DOI: 10.2478/auom-2013-0053
An. Şt. Univ. Ovidius Constanţa
Vol. 21(3),2013, 209–221
Kantrovich Type Generalization of
Meyer-König and Zeller Operators via
Generating Functions
Ali Olgun, H. Gül İnce and Fatma Taşdelen
Abstract
In the present paper, we study a Kantorovich type generalization of
Meyer-König and Zeller type operators via generating functions. Using
Korovkin type theorem we first give approximation properties of these
operators defined on the space C [0, A] , 0 < A < 1. Secondly, we compute the rate of convergence of these operators by means of the modulus
of continuity and the elements of the modified Lipschitz class. Finally,
we give an r-th order generalization of these operators in the sense of
Kirov and Popova and we obtain approximation properties of them.
1
Introduction
For a function f on [0, 1) , the Meyer -König and Zeller operators ( MKZ )
(see[11]) are given by
n+1
Mn (f ; x) = (1 − x)
∞
X
f
k=0
k
n+k+1
n+k
xk , n ∈ N,
k
(1)
Where x ∈ [0, 1) . The approximation properties of the operators have
been studied by Lupaş and Müller [12]. A slight modification of these operatos, called Bernstein power series, was introduced by Cheney and Sharma[3]
Key Words: Positive Linear operators,Kantorovich-type operators, Meyer-König and
Zeller operators, Modulus of contiunity, Modified Lipschitz class, r-th order generalization.
2010 Mathematics Subject Classification: Primary 41A25; Secondary 41A36.
Received: April, 2011.
Revised: April, 2011.
Accepted: February, 2012.
209
210
Ali Olgun, H. Gül İnce and Fatma Taşdelen
and Khan[8] obtained the rate of convergence of Bernstein power series for
functions of bounded variation. The Meyer-König and Zeller operators were
also generalized in [4] by Doğru. A Stancu type generalization of the operators
(1) have been studied by Agratini [2]. Doğru and Özalp [5] studied a Kantorovich type generalization of the operators. Using statistically convergence,
a Kantorovich type generalization of Agratini’s operators have been studied by
Doğru, Duman and Orhan in [6]. Furthermore Altın, Doğru and Taşdelen [1]
introduced a generalization of the operators (1) via linear generating functions
as follows:
∞
X
1
f
Ln (f ; x) =
hn (x, t)
k=0
ak,n
ak,n + bn
Γk,n (t) xk ,
(2)
a
where 0 < ak,nk,n
+bn < A , A ∈ (0, 1), and {hn (x, t)}n∈N is the generating
function for the sequence of function {Γk,n (t)}n∈N0 in the form
hn (x, t) =
∞
X
Γk,n (t) xk ,
t ∈ I,
(3)
k=0
where I is any subinterval of R , N0 = N ∪ {0} . The Authors studied approximation properties of the operators (2) under the following conditions (see
[1]):
i) hn (x, t) = (1 − x) hn+1 (x, t)
ii) bn Γk,n+1 (t) = ak+1,n Γk+1,n (t)
iii) bn → ∞, bn+1
bn → 1, bn 6= 0, f or all n ∈ N
iv) Γk,n (t) ≥ 0, f or all I ⊂ R
v) ak+1,n = ak,n+1 + ϕn , |ϕn | ≤ m < ∞, a0,n = 0
On the other hand, Korovkin type theorems (see [10]) on some general Lipschitz type maximal functions spaces were
and
given
by Gadjiev
Çakar [7] and
υ
υ
an x
x
Doğru [4], including the test functions 1−x
and 1+a
(υ = 0, 1, 2) ,
nx
respectively. The space of Lipschitz type maximal functions was defined by
Lenze in [11]. Altın, Doğru
and Taşdelen [1] obtained a Korovkin type theorem
using the test functions
x
1−x
υ
(υ = 0, 1, 2) , for the investigation of the apa
proximation properties of the operators (2). They used the nodes s = ak,nk,n
+bn ,
ak,n
s
with 1−s = bn .
We known from [1] that the operators Ln given by (2) satisfy the the
following equalities:
Ln (1; x) = 1,
(4)
Ln (θ (s) ; x) = θ (x) ,
(5)
KANTOROVICH TYPE GENERALIZATION...
ϕn
2 bn+1
Ln θ2 (s) ; x = (θ (x))
+
θ (x) ,
bn
bn
211
(6)
s
, x ∈ [0, A] , 0 < A < 1, t ∈ I, n ∈ N.
where θ (s) = 1−s
Throughout our present investigation, by k.k , we denote the usual supremum norm on the space C [0, A] which is the space of all continuous functions
on [0, A].
