International Journals of Advanced Research in
Computer Science and Software Engineering
ISSN: 2277-128X (Volume-7, Issue-6)
Research Article
June
2017
Local Approximation Estimates for a Certain Family of
Summation-Integral Type Operators
Manoj Kumar
Department of Mathematics and Statistics, Gurukula Kangri Vishwavidyalaya, Haridwar,
Uttrakhand-249404, India
Abstract: In 1988, Felten obtained local and global approximation theorems for positive linear operators. In the last
decade, Finta studied direct local and global approximation theorems for Baskakov type operators and SzaszMirakian type operators. In the present paper, we establish some local approximation estimates for a new family of
beta operators, by using Ditzian-Totik modulus of smoothness.
Key Words and Phrases: Local approximation, Linear positive operators, Beta operators, K-functional, Ditzian-Totik
modulus of smoothness.
MSC : 41A17, 41A36.
I. INTRODUCTION
In 1985, Upreti[10] studied approximation properties of beta operators[4, 5, 8]. Zhou [12] obtained direct and
inverse theorems for these operators. Gupta and Dogru [6] obtained some direct results for beta operators. Recently,
Kumar [9] studied direct results for Beta-Szasz operators in simultaneous approximation. Motivated by the work on beta
operators, Gupta et al. [7] introduced a new family of beta operators to approximate lebesgue integrable functions on
[0, ) as


1
Bn ( f , x) 
bn,v ( x)  bn,v (t ) f (t ) dt ,

(n  1) v1
0
where
n  N (theset of natural numbers) ,
bn,v (t ) 
x  [0, )
(1.1)
1
t v1 (1  t ) nv1 and
 (v, n  1)
 (v, n  1)  (v  1)!n!/( n  v)! the beta function.
Some basic properties of
bn,v ( x) are as follows

(i).
b
n  r ,v  r
(t ) dt  1
(1.2)
0

(ii).
tb
n ,v
(t ) dt 
0
v
n

t
(iii).
2
bn,v (t ) dt 
0

b
(iv).
v 1
n ,v
vb
n ,v
v 1

(vi).
v
v 1
(vii).
2
v(v  1)
n(n  1)
( x)  (n  1)

(v).
(1.3)
( x)  (n  1)[1  (n  2) x]
bn,v ( x)  (n  1)[1  3(n  2) x  (n  2)(n  3) x 2 ]
x(1  x)bn ,v ( x)  [(v  1)  (n  2) x] bn,v ( x)
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(1.4)
(1.5)
(1.6)
(1.7)
(1.8)
Page | 616
where
n N
Manoj International Journals of Advanced Research in Computer Science and Software Engineering
ISSN: 2277-128X (Volume-7, Issue-6)
and x  [0, ) .
It can be easily verified that the operators (1.1) are linear positive operators and the order of approximation by
these operators is at best O(n-1) as n   , howsoever smooth the function may be.
Let £1 [0, ) be the class of functions g defined by
r
£1r [0, )  { g : g(r)  L1[0, a] for every a  (0, ) and g(r) (t )  M (1  t ) m } ,
where the constants M and m depend on g, and
It is obvious that
L p [a, b], 0  p   stands for the Lp -space.
Lrp [0, ) is not contained in £1r [0, ) .
Due to Ditzian and Totik[1], the modulus of smoothness of a function f is defined by
2 ( f , t )  sup 2h f
0ht
 2 ( x)  x(1  x)
where
(1.9)
p
and
 f ( x  h)  2 f ( x)  f ( x  h) if [ x  h, x  h]  [0, )
2h f ( x)  
0
otherwise



W p   , [0, )  g  L p [0, ) : g  ACloc[0, ) and  2 g  L p [0, ) .
2
Let
Following [14], it can be easily verified that the modulus of smoothness [1] defined by (1.9) , is equivalent to
the modified K-functional given by
2


K  f , t 2  inf
 f -g

 t 2  2 g  t 4 g p : g W p   , [0, ) .
2
p
p
The main object of the present paper is to establish some local approximation estimates [2, 3] for the operators
(1.1) in terms of Ditzian-Totik modulus of second order.
II. PRELIMINARY RESULTS
This section consists of some auxiliary results, which will be helpful in proving the main results of next section.
Lemma 2.1. For m, r  N (the set of non-negative integers), let the function
0
Tr ,n,m ( x) be defined as

Tr ,n,m ( x) 
Then

1
bnr ,v ( x)  bnr ,vr (t ) (t  x) m dt .

