≥ 1
[k]! =
1
;k = 0
[k] =
and
k=0
[n + k]!
n+k
=
k
[n]! [k]!
(k, n ∈ N) for q ∈ (0, 1].
In [13], Özarslan defined the q-analogue for Pn (f ; x) operators as follow:
∞
X
1
[k]
(n)
Pn,q (f ; x) =
f
Lk (t; q) xk
(1.2)
Fn (x, t)
[k + n]
k=0
q − Laguerre Kantorovich Operators
89
where x ∈ [0, 1], t ∈ (−∞, 0], q ∈ (0, 1] and {Fn (x, t)}
n∈N is
the generating functions
n
Q
n+k
(n)
for the q-Laguerre polynomials. Since Lk (0; q) =
and Fn (x, 0) =
1 − qj x ,
k
j=0
then Mn,q (f ; x) is the special case of the operators Pn,q (f ; x).
Let us recall the concepts of q-differential, q-derivative and q-integral respectively.
For an arbitrary function f (x), the q-differential is given by
dq f (x) = f (qx) − f (x) .
For an arbitrary function f (x), the q-derivative is defined as
Dq f (x) =
dq f (x)
f (qx) − f (x)
=
.
dq x
(q − 1) x
Now suppose that 0 < a < b, 0 < q < 1 and f is a real-valued function. The qJackson integral of f over the interval [0, b] and a general interval [a, b] are defined
by (see [9])
Za
∞
X
f (t) dq t = (1 − q) a
f qj a qj
j=0
0
and
Zb
f (t) dq t =
a
Zb
0
f (t) dq t −
Za
f (t) dq t
0
respectively.
It is clear that q-Jackson integral of f over an interval [a, b] contains two infinite
sums, so some problems are encountered in deriving the q-analogues of some wellknown integral inequalities which are used to compute order of approximation of
linear positive operators containing q-Jackson integral. In order to overcome these
problems Gauchman [5] and Marinković et al. [10] introduced a new type of qintegral. This new q-integral is called as Riemann type q-integral and defined by
Zb
a
f (t) dR
q t = (1 − q) (b − a)
∞
X
j=0
f a + (b − a) q j q j
where a, b and q are some real numbers such that 0 < a < b and 0 < q < 1.
Contrary to the classical definition of q-integral, this definition includes only points
within the interval of integration.
Now, we give a Kantorovich type generalization of operators Pn , Mn∗ , Mn,q and
Pn,q . This Kantorovich type generalization was studied by Dalmanoğlu [3], Radu
[15] and etc. We consider the sequence of Kantorovich type linear positive operators
as follow:


[k+1]/[n+k]
Z
∞
X
1

 −k
(n)
f (t) dR
[n + k] Lk (t; q) xk ,
(Kn,q f ) (x, t) =

q t q
Fn,q (x, t)
k=0
[k]/[n+k]
(1.3)
where x ∈ [0, 1] , t ∈ (−∞, 0] , q ∈ (0, 1], n > 1 and {Fn (x, t)}n∈N is the generating
functions for the q-Laguerre polynomials which was given in (1.1).
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GÜRHAN İÇÖZ, SERHAN VARMA AND FATMA TAŞDELEN
2. Approximation Properties of the (Kn,q f ) (x, t) Operators
We have the following theorem for the convergence of (Kn,q f ) (x, t) operators.
Theorem 1. Let q := qn be a sequence satisfying limqn = 1 and 0 < qn < 1. If
n
f ∈ C [0, 1] and
(0 < b < 1) .
|t|
[n]
→ 0 (n → ∞) then (Kn,q f ) converges to f uniformly on [0, b]
Proof. By Korovkin’s theorem, it is sufficient for us to prove that (Kn,q f ) is a
positive linear operator and that the desired convergence occurs whenever f is a
quadratic function. It is obvious that (Kn,q f ) is linear and positive operators.