Clearly by (4), (5) and (6) it follows that
lim kLn (θυ ) − θυ k = 0; υ = 0, 1, 2.
n→∞
(7)
The objective of this paper is to investigate the Kantorovich type generalization of the operators Ln . In the next section, we present the integral type
extension of Ln . In section 3, we study the rate of convergence of our operators by means of the modulus of continuity and the elements of the modified
Lipschitz class. Finally, in section 4, we give an r-th order generalization for
these operators and approximation properties of them.
2
Construction of the Kantorovich-type Operators
Now, we set
s
.
1−s
In this section, we construct a Kantorovich type generalization of MKZ type
operators via generating functions and we investigate the approximation properties of these operators with the aid of the test functions θυ , υ = 0, 1, 2. We
should note here that in the sequel we shall use the letter θ for the test function
θ (s) .
Let ω be the modulus of continuity of f defined as
θ (s) :=
ω (f, δ) := sup {|f (s) − f (x)| ; s, x ∈ [0, A] , |s − x| < δ}
for f ∈ C [0, A] .
Let ω = ω (t) be arbitrary modulus of continuity defined on [0, A] , which
satisfies the following conditions [14]
a) ω is non- decreasing,
b) ω (δ1 + δ2 ) ≤ ω (δ1 ) + ω (δ2 ) for δ1 , δ2 ∈ [0, A]
c) lim+ ω (δ) = 0.
δ→0
Inspired of the well-known Hω class (see for ex. p 108 of [16]), we now
define the following Wω class. We say that
a function f ∈ C [0, A] belongs to
t1
t2 the class Wω [0, A] if |f (t1 ) − f (t2 )| ≤ ω 1−t
− 1−t
for all t1 , t2 ∈ [0, A] .
1
2
Clearly, if f ∈ Wω [0, A] , then ω (f, δ) ≤ ω (δ) for all δ ∈ [0, A] . Now, fix
212
Ali Olgun, H. Gül İnce and Fatma Taşdelen
ω (t) = t, an example of the function belonging to the W
given
ω class can be
s
|s−x|
1
x by f (x) = 1−x . Indeed, |f (s) − f (x)| ≤ (1−x)(1−s) = ω 1−s − 1−x .
Let Wω be the space of real valued functions f ∈ C [0, A] satisfying
s
x |f (s) − f (x)| ≤ ω , f or any x, s ∈ [0, A] .
−
1 − s 1 − x
It can easily be obtained that ω satisfies
ω (nδ) ≤ nω (δ) ,
n ∈ N,
and from the property b) of the modulus of continuity ω we have
ω (λδ) ≤ ω ((1 + kλk) δ) ≤ (1 + λ) ω (δ) ,
λ > 0,
where kλk is the greatest integer of λ.
Let us assume that the properties (i) − (v) are satisfied. We also assume
that {bn } is a positive increasing sequence.
a
Now we modify the operators Ln by replacing f ak,nk,n
in (2) by an
+bn
integral mean of f over an interval Ik,n = [ak,n , ak,n + ck,n ] as follows:
L∗n (f (s) ; x) = L∗n (f ; x) =
1
hn (x,t)
∞
P
k=0
Γk,n (t)
ck,n
xk
ak,nR+ck,n
ak,n
f
ξ
ξ+bn
dξ
(8)
where f is an integrable function on the interval (0, 1) and {ck,n } is a sequence
such that
0 < ck,n ≤ 1, f or every k ∈ N.
(9)
Clearly, L∗n is a positive linear operator and also by (4)
L∗n (1; x) = 1.
(10)
Now, we give the following Korovkin type theorem for the test functions
proved by Altın, Doğru and Taşdelen [1].
Theorem A. Let {An } be a sequence of linear positive operators from
Wω into C [0, A] and satisfies the three conditions (7). Then for all f ∈ Wω ,
we have
lim kAn (f ) − f k = 0
(11)
n→∞
[1] .
To prove the main result for the sequence of linear positive operators L∗n ,
we need the following two lemmas.
Lemma 1. We have
213
KANTOROVICH TYPE GENERALIZATION...
lim kL∗n (θ) − θk = 0.
n→∞
Proof. From (iii) and (2) we get
L∗n (θ; x)
=
=
1
hn (x,t)
1
hn (x,t)
∞
P
k=0
∞
P
k=0
= Ln (θ; x) +
Γk,n (t) k
ck,n x
ak,n
bn Γk,n
ak,nR+ck,n
ak,n
(t) xk +
1
1
2bn hn (x,t)
∞
P
ξ
bn dξ
1
hn (x,t)
∞
P
k=0
ck,n
2bn Γk,n
(t) xk
ck,n Γk,n (t) xk .
k=0
In this way, we obtain from (5)
∞
L∗n (θ; x) − θ (x) =
X
1
1
ck,n Γk,n (t) xk .