(n  r  1) v1
0
Tr ,n,0 ( x)  1 ,
and for all
Tr ,n,1 ( x) 
(1  r )(1  2 x)
(n  r )
(n  r ) ,
n  m  r , there holds the recurrence relation
(n  m  r ) Tr ,n,m1 ( x)  x(1  x)[Tr,n,m ( x)  2mTr ,n,m1 ( x)]  (m  r  1)(1  2 x) Tr ,n,m ( x) .
Consequently, for each
x  [0, ) , we have


Tr ,n,m ( x)  O n [( m1) / 2] ,
where
[ ] denotes the integral part of  .
Proof. Using the definition of
Tr ,n,m ( x) and basic properties of bn,v ( x) , we obtain
Tr ,n,0 ( x)  1
Tr ,n,1 ( x) 
and
(1  r )(1  2 x)
(n  r )
Now, we have
x(1  x)Tr,n,m ( x) 

x(1  x) 
 bnr,v ( x)  bnr,vr (t ) (t  x) m dt  mx(1  x)Tr,n,m1 ( x)
(n  r  1) v1
0
Therefore, using (1.8) we get
x(1  x)[Tr,n,m ( x)  mTr ,n,m1 ( x)] 
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
1
[(v  1)  (n  r  2) x] bnr,v ( x)
(n  r  1) v1
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Manoj International Journals of Advanced Research in Computer Science and Software Engineering
ISSN: 2277-128X (Volume-7, Issue-6)


b
n  r ,v  r
(t ) (t  x) m dt
0



1
 bnr,v ( x)    (v  r  1)  (n  r  2) t  (n  r  2)(t  x)  r (1  2 x)
(n  r  1) v1
0
 bnr ,vr (t ) (t  x) m dt


1

bnr ,v ( x)  t (1  t ) bn r ,vr (t ) (t  x) m dt

(n  r  1) v1
0
 (n  r  2) Tr ,n,m1 ( x)  r (1  2 x) Tr ,n,m ( x)



1
bnr ,v ( x)  [(t  x) 2  (1  2 x)(t  x)  x(1  x)] bn r ,vr (t ) (t  x) m dt

(n  r  1) v1
0
 (n  r  2) Tr ,n,m1 ( x)  r (1  2 x) Tr ,n,m ( x)




1
m 2

b
(
x
)
b
(
t
)
(
t

x
)
dt

(
1

2
x
)
bn r ,vr (t ) (t  x) m1 dt
  n  r ,v  r

n  r ,v

(n  r  1) v1
0
0


 x(1  x)  bn r ,vr (t ) (t  x) m dt   (n  r  2) Tr ,n,m1 ( x)  r (1  2 x) Tr ,n,m ( x)
0

 (m  2) Tr ,n,m1 ( x)  (m  1)(1  2 x) Tr ,n,m ( x)  mx(1  x) Tr ,n,m1 ( x)
 (n  r  2) Tr ,n,m1 ( x)  r (1  2 x) Tr ,n,m ( x)
Thus, collecting the like terms, we get the desired recurrence relation
(n  m  r ) Tr ,n,m1 ( x)  x(1  x)[Tr,n,m ( x)  2mTr ,n,m1 ( x)]  (m  r  1)(1  2 x) Tr ,n,m ( x) .
The other consequence easily follows from the above recurrence relation.
Lemma 2.2. For
Bn( r ) ( f , x) 
f  Lrp [0, )  £1r [0, ), 1  p  , n  r (m  1) and x  [0, ) , we have
 (n, r )
(n  1)


v 1
0
 bnr,v ( x) bnr,vr (t ) f (r ) (t ) dt
(2.1)
r
(n  r )!( N  r )!
(n  j )
 (n, r ) 