[k+1]/[n+k]
Z
∞
X
1

 −k
(n)
(Kn,q e0 ) (x, t) =
dR
[n + k] Lk (t; q) xk .

q t q
Fn,q (x, t)
k=0
Since
[k]/[n+k]
[k+1]/[n+k]
Z
dR
q t=
qk
[n + k]
[k]/[n+k]
and from (1.1), we get
(Kn,q e0 ) (x, t) = 1.
(2.1)
By considering the function f (s) = e1 (s) = s, we obtain


[k+1]/[n+k]
Z
∞
X
1

 −k
(n)
(Kn,q e1 ) (x, t) =
tdR
[n + k] Lk (t; q) xk .

q t q
Fn (x, t)
k=0
[k]/[n+k]
One can easily compute that
[k+1]/[n+k]
Z
tdR
q t
=
[k]/[n+k]
qk
2
[n + k]
qk
[k] +
[2]
,
then we have
∞
X
1
1
(Kn,q e1 ) (x, t) =
Fn (x, t)
[n + k]
k=0
k
qk
(n)
[k] +
Lk (t; q) xk .
[2]
Since q < 1 for 0 < q < 1 and [k + n] ≥ [n], we can write
1
(Kn,q e1 ) (x, t) ≤ Pn,q (e1 ; x) +
Pn,q (e0 ; x) .
[2] [n]
tx
If we use Pn,q (e0 ; x) = 1 and Pn,q (e1 ; x) ≤ x −
from [13], then we
[n] (1 − bq n+1 )
get
tx
1
(Kn,q e1 ) (x, t) − x ≤ −
+
.
(2.2)
n+1
[n] (1 − bq
) [2] [n]
On the other hand, we see
(Kn,q e1 ) (x, t) ≥ Pn,q (e1 ; x) .
If we use Pn,q (e1 ; x) ≥ x from [13], we obtain
(Kn,q e1 ) (x, t) ≥ x.
(2.3)
q − Laguerre Kantorovich Operators
91
From (2.2) and (2.3), we have
0 ≤ (Kn,q e1 ) (x, t) − x ≤ −
tx
1
+
.
n+1
[n] (1 − bq
) [2] [n]
(2.4)
From (2.4) , it is obvious that
k(Kn,q e1 ) (x, t) − xkC[0,b] ≤
|t| b
1
+
.
n+1
[n] (1 − bq
) [2] [n]
(2.5)
We proceed with the consideration of the function f (s) = e2 (s) = s2 .