2bn hn (x, t)
(12)
k=0
Following (iii) , (iv) and (9), each term of the right hand side is non- negative,
we have
L∗n (θ; x) − θ (x) ≥ 0.
(13)
Hence from (9) and (4) we can write
0 ≤ L∗n (θ; x) − θ (x) ≤
1
.
2bn
So we have
kL∗n (θ) − θk ≤
1
.
2bn
Taking limit for n → ∞ in (14), (iii) yields that
lim kL∗n (θ) − θk = 0.
n→∞
Lemma 2. we have
lim L∗n θ2 − θ2 = 0.
n→∞
(14)
214
Ali Olgun, H. Gül İnce and Fatma Taşdelen
Proof. From (8) and (2) we have
L∗n θ2 ; x
=
1
hn (x,t)
∞
P
k=0
n
1
hn (x,t)
ak,n
1
1
bn hn (x,t)
= Ln θ2 ; x +
+ 3b12
ak,nR+ck,n
Γk,n (t) k
ck,n x
∞
P
k=0
∞
P
k=0
ξ2
b2n dξ
ak,n
bn ck,n Γk,n
(t) xk
c2k,n Γk,n (t) xk .
Hence by (9) we get
L∗n θ2 ; x − θ2 (x) ≤ Ln θ2 ; x − θ2 (x)
+ b1n Ln (θ; x) +
1
3b2n Ln
(1; x) .
So, using (4), (5) and (6) we have
−
1
L∗n θ2 ; x − θ2 (x) ≤ θ2 (x) bn+1
bn
(15)
+
ϕn
bn
+
1
bn
θ (x) +
1
3b2n .
Since
s
1−s
2
=
s
x
−
1−s 1−x
2
+
2x
s
−
1−x1−s
x
1−x
2
we get
L∗n θ2 ; x − θ2 (x)
2
= L∗n (θ (s) − θ (x)) ; x
(16)
+2θ (x) L∗n
(θ (s) − θ (x) ; x) ≥ 0.
L∗n
Here we have also used the positivity of
and (13). By taking the relations
(15) and (16) into account we obtain
ϕn +1
1
0 ≤ L∗n θ2 − θ2 ≤ B ∗ bn+1
−
1
+
+
,
(17)
2
bn
bn
3b
n
A
where B ∗ = max 1, 1−A
,
A
1−A
2 .
Now taking limit for n → ∞ in (17), (iii) and (v) yield
lim L∗n θ2 − θ2 = 0.
n→∞
215
KANTOROVICH TYPE GENERALIZATION...
Then by using (10), Lemma 1, Lemma 2 and Theorem A we can state the
following approximation theorem for the operators L∗n at once.
Theorem 1. Let L∗n be the operator given by (8). Then for all f ∈ Ww ,
we have
lim kL∗n (f ) − f k = 0.
n→∞
3
Rates of Convergence Properties
In this section, we compute the rate of convergence of the sequence{L∗n (f ; .)}
to function f by means of the modulus of continuity and the elements of
modified Lipschitz class.
Theorem 2. Let L∗n be the operator given by (8). Then for all f ∈ Wω ,
we have
√ kL∗n (f ) − f k ≤ 1 + B ∗ ω (δn ) ,
n
−
1
where δn : = bn+1
+
bn
ϕn +1
bn
+
1
3b2n
o 12
and B ∗ is given as in Lemma 2.
Proof. We suppose that f ∈ Wω ; then, by (10),
|L∗n (f ; x) − f (x)|
≤ ω (δn ) L∗n
|s−x|
δn
≤ ω (δn ) L∗n 1 +
|L∗n (f ; x) − f (x)|
h
= ω (δn ) L∗n (1; x) +
"
= ω (δn ) 1 +
1
1
δn hn (x,t)
∞
P
k=0
+ 1; x
1
δn
1 ∗
δn Ln
Γk,n (t)
ck,n
|θ (s) − θ (x)| ; x
i
(|θ (s) − θ (x)| ; x)
xk
#
ξ
bn − θ (x) dξ .