.
2
(n!)
j 1 [ n  ( j  1)]
where
Proof. By Leibnitz theorem, with the notation D

d
, we have
dx


1
B ( f , x) 
bn( r,v) ( x)  bn,v (t ) f (t ) dt

(n  1) v1
0
(r)
n
r
1


(n  1) i 0

(n  v)!  r  i v1
  D x
D r i (1  x) ( nv 1)

v i 1 (v  1)! n!  i 


 b

n ,v
(t ) f (t ) dt
0

1  r
x v1
r i  r  ( n  r )! (n  v  r )!



 (1)  i  n!(v  1)!(n  r )! (1  x) nr v1  bn,vi (t ) f (t ) dt
(n  1 v1 i 0
 
0


 r

(n  r )! 
r
ir


b
(
x
)
(

1
)
(

1
)
b
(
t
)

 f (t ) dt


n  r ,v
n
,
v

i
0
i
n!(n  1) v1
i

0




(2.2)
Again by Leibnitz theorem, we have
r
bn( r )r ,vr ( x)  
i 0

 r  r-i vr 1
(n  v)!
  D x
D i (1  x) ( nv 1)
(n  r )!(v  r  1)!  i 
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

Page | 618
Manoj International Journals of Advanced Research in Computer Science and Software Engineering
ISSN: 2277-128X (Volume-7, Issue-6)
r
 r  n!
(n  v  i)!
x vi 1
  (1) i  
n  v  i 1
i 0
 i  (n  r )! (v  i  1)! n! (1  x)
r
r
n!

(1) i   bn,vi ( x)

(n  r )! i 0
i
(2.3)
Thus, from (2.2) and (2.3), we get
 (n, r )
B ( f , x) 
(r)
n
(n  1)


b
n  r ,v
v 1
( x)  (1) r bn( r-r,) vr (t ) f (t ) dt
0
Further integrating by parts r times, taking f (t) as first function, we get the required result (2.1).
Here, it is important to note that the operators defined in (2.1) by
Bn( r ) ( f , x)  ( Bn f ) ( r ) ,
f  Lrp [0, )  £1r [0, )
are not positive. To make these operators positive, we introduce the operators
Bn,r f  D r Bn I r f
f  L p [0, )  £1[0, )
,
where D and I stand for differentiation and integration operators respectively.
Thus, for all
f  L p [0, )  £1[0, ) and n>r(1+m), the operators (2.1) can now be identified as
( Bn,r f )( x) 
 (n, r )
(n  1)


v 1
0
 bnr,v ( x) bnr,vr (t ) f (t ) dt .
Obviously, the operators Bn,r are positive and the estimation
( Bn f ) ( r )  f ( r )
p
f  Lrp [0, )
,
is equivalent to
Bn,r f  f
f  L p [0, ) .
,
p
III. LOCAL APPROXIMATION RESULTS
In this section, we establish direct local approximation theorems for the operators (1.1). Let
space of all real valued continuous and bounded functions f on
CB [0, ) be the
[0,) equipped with the norm f  sup f ( x) .
x[ 0, )
Also let
W2  { g C B [0, ) : g , g C B [0, ) } .
Then, for
If
  0 , the K-functional are defined as
K 2 ( f ,  )  inf { f  g   g : g  W2 } .
 ( f , ) is the usual modulus of continuity of the function f  C B [0, )
 ( f ,  )  sup
h( 0 ,  ]
then
sup
x[ 0 ,  )
defined by
f ( x  h)  f ( x) ,
 2 ( f , 1/ 2 ) the second order modulus of smoothness is defined as
 2 ( f , 1/ 2 )  sup
sup
h( 0 ,  1 / 2 ]
x[ 0 ,  )
f ( x  2h)  2 f ( x  h)  f ( x) .
By Theorem 2.4 [11, p. 177 ], we can find a positive constant C1 such that
K 2 ( f ,  )  C1 2 ( f , 1/ 2 )
(3.1)
Now we start to study the main results of this section, one of which stated as
Theorem 3.1. Let
f  C B [0, ) and n  2 . Then for every x  [0, ) , there exists an absolute constant
C2  0 such that