[k+1]/[n+k]
Z
∞
X
1

 −k
(n)
t2 dR
[n + k] Lk (t; q) xk .
(Kn,q e2 ) (x, t) =

q t q
Fn (x, t)
k=0
[k]/[n+k]
One can easily see that
[k+1]/[n+k]
Z
t2 dR
q t
[k]/[n+k]
qk
q 2k
2q k
2
=
[k] +
.
[k] +
3
[2]
[3]
[n + k]
So, we acquire
(Kn,q e2 ) (x, t) =
∞
X
1
1
2
Fn (x, t)
[n + k]
k=0
2q k
q 2k
(n)
2
[k] +
[k] +
Lk (t; q) xk .
[2]
[3]
Since q k < 1 for 0 < q < 1 and [k + n] ≥ [n], we can write
(Kn,q e2 ) (x, t) ≤ Pn,q (e2 ; x) +
2
1
Pn,q (e1 ; x) +
Pn,q (e0 ; x) .
[2] [n]
[3] [n]2
tx
and
[n] (1 − bq n+1 )
t x2 + x
x
Pn,q (e2 ; x) ≤ x2 −
+
,
[n] (1 − bq n+1 ) [n]
If we use Pn,q (e0 ; x) = 1, Pn,q (e1 ; x) ≤ x −
from [13], then we get
2
(Kn,q e2 ) (x, t) − x
≤
t x2 + x
x
1
−
+
+
n+1
[n] (1 − bq
) [n] [3] [n]2
2
tx
+
x−
.
[2] [n]
[n] (1 − bq n+1 )
(2.6)
On the other hand, using the equality
2
s2 = (s − x) + 2xs − x2
we may write
(Kn,q e2 ) (x, t) − x2 = Kn,q (e1 − x)2 (x, t) + 2x (Kn,q (e1 − x)) (x, t) .
By (2.4) and positivity of Kn,q , it follows that
(Kn,q e2 ) (x, t) − x2 ≥ 0.
(2.7)
92
GÜRHAN İÇÖZ, SERHAN VARMA AND FATMA TAŞDELEN
Thus from (2.6) and (2.7), we have
0 ≤
t x2 + x
x
1
+
+
(Kn,q e2 ) (x, t) − x ≤ −
n+1
[n] (1 − bq
) [n] [3] [n]2
2
tx
x−
.
+
[2] [n]
[n] (1 − bq n+1 )
2
(2.8)
From (2.8), it is clear that
(Kn,q e2 ) (x, t) − x2
≤
C[0,b]
|t| b2 + b
2 |t| b
+
[n] (1 − bq n+1 ) [2] [n]2 (1 − bq n+1 )
2
b
1
+ 1+
+
.
(2.9)
[2] [n] [3] [n]2
After replacing q by a sequence qn such that lim qn = 1, we have from (2.1), (2.5)
n
and (2.9) (Kn,q ei ) (x, t0 ) ⇒ ei (x) = xi (i = 0, 1, 2.) on [0, b] .
3. Rates of Convergence
In this section, we compute the rates of convergence by means of modulus of
continuity, elements of Lipschitz class and second order modulus of smoothness.
Let f ∈ C [0, b]. The modulus of continuity of f denotes by ω (f, δ), is defined
to be
ω (f, δ) = sup |f (s) − f (x)| .
s,x∈[0,b]
|s−x|<δ
It is well known that a necessary and sufficient condition for a function f ∈ C [0, b]
is
lim ω (f, δ) = 0.
δ→0
It is also well known that for any δ > 0 and each s ∈ [0, b]
|s − x|
|f (s) − f (x)| ≤ ω (f, δ) 1 +
.
δ
(3.1)
Before giving the theorem on the rate of convergence of the operator Kn,q f , let us
first examine its second moment:
2
Kn,q (e1 − x) (x, t) = (Kn,q e2 ) (x, t) − x2 − 2x [(Kn,q e1 ) (x, t) − x] ,
Kn,q (e1 − x)2 (x, t)
C[0,b]
≤
(Kn,q e2 ) (x, t) − x2
C[0,b]
+2 kxkC[0,b] k(Kn,q e1 ) (x, t) − xkC[0,b] .
Using (2.9) and (2.5), we can write
Kn,q (e1 − x)2 (x, t)
≤
C[0,b]
|t| 3b2 + b
2 |t| b
+
[n] (1 − bq n+1 ) [2] [n]2 (1 − bq n+1 )
4
b
1
+ 1+
+
.
(3.2)
[2] [n] [3] [n]2
The following theorem gives the rate of convergence of the operator Kn,q f to the
function f by means of modulus of continuity.
q − Laguerre Kantorovich Operators
93
Theorem 2. Let q := qn be a sequence satisfying limqn = 1 and 0 < qn < 1. For
n
all f ∈ C [0, b] and
|t|
[n]
→ 0 (n → ∞)
k(Kn,q f ) (x, t) − f (x)kC[0,b] ≤ 2ω (f, δn )
(3.3)
where
#1/2
|t| 3b2 + b
4
2 |t| b
b
1
δn =
+
+ 1+
+
.
[n] (1 − bq n+1 ) [2] [n]2 (1 − bq n+1 )
[2] [n] [3] [n]2
"
Proof. Let f ∈ C [0, b]. By using (3.1), linearity and monotonicity of Kn,q f , we
obtain
|(Kn,q f ) (x, t) − f (x)| ≤
(Kn,q |f (s) − f (x)|) (x, t)
|s − x|
ω (f, δ) Kn,q 1 +
(x, t)
δ
1
ω (f, δ) 1 + (Kn,q |s − x|) (x, t) .
δ
≤
=
(3.4)
In [2], Dalmanoğlu and Doğru show that the Riemann type q-integral is a positive
operator and it satisfies the following Hölder’s inequality:
1
+ n1 = 1. Then
Let 0 < a < b, 0 < q < 1 and m
1/m
Rq (|f g| ; a; b) ≤ (Rq (|f |m ; a; b))
(Rq (|g|n ; a; b))
1/n
.
(3.5)
Therefore, by using the Cauchy-Schwarz inequality for the Riemann type q-integral
with m = 2 and n = 2 in (3.5), we have