ak,nR+ck,n ak,n
By applying the Cauchy-Schwarz inequality to the integral first and to the
216
Ali Olgun, H. Gül İnce and Fatma Taşdelen
sum next, we obtain
|L∗n (f ; x) − f (x)|

≤ ω (δn ) 1 +
1
1
δn hn (x,t)
∞
P
k=0

1
δn
≤ ω (δn ) 1 +
×
1
hn (x,t)
∞
P
k=0
1
hn (x,t)
Γk,n (t) k
ck,n x
≤ ω (δn ) 1 +
1
δn
1
δn
1
hn (x,t)
L∗n
∞
P
k=0
ak,n
Γk,n (t) k
ck,n x
ξ
bn
− θ (x)
ak,nR+ck,n ak,n
2
! 12
ak,nR+ck,n
dξ
! 12 
dξ
ak,n
ξ
bn
2
− θ (x) dξ
ξ
bn
2
− θ (x) dξ
! 21
12 #

≤ ω (δn ) 1 +
ak,nR+ck,n Γk,n (t) k
ck,n x
∞
P
k=0
Γk,n (t) k
ck,n x
ak,nR+ck,n ak,n
! 21 

21 2
(θ (s) − θ (x)) ; x
.
(18)
This implies that

kL∗n (f ) − f k ≤ w (f, δn ) 1 +
1
δn
2
sup L∗n (θ (s) − θ (x)) ; x
! 21 
 .
x∈[0,A]
(19)
By using the equality (16), from (14) and (17) we get that
∗
2 Ln (θ (s) − θ (x)) ≤ L∗n θ2 − θ2 + max 2θ (x) k(L∗n (θ) − θ)k
(20)
x∈[0,A]
h
≤ B ∗ bn+1
−
1
+
bn
ϕn +1
bn
+
1
3b2n
i
.
So, combining (19) with (20) we can write
(
21 )
ϕn + 1
1
1
∗
∗ bn+1
B kLn (f ) − f k ≤ ω (δn ) 1 +
− 1 +
+ 2
.
δn
bn
bn
3bn

217
KANTOROVICH TYPE GENERALIZATION...
h
i 21
ϕn +1
1
For δ : = δn = bn+1
the proof is completed.
bn − 1 + bn + 3b2n
Now we will study the rate of convergence of the positive linear operators
g (α) .
L∗n by means of the elements of the modified Lipschitz class Lip
M
g (α) , as follows:
Let us consider the class of functions f, denoted by Lip
M
α
|f (s) − f (x)| ≤ M |θ (s) − θ (x)|
, s, x ∈ [0, A] , 0 < A < 1,
g (α) as
where M > 0, 0 < α ≤ 1 and f ∈ C [0, A] . We can call the class Lip
M
”the modified Lipschitz class”.
Theorem 3. Let L∗n be the operator given by (8). Then for all f ∈
g
LipM (α) , we have
α
kL∗n (f ) − f k ≤ M (B ∗ ) 2 δnα ,
where δn and B ∗ are the same as in Theorem 2.
g (α) and 0 < α ≤ 1. By linearity and monotonicity
Proof. Let f ∈ Lip
M
of L∗n , we have
|L∗n (f ; x) − f (x)|
≤ L∗n (|f (s) − f (x)| ; x)
α
≤ M L∗n (|θ (s) − θ (x)| ; x)
∞
P
1
= M hn (x,t)
k=0
Γk,n (t)
ck,n
xk
ak,nR+ck,n α
ξ
bn − θ (x) dξ.
ak,n
2
= 2−α
, to the integral first
By applying the Hölder inequality with p =
and to the sum next , then last formula is reduced to
(
) α2
ak,nR+ck,n 2
∞
P
Γk,n (t) k
ξ
1
∗
|Ln (f ; x) − f (x)| ≤ M hn (x,t)
dξ
ck,n x
bn − θ (x)
2
α, q
k=0
×
1
hn (x,t)
∞
P
Γk,n (t) x
ak,n
k
2
2−α
k=0
n o α2
2
= M L∗n (θ (s) − θ (x)) ; x
.
By taking into account the inequality (20) in the last inequality we have
α
2 2
kL∗n (f ) − f k ≤ M L∗n (θ (s) − θ (x)) n h
≤ M B ∗ bn+1
−
1
+
bn
Thus we obtain the desired result.
ϕn +1
bn
+
1
3b2n
io α2
.
218
Ali Olgun, H. Gül İnce and Fatma Taşdelen
An r-th order generalization of the operators L∗n
4
By C r [0, A] (0 < A < 1, r ∈ No ) we denote the space of all functions of having
continuous r-th order derivative f (r) on the segment [0, A] , (0 < A < 1) , where
as usual, f (0) (x) = f (x) .