Bn ( f , x)  f ( x)  C2 2  f ,

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( x  1) ( x  2)
n

1  2x 
    f ,


n 


(3.2)
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Manoj International Journals of Advanced Research in Computer Science and Software Engineering
ISSN: 2277-128X (Volume-7, Issue-6)
Proof. To derive (3.2), we introduce a new operator
Bˆ n : CB [0, )  CB [0, )
defined by
 1  (n  2) x 
Bˆ n ( f , x)  Bn ( f , x)  f ( x)  f 

n


ˆ ( (t  x), x)  0 .
Using Lemma 2.1 for r =0, we get B
(3.3)
n
For
t , x  [0, ) , by Taylor’s expansion of g W2 , we have
t
g(t )  g( x)  (t  x)g( x)   (t  y)g( y) dy
x
Consequently, we have
 t
ˆ
ˆ
Bn ( g, x)  g( x )  Bn   (t  y )g( y ) dy ,
 x

 Bˆ n   (t  y )g( y ) dy ,
 x
t

x  


x 

1 ( n  2 ) x
n

x
 1  (n  2) x

 y  g( y ) dy

n


(3.4)
Since
t
 (t  y) g( y) dy
 (t  x) 2 g
(3.5)
x
and
1 ( n  2 ) x
n

x
 1  (n  2) x

 1  (n  2) x

 y  g( y ) dy  
 x  g

n
n




2
 1  2 x  
 1  x  

 g  4
 g
 n 
 n 
4( x  1) ( x  2)

g
n2
2
2
(3.6)
therefore, from (3.4), (3.5) and (3.6), we get
4( x  1) ( x  2)
Bˆ n ( g , x)  g( x)  Bn (t  x) 2 , x g 
g .
n2


Thus, using Lemma 2.1 for r =0, we obtain
 2(n  5) x 2  2(n  5) x  2 4( x  1) ( x  2) 
ˆ
Bn ( g , x)  g( x)  

 g
2
n
(
n

1
)
n


 2(n  5) 4  ( x  1) ( x  2)

 
g
n
 (n  1) n 

16
( x  1) ( x  2) g
n
(3.7)
Further applying Lemma 2.1 for r =0, we get

Bn ( f , x) 
1 
 bn,v ( x) bn,v (t ) f (t ) dt  f
n  1 v1
0
which implies that Bn is a contraction i.e.
Bn f  f
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for all
f  C B [0, ) .
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Manoj International Journals of Advanced Research in Computer Science and Software Engineering
ISSN: 2277-128X (Volume-7, Issue-6)
Now from (3.3), we get
Bˆ n f  Bn f  2 f  3 f
for all
f  C B [0, )
(3.8)
Hence, in view of (3.3), (3.7) and (3.8), we have
 1  (n  2) x 
Bn ( f , x)  f ( x)  Bˆ n ( f , x)  f ( x)  f ( x)  f 

n


 1  (n  2) x 
 Bˆ n ( f  g, x)  ( f  g)( x)  Bˆ n ( g , x)  g( x)  f ( x)  f 

n


 4 f g 
16
 1  (n  2) x 
( x  1) ( x  2) g  f ( x)  f 

n
n


( x  1) ( x  2)


 16 f  g 
g  
sup
n

 t , t [(12 x ) / n ][ 0, )
 (1  2 x) 
f t 
  f (t )
n 

( x  1) ( x  2)
1  2x 



 16 f  g 
g     f ,

n
n 



(3.9)
Finally, taking the infimum on the right hand side of (3.9), over all g W and applying (3.1), we get the
2
desired result (3.2).
Our next result of this section is the following
Theorem 3.2. Let
n  r  1  2 and f (i )  CB [0, ) , i  0 (1) r. Then for every x  [0, ) , we have
 (n  r  1)!(n  r )!  ( r )
Bn( r ) ( f , x)  f ( r ) ( x)  
 1 f
(n  1)! n!