1/2 
1/2
[k+1]/[n+k]
[k+1]/[n+k]
[k+1]/[n+k]
Z
Z
Z

 

2
|t − x| dR
(t − x) dR
dR
.

q t≤
q t
q t
[k]/[n+k]
[k]/[n+k]
[k]/[n+k]
Now applying the Cauchy-Schwarz inequality for the sum with p = 12 and q =
and taking into consideration (3.2), one can write
(Kn,q |s − x|) (x, t) ≤

1/2


[k+1]/[n+k]


Z
∞
X

1

(n)
2 R  −k
k
(t − x) dq t q [n + k] Lk (t; q) x



Fn,q (x, t)


k=0
×
=
≤
[k]/[n+k]


∞
X

1



F
(x,
t)
n,q
k=0
Kn,q (e1 − x)
1
2
2
[k+1]/[n+k]
Z
[k]/[n+k]
(x, t)
1/2

 −k
(n)
dR
[n + k] Lk (t; q) xk
q t q
1/2
((Kn,q e0 ) (x, t))
1/2




#1/2
t 3b2 + b
2tb
4
b
1
−
−
+ 1+
+
(3.6)
.
[n] (1 − bq n+1 ) [2] [n]2 (1 − bq n+1 )
[2] [n] [3] [n]2
"
If we write (3.6) in (3.4) and choose δ = δn , then we arrive at the desired result.
94
GÜRHAN İÇÖZ, SERHAN VARMA AND FATMA TAŞDELEN
Next, we compute the approximation order of operator Kn,q f in term of the
elements of the usual Lipschitz class.
Let f ∈ C [0, b] and 0 < α ≤ 1. We recall that f belongs to LipM (α) if the
inequality
α
|f (ζ) − f (η)| ≤ M |ζ − η| ; ζ, η ∈ [0, b]
(3.7)
holds.
Theorem 3. Let q := qn be a sequence satisfying limqn = 1 and 0 < qn < 1. For
n
all f ∈ LipM (α) and
|t|
[n]
→ 0 (n → ∞)
k(Kn,q f ) (x, t) − f (x)kC[0,b] ≤ M δnα
(3.8)
where δn is the same as in Theorem 2.
Proof. Let f ∈ C [0, b]. By (3.7), linearity and monotonicity of Kn,q f , we have
|(Kn,q f ) (x, t) − f (x)| ≤
(Kn,q |f (s) − f (x)|) (x, t)
M
Fn,q (x, t)


[k+1]/[n+k]
Z
∞
X

 −k
(n)
α
×
|t − x| dR
[n + k] Lk (t; q) xk .

q t q
≤
k=0
[k]/[n+k]
(3.9)
On the other hand, by using the Hölder inequality for the Riemann type q-integral
2
with m = α2 and n = 2−α
, we have

[k+1]/[n+k]
Z
[k]/[n+k]

α
|t − x| dR
q t≤ 
α/2 
[k+1]/[n+k]
Z

2
(t − x) dR
q t
[k]/[n+k]


[k+1]/[n+k]
Z
[k]/[n+k]
(2−α)/2

dR
q t
.
If we write above inequality in (3.9) and then apply the Hölder inequality for the
2
sum with p = α2 and q = 2−α
, we get
|(Kn,q f ) (x, t) − f (x)| ≤



α/2
[k+1]/[n+k]
Z
∞
1
X

 −k

(n)
2
(t − x) dR
[n + k] Lk (t; q) xk 
M

q t q
Fn,q (x, t)
k=0

∞
X
×
k=0
[k]/[n+k]

