Kantorovich Type Generalization for Certain Class of Positive Linear
Operators
ABDULLAH ALTIN, OGÜN DOĞRU and M. ALİ ÖZARSLAN
Department of Mathematics
Ankara University, Faculty of Science
06100 Tandoğan, Ankara
TURKEY
Abstract: - In this study, we introduced the integral type generalization of positive linear operators given in [4].
Approximation properties of these operators are obtained with the help of Bohman-Korovkin Theorem. The
rate of convergence of these integral type operators computed by means of modulus of continuity. We
investigate approximation properties of these operators in Lp (0, b) (0  b  1, 1  p) space.
Key-Words: - Positive linear operators, the Bohman-Korovkin theorem, Kontorovich type operators, Kfunctional of Peetre, modulus of continuity.
1 Introduction
In [4], the following sequence of linear positive
operators were considered.
  v 
1
Cv( n ) (t ) x v ,
Ln ( f ; x) 
 f 
(1)
Fn ( x, t ) v  0  an (v) 
x  [0,1), t  (,0]
where {Fn ( x, t )}, (n  N ) is the generating
functions for the sequence of functions
{Cvn (t )}vN 0 in the form

Fn ( x, t )   Cv( n) (t ) xv
 0 for t  (,0] . The approximation
and
properties for the operators Ln were also be given
under the following conditions.
1 Fn 1 ( x, t )  p( x) Fn ( x, t ), p( x)  M  ,
x  (0,1).
Cv( n ) (t )
AtCv(n11) (t )  an (v)Cv(n1) (t )  vCv(n) (t ),
A  [0, a], Cv( n ) (t )  0 for
which
was
introduced
in
[2].
Since
n

v


 then, for t=0 in Pn ( f ; x) we get
L(vn ) (0)  
v


  v  n  v 
 xv .
M n ( f ; x)  (1  x)n 1  f 

v
v

n


v 0


These operators are called as Bernstein power
series by Cheney and Sharma [2].
(2)
v 0
2
 xt    v  ( n)
v
Pn ( f ; x)  (1  x)n 1 exp
 f 
 Lv (t ) x
 1  x v 0  v  n 
v  Z.
max{v, n}  an (v)  an (v  1).
3
With the help of the appropriate choices, it can be
easily seen that the operators Ln , defined in (1) is
the generalization of many well-known operators.
By choosing an (v)  h(n) and Cv( n) (t ) 
(h(n))v
in
v!
(1), we have
  v  ( h( n) x ) v

Dn ( f ; x)  e h( n) x  f 
v!
v  0  h( n) 
defined in [3]. Notice that choosing h(n)  n then
the operators Dn ( f ; x) turns out to be the SzaszMirakjan operators [5].
For the approximation properties of the operators
Ln , the following theorem was given in [4].
t
If we choose
an (v)  v  n, Cv( n ) (t )  L(vn ) (t )
0
n
then Ln ( f ; x) converges to f (x) uniformly on
[0, b] .
(where L(vn ) (t ) is the Laguerre polynomials) then
our operators turn out to be
In the proof of Theorem A , the following results:
Theorem A. If f is continuous on [0,1],
0  Ln ( s; x)  x 
bMa t
(3)
,
n
1
0  Ln (s 2 ; x)  x2  b  a t bM (1  b)
n
are obtained.
(4)
The goal of this paper is to investigate the
Kantorovich type generalization of the operators
Ln . In the next section we present the integral
extension of Ln . In the third section, we study the
approximation properties of these operators in
C[0, b] norm. In the fourth section we will give the
order of approximation by using modulus. In
section five, we obtain approximation properties of
these operators in Lp [0, b] , p  1 with the help of
Peetre K-functional.
2 Construction of the Operators
Let us denote by M [0, b], (0  b  1) the class of
functions measurable on [0,b].
Assume that in addition to the properties 1 , 2 , 3 ,
the following holds
4 van (v  1)  n  (v  1)an (v) .
Now we modify the operators Ln by replacing
 v 
 in (1) with an integral mean of f (x)
f 

 an v  
 v
v 1 
,
over a small interval I  
 as
n,v
 an (v) an (v  1) 
follows:

1
a (v  1)
L*n ( f ; x) 
 Cv( n) (t ) xv n
Fn ( x, t ) v  0
n
(5)
n
v
a n ( v 1)
 u 

 f  a (v) du
 n 
v
for x  [0,1), t  (,0] .
Notice that this modification is the Kantorovich
type generalization of the operators Pn , M n , S n
and Dn .
t
0
n
then L*n ( f ; x) converges to f (x) uniformly on
[0, b] .
Proof. Since we are using the well-known Bohman-
Theorem 3.1. If f is continuous on [0, b] ,
Korovkin theorem, it will suffice to prove that L*n
is a positive linear operator and that the desired
convergence occurs whenever f is a quadratic
function. It is clear that L*n is positive and linear.
By (2), it is obvious that L*n (1; x)  1 . Consider then
the function f ( s)  s . From (1), (5) and (3) we
have
1
0  L*n ( s; x)  x  ( Ln ( s; x)  x) 
2 Fn ( x, t )
(6)

n
( n)
v

Cv (t ) x .
v  0 an (v)an (v  1)
By 3 one can easily verify that
1
1

,
2
an (v)an (v  1)
n
so we get

1
n
1
Cv( n) (t ) xv 
.

2 Fn ( x, t ) v 0 an (v)an (v  1)
2n
Thus, using (7) and (3) in (6) we have
bMa t
1
L*n (s; x)  x

 .
C[0,b]
n
2n
(7)
(8)
Then L*n ( s; x) converges uniformly to f ( x)  x on
[0,b].
We proceed, then, to a consideration of the function
f ( s )  s 2 . We have, using (5) and (1),
L*n ( s 2 ; x)  Ln ( s 2 ; x) 

n
v

Fn ( x, t ) v  0 an 1 (v)(an (v))2
(9)

n2
1
 Cv( n ) (t ) x v 
Cv( n ) (t ) x v .

3Fn ( x, t ) v  0 (an 1 (v)an (v))2
Using the positivity of the last two member of (9)
and (4), it follows that
0  L*n ( s 2 ; x)  x 2 .
(10)
If we use 3 in (9) and with (4), (10), we have
3
Approximation Properties in
C[0,b]
In this section in order to obtain uniform
L*n ,
convergence of the positive linear operators
we
will use the classical Bohman-Korovkin
approximation theorem.
L*n ( s 2 ; x)  x 2
C [ 0, b ]
 Ln ( s 2 ; x)  x 2

1
n
C [ 0,b ]

1
n
b  a t bM (1  b) 
2

4
3n 2
1
3n 2
.
(11)
So, L*n ( s 2 ; x) converges uniformly to f ( x )  x 2 on
[0,b]. Thus the proof is completed.
4 Order of Approximation
In this section, we compute the rates of
convergence in Theorem 3.1 by the means of the
modulus of continuity. Let f  C[0, b] . The
modulus of continuity of f denoted by w( f , ) , is
defined to be
w( f , ) 
sup
f ( s )  f ( x) .
s  x  ; s , x[ 0,b ]
It is well known
condition
for
lim 0 w( f , )  0.
any   0 and each
that necessary and sufficient
a
function
f  C[0, b] is
It is also well known that for
s  [0, b]
 sx

(12)
f ( s)  f ( x)  w( f , )
 1.
 

The next result gives the rate of convergence of the
sequence {L*n ( f ; x)} to f (x) by means of the
modulus of continuity.
Theorem 4.1. For all f  C[0, b] , we have
L*n ( f ; x)  f ( x)
C[ 0,b ]
 2w( f , n )
where
2n 
Proof.
1
2b  a t bM (1  3b)  42 .
n
3n
Let
f  C[0, b] . By linearity
(13)
and
monotonicity of L*n and using (12), we obtain

1
L*n ( f ; x)  f ( x)  w( f ,  n ) 1 
  n
v

n
a n ( v 1)

v

L*n ( s 2 ; x) 
2
x  2x
2
L*n ( s; x) 
(14)
x.
C [ 0,b ]
 w( f , n )
1


 2  (15)
1 

 1   sup An ( x, t )  .
 
 n  x[ 0,b ]


So, by (8) and (11) we have
sup An ( x, t )  L*n ( s 2 ; x)  x 2
x[ 0,b ]
C [ 0, b ]
 2b L*n ( s; x)  x
(16)
C [ 0, b ]
 2n .
Combining (16) with (15) we get the result.
5 Approximation Properties in
Norm
Lp [0, b]
Since the operators L*n are constructed based on
integral means of f over small intervals, then we
may obtain better results in approximating the
integrable functions. In this part we give the rate of
approximation for the operators L*n ( f ; x) by using
Peetre's K-functional in the L p -metrics. In [6],
Totik obtained the rate of convergence properties of
the operators M n* ( f ; x ) with the help of Kfunctional in the L p -metrics for p  1 .
Let D=[0,b]. L2p ( D ) denotes the space of functions
are in L p (D ) .
n
a n ( v 1)
 u



 a (v)  x  du
 n

L*n ( f ; x)  f ( x)
in the L p (D ) such that the functions g, g ' and g ' '
 1

 Fn ( x, t )


u

(n)
v an (v  1)
  Cv (t ) x
 x du 

n
an (v)
v 0
v
 

By the Cauchy - Schwarz inequality we have
1

1
L*n ( f ; x)  f ( x)  w( f , n )1  ( An ( x, t )) 2 .
 n

where

1
a (v  1)
An ( x, t ) 
 Cv( n) (t ) x v n
Fn ( x, t ) v  0
n
v
This implies that
Also we use the K-functional as follows:

K p ( f ; )  inf 
 f  g L p ( D )   g L2p ( D )  (17)
2


gL p ( D )
where
g L2 ( D) : g
p
L p ( D)
 g' L
p ( D)
 g'' L
p ( D)
. (18)
This Peetre’s type K-functional and the norm
 L2 ( D ) are introduced by Bleimann, Butzer and
p
Hahn [1] for the space CB (D) , the space of
uniformly continuous and bounded functions on D.
Lemma 5.1.
t  (,0] ,

 dn,v (t )
v 0
Assume
that
for
every
fixed
an (v)an 1(v) ( n)
Cv (t )  m  ,
n
(19)
b
where dv,n (t ) : 0
every n  N ,
xv
dx. Then we have, for
Fn ( x, t )
L*n ( f ; x)  L*n ( g ; x)
L p ( D)
m
1
p
f g
L p ( D)
On the other hand, by Lemma 5.1 and inequality
(22), we get
L*n ( f ; x)  f ( x)
Proof. By using Hölder and Jensen inequality and
also (19), we get
xv
dx

0 Fn ( x, t )
b
 Cv( n ) (t ) 
m
1
p
f
L p ( D)
1
.
Fn ( x, t )  (1  x)n 1,
operators M n* .
Theorem 5.2. For every n  N and f  L p (D) ,
the following inequality
1
 (m p  1)
(20)
 K p ( f ; 1n (2b  a t bM (2  3b)  12  34n ))
holds, where the constant m is the same as in
Lemma 5.1, K p ( f ; n )
is the sequence


of K  functional defined as in (5.1)
.
Proof. If we use the Taylor expansion then we
obtain for g  L2p ( D) that
L*n ( g ; x)  g ( x)
L p ( D)
 L*n ( s  x; x)
 L*n ((s  x)2 ; x)
C ( D)
C ( D)
g
L p ( D)
g  L p ( D)
.
(21)
By combining inequalities (8) and (13) in (21) we
have
1
L*n ( g ; x)  g ( x)
 (2b  a t bM (2  3b)
L p ( D)
n
(22)
 12  34n )) g
Therefore by taking infimum over g  L2p ( D) , we
n 
[6]). Totik obtained the constant m  2 p for the
L p ( D)
L p ( D)
convergence of the sequence of the operators
v  n
 then L*n turns out to be M n* (see
Cv( n ) (t )  
 v 
L*n ( f ; x)  f ( x)
 f g
obtain inequality (20) whence the result. Since
lim K p ( f ; n )  0 , Theorem 5.2 gives us
Replacing f by f  g , then the proof is completed
immediately.
By choosing an (v)  v  n,
L p ( D)
L p ( D)
 (m p  1) f  g L ( D )  1n (2b  a t bM
p

1 4

 (2  3b)   ) g L2 ( D ) 
p
2 3n

1
1
 p
 L*n ( f ; x)  L*n ( g ; x)
 L*n ( g ; x)  g ( x)
.
p
b *
 p  
an (v)an 1 (v)
p
 Ln ( f ; x) dx     f ( s ) ds
n
v  0I n,
0

L p ( D)
L2p ( D )
.
L*n ( f ; x) to f (x) in the L p metric spaces.
References:
[1] Bleimann, G., Butzer, P.L. and Hahn, L., A
Bernstein-type
operator
approximating
continuous functions on the semi-axis, Math.
Proc., Vol. 83, No.3, 1980, pp.255-262.
[2] E.W. Cheney and A. Sharma, Bernstein power
series, Canad. J. Math., Vol. 16, 1964, pp.241253.
[3] O. Doğru, Weighted approximation properties
of Szasz-type operators, Intern. Math. J., Vol.
2, No. 9, 2002, pp.889-895.
[4] O. Doğru, M.A. Özarslan and F. Tasdelen, On
positive operators involving a certain class of
generating functions, Studia Sci. Math.
Hungar. (to appear).
[5] O. Szasz, Generalization of S. Bernstein's
polynomials to the infinite interval, J.
Research Nat. Bur. Standards, Vol. 45, 1950,
239-245.
[6] V. Totik, Approximation by Meyer-König and
Zeller type operators, Math. Z., Vol. 182,
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