(n  r  1)!(n  r )! 
2{(n  1)  2(r  1)(r  2)}x( x  1)  (r  1) (r  2) 
1 

(n  1)! n!
(n  r  1)




  f ( r ) , (n  r ) 1/ 2
Proof. Using the basic properties of
bn,v ( x) and Lemma 2.2, we have

Bn( r ) ( f , x)  f ( r ) ( x) 
(3.10)


(n  r )!(n  r )! 
bnr ,v ( x)  bnr ,vr (t ) f ( r ) (t )  f ( r ) ( x) dt

(n  1)! n! v1
0
 (n  r  1)!(n  r )!  ( r )

 1 f ( x)
(n  1)! n!


Since for every
  0 , we have

f ( r ) (t )  f ( r ) ( x)   f ( r ) , t  x
  1 
tx 
  f (r) , 
 



,
therefore, we get

B ( f , x)  f
(r)
n
(r)
(n  r )!(n  r )! 
( x) 
bnr ,v ( x)  bnr ,vr (t ) f ( r ) (t )  f ( r ) ( x) dt

(n  1)! n! v1
0
 (n  r  1)!(n  r )!  ( r )

 1 f
(n  1)! n!





 
(n  r )!(n  r )! 
 bnr,v ( x)  bnr,vr (t ) 1   1 t  x  f (r ) ,  dt
(n  1)! n! v1
0
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
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Manoj International Journals of Advanced Research in Computer Science and Software Engineering
ISSN: 2277-128X (Volume-7, Issue-6)
 (n  r  1)!(n  r )!  ( r )

 1 f
(
n

1
)!
n
!


(3.11)
Applying Cauchy-Schwarz inequality for integration and then summation, we obtain


bnr ,v ( x)  bnr ,vr (t ) t  x dt   bnr ,v ( x)  bnr ,vr (t ) dt 

v 1
v 1
0
0






  bnr ,v ( x) 
 v1

 (n  r  1)
1/ 2
1/ 2
1/ 2


2
 bnr ,vr (t ) (t  x) dt 
0




2
b
(
x
)
b
(
t
)
(
t

x
)
dt
  n  r ,v  n  r ,v  r

0
 v1




2
b
(
x
)
b
(
t
)
(
t

x
)
dt
  n  r ,v  n  r , v  r

0
 v1

1/ 2
1/ 2
1/ 2
(3.12)
From straight computations, we get

b
n  r ,v  r
(t ) (t  x) 2 dt 
0
(v  r  1) (v  r )
(v  r ) 2
 2x
x
(n  r ) (n  r  1)
(n  r )
Thus, we obtain


b
v 1
n  r ,v
( x)  bnr ,vr (t ) (t  x) 2 dt
0

(n  r  1) 2{(n  1)  2(r  1) (r  2)} x( x  1)  (r  1) (r  2)
(n  r ) (n  r  1)
(3.13)
Hence, collecting the estimates of (3.11), (3.12) and (3.13), we get
 (n  r  1)!(n  r )!  ( r )
Bn( r ) ( f , x)  f ( r ) ( x)  
 1 f
(
n

1
)!
n
!


+

(n  r  1)!(n  r )! 
1 2{(n  1)  2( r  1)(r  2)}x( x  1)  ( r  1) ( r  2)
(r)
1  
  f ,
(n  1)! n!
(n  r ) (n  r  1)



Finally, choosing

  (n  r ) 1/ 2 , we get the required result (4.10).
This completes the proof of the Theorem 3.2.
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