1


Fn,q (x, t)
and so we have
[k+1]/[n+k]
Z
[k]/[n+k]
(2−α)/2

 −k

(n)
dR
[n + k] Lk (t; q) xk 
q t q
|(Kn,q f ) (x, t) − f (x)| ≤ M
Kn,q (e1 − x)
2
α/2
(x, t)
.
If we use (3.2) and choose δ = δn , then the proof is completed.
q − Laguerre Kantorovich Operators
95
Finally, we establish a local approximation theorem for the operator Kn,q f .
Let Ω2 := {g ∈ C [0, b] : g ′ , g ′′ ∈ C [0, b]}. For any δ > 0, the Peetre’s Kfunctional is defined by
K2 (ϕ; δ) = inf 2 {kϕ − gk + δ kg ′′ k}
g∈Ω
where k.k is the uniform norm on C [0, b] (see [6]). From [4](p.177, Theorem 2.4),
there exists an absolute constant C > 0 such that
√ K2 (f ; δ) ≤ Cω2 f ; δ
(3.10)
where the second order modulus of smoothness of f ∈ C [0, b] is denoted by
√ ω2 f ; δ =
sup
√
0<h≤ δ
sup
x,x+2h∈[0,b]
|f (x + 2h) − 2f (x + h) + f (x)| .
We recall the usual modulus of continuity of f ∈ C [0, b] by
√ ω f; δ =
sup
√
0<h≤ δ
sup
x,x+h∈[0,b]
Now consider the following operator
(Ln,q f ) (x, t) = (Kn,q f ) (x, t) − f x −
|f (x + h) − f (x)| .
tx
1
+
[n] (1 − bq n+1 ) [2] [n]
+ f (x)
(3.11)
for x ∈ [0, 1] .
Lemma 4. Let g ∈ Ω2 . Then we have
(
−t 3x2 + x
2tx
x
4
|(Ln,q g) (x, t) − g (x)| ≤
−
+
1
+
[n] (1 − bq n+1 ) [2] [n]2 (1 − bq n+1 )
[2] [n]
2 )
1
−tx
1
+
+
kg ′′ k . (3.12)
2 +
[n] (1 − bq n+1 ) [2] [n]
[3] [n]
Proof. By definition of the operator Ln,q f , (2.1) and (2.4), it is seen that
(Ln,q (s − x)) (x, t) =
=
(Kn,q (s − x)) (x, t) +
0.
tx
1
−
[n] (1 − bq n+1 ) [2] [n]
Let x ∈ [0, 1] and g ∈ Ω2 . Then by using the Taylor formula
′
g (s) − g (x) = (s − x) g (x) +
and (3.13), we have
Zs
x
(s − u) g ′′ (u) du
(3.13)
96
GÜRHAN İÇÖZ, SERHAN VARMA AND FATMA TAŞDELEN
(Ln,q g) (x, t) − g (x) =
Zs
g (x) (Ln,q (s − x)) (x, t) + (Ln,q ( (s − u) g ′′ (u) du)) (x, t)
′
x
Zs
≤ (Ln,q ( (s − u) g ′′ (u) du)) (x, t)
x
Zs
= (Kn,q ( (s − u) g ′′ (u) du)) (x, t)
x
−
tx
x− [n] 1−bq
+ 1
( Zn+1 ) [2][n]
(x −
x
tx
1
+
− u)g ′′ (u) du.
[n] (1 − bq n+1 ) [2] [n]
The monotonicity of Kn,q f gives
|(Ln,q g) (x, t) − g (x)| ≤
x− [n]
tx
(1−bq
Zn+1 )
1
+ [2][n]
(x −
x
+(Kn,q
Zs
x
tx
1
+
− u)g ′′ (u) du
[n] (1 − bq n+1 ) [2] [n]
(s − u) g ′′ (u) du ) (x, t) .
(3.14)
On the other hand, it is clear that
Zs
x
2
(s − u) g ′′ (u) du ≤ (s − x) kg ′′ k .
(3.15)
Now let
x−
tx
1
+ [2][n]
[n](1−bqn+1 )
I :=
Z
x
tx
1
x−
+
− u g ′′ (u) du.
[n] (1 − bq n+1 ) [2] [n]
Then we may write
I≤
−