We consider the following r-th order generalization of the positive linear
operators L∗n defined by (8):
1
L∗n,r (f ; x) = hn (x,t)
∞
P
Γk,n (t)
ck,n
k=0
xk
ak,nR+ck,n r
P
f (j)
j=0
ak,n
ξ
ξ+bn
j
ξ
(x− ξ+b
)
n
j!
dξ,
(21)
where f ∈ C r [0, A] (0 < A < 1, r ∈ No ) , n ∈ N.
The r-th order generalization of the positive linear operators was given
in [9]. But we remark that the r-th order generalization for the Kantorovichtype operators are first introduced by Özarslan, Duman and Srivastava in [15].
Note that taking r = 0, we obtain the operators L∗n (f ; x) defined by (8).
We recall that a function f ∈ [0, A] belongs to LipM (α) if the following
inequality holds:
α
|f (y) − f (x)| ≤ M |y − x| , (x, y ∈ [0, A]) .
Theorem 4. For any f ∈ C r [0, A] such that f (r) ∈ LipM (α) we have
∗
M
α
α+r
Ln,r (f ) − f ≤
B
(α,
r)
,
|s
−
x|
;
x
L
n
C[0,A]
(r − 1)! α + r
C[0,A]
(22)
where B (α, r) is the beta function and r ∈ No , n ∈ N.
Proof. We can write from (21) that
f (x) − L∗n,r (f ; x)
=
×
1
hn (x,t)
∞
P
k=0
ak,nR+ck,n
Γk,n (t)
ck,n
"
f (x) −
xk
r
P
f
(j)
j=0
ak,n
ξ
ξ+bn
ξ
)
(x− ξ+b
n
j
j!
#
dξ.
(23)
It is also known from Taylor’s formula that
x− ξ j
r
P
( ξ+bn )
ξ
f (x) −
f (j) ξ+b
j!
n
j=0
=
r
ξ
(x− ξ+b
) R1
n
(r−1)!
0
r−1
(1 − s)
h
f (r)
ξ
ξ+bn
+s x−
ξ
ξ+bn
− f (r)
ξ
ξ+bn
i
ds.
(24)
219
KANTOROVICH TYPE GENERALIZATION...
Because of f (r) ∈ LipM (α) , one can write
α
(r)
ξ ξ
ξ
ξ
α
(r)
f
.
≤ M s x −
+s x−
−f
ξ + bn
ξ + bn
ξ + bn ξ + bn (25)
From the well known beta function, we can write
Z1
sα (1 − s)
r−1
ds = B (1 + α, r) =
α
B (α, r) .
α+r
(26)
0
Substituting (25) and (26) in (24), we conclude that
x− ξ j r
P
( ξ+bn ) ξ
M
α
(j)
B (α, r) x −
f
≤ (r−1)!
f (x) −
ξ+b
j!
α+r
n
j=0
α+r
ξ ξ+bn .
(27)
By taking (23) and (27) into consideration, we arrive at (22).
Now, we consider the function g ∈ C [0, A] defined by
r+α
g (s) = |s − x|
.
α+r Since g (x) = 0 we can write Ln |s − x|
(28)
= 0. Theorem 4 yields
C[0,A]
that for all f ∈ C r [0, A] such that f (r) ∈ LipM (α) , we have
(f
)
−
f
lim L∗[r]
= 0.
n
n→∞
Finally, in view of Theorem 2, Theorem 3 and g ∈ LipAr (α) , one can
deduce the following result from Theorem 4 immediately:
Corollary 1. For all f ∈ C r [0, A] , such that f (r) ∈ LipM (α), we have
kL∗n (f ) − f k ≤
√
α
M
B (α, r) 1 + B ∗ ω (g, δn ) .
(r − 1)! α + r
where B ∗ is the same as in Lemma 2, δn is the same as in Theorem 2 and g
is defined by (28).
Corollary 2. For all f ∈ C r [0, A] such that f (r) ∈ LipM (α), we have
α
α
M Ar
[r]
B (α, r) (B ∗ ) 2 δnα
Ln (f ) − f ≤
(r − 1)! α + r
where B ∗ is the same as in Lemma 2 and δn is the same as in Theorem 2.
220
Ali Olgun, H. Gül İnce and Fatma Taşdelen
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Ali Olgun
Kırıkkale University
Faculty of Sciences and Arts,
Department of Mathematics,
Kırıkkale-TURKEY
Email: aliolgun71@gmail.com
H. Gül İnce
Gazi University
Faculty of Sciences,
Department of Mathematics,
Ankara-TURKEY
Email: ince@gazi.edu.tr
Fatma Taşdelen
Ankara University
Faculty of Sciences,
Department of Mathematics
Ankara-TURKEY
Email: tasdelen@science.ankara.edu.tr
222
Ali Olgun, H. Gül İnce and Fatma Taşdelen