tx
1
+
n+1
[n] (1 − bq
) [2] [n]
2
kg ′′ k .
(3.16)
Substituting (3.15) and (3.16) into (3.14), we have
n
2
|(Ln,q g) (x, t) − g (x)| ≤
Kn,q (s − x) (x, t)
2 )
tx
1
+ −
+
kg ′′ k . (3.17)
[n] (1 − bq n+1 ) [2] [n]
q − Laguerre Kantorovich Operators
97
Using (3.2) in (3.17), it follows that
(
|(Ln,q g) (x, t) − g (x)| ≤
−t 3x2 + x
2tx
−
[n] (1 − bq n+1 ) [2] [n]2 (1 − bq n+1 )
4
1
x
+
+
1
+
2
[2]
[n]
[3] [n]
2 )
−tx
1
+
+
kg ′′ k .
[n] (1 − bq n+1 ) [2] [n]
This completes the proof.
Theorem 5. Let q := qn be a sequence satisfying limqn = 1 and 0 < qn < 1. For
n
each f ∈ C [0, 1] and x ∈ [0, 1], we have
p
|(Kn,q f ) (x, t) − f (x)| ≤ Cω2 f ; δn (x) + ω f ;
where
δn (x)
=
1
−tx
+
n+1
[n] (1 − bq
) [2] [n]
−t 3x2 + x
2tx
4
x
−
+
1
+
[n] (1 − bq n+1 ) [2] [n]2 (1 − bq n+1 )
[2] [n]
2
1
−tx
1
+
+
+
2
[n] (1 − bq n+1 ) [2] [n]
[3] [n]
and C is a positive constant.
Proof. From (3.11), we have
|(Ln,q f ) (x, t)| ≤ 3 kf k .
(3.18)
In view of (3.12) and (3.18), the equality (3.11) implies that
|(Kn,q f ) (x, t) − f (x)| ≤
≤
≤
≤
|(Ln,q (f − g)) (x, t)| + |(f − g) (x)| + |(Ln,q g − g (x)) (x, t)|
tx
1
+ f x−
+
− f (x)
[n] (1 − bq n+1 ) [2] [n]
4 kf − gk + |(Ln,q g) (x, t) − g (x)|
tx
1
+ω f ; −
+
[n] (1 − bq n+1 ) [2] [n]
(
−t 3x2 + x
2tx
4 kf − gk +
−
2
n+1
[n] (1 − bq
) [2] [n] (1 − bq n+1 )
4
x
1
+
+ 1+
[2] [n] [3] [n]2
2 )
−tx
1
+
+
kg ′′ k
[n] (1 − bq n+1 ) [2] [n]
tx
1
+ω f ; −
+
[n] (1 − bq n+1 ) [2] [n]
1
−tx
′′
4 kf − gk + 4δn (x) kg k + ω f ;
+
.
[n] (1 − bq n+1 ) [2] [n]
98
GÜRHAN İÇÖZ, SERHAN VARMA AND FATMA TAŞDELEN
Hence taking infimum on two-hand side of above inequality over all g ∈ Ω2 and
considering (3.10), we get
−tx
1
+
|(Kn,q f ) (x, t0 ) − f (x)| ≤ 4K2 (f ; δn (x)) + ω f ;
[n] (1 − bq n+1 ) [2] [n]
p
−tx
1
≤ Cω2 f ; δn (x) + ω f ;
+
[n] (1 − bq n+1 ) [2] [n]
which is the desired result.
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Gürhan İÇÖZ
Gazi University, Faculty of Science, Department of Mathematics, Teknikokullar 06500,
Ankara, Turkey.
E-mail address: gurhanicoz@gazi.edu.tr
Serhan VARMA
Ankara University, Faculty of Science , Department of Mathematics, Tandoğan 06100,
Ankara, Turkey.
E-mail address: svarma@science.ankara.edu.tr
Fatma TAŞDELEN
Ankara University, Faculty of Science, Department of Mathematics, Tandoğan 06100,
Ankara, Turkey.
E-mail address: tasdelen@science.ankara.edu.tr