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q-analogue of a new sequence of linear positive operators

Journal of Inequalities and Applications20122012:144

https://doi.org/10.1186/1029-242X-2012-144

Received: 7 May 2012

Accepted: 4 June 2012

Published: 22 June 2012

Abstract

This paper deals with Durrmeyer type generalization of q-Baskakov type operators using the concept of q-integral, which introduces a new sequence of positive q-integral operators. We show that this sequence is an approximation process in the polynomial weighted space of continuous functions defined on the interval [ 0 , ) . An estimate for the rate of convergence and weighted approximation properties are also obtained.

MSC:41A25, 41A36.

Keywords

Durrmeyer type operatorsweighted approximationrate of convergenceq-integral

1 Introduction

In the year 2003 Agrawal and Mohammad [1] introduced a new sequence of linear positive operators by modifying the well-known Baskakov operators having weight functions of Szasz basis function as
D n ( f , x ) = n k = 1 p n , k ( x ) 0 s n , k 1 ( t ) f ( t ) d t + p n , 0 ( x ) f ( 0 ) , x [ 0 , ) ,
(1.1)
where
p n , k ( x ) = ( n + k 1 k ) x k ( 1 + x ) n + k , s n , k ( t ) = e n t ( n t ) k k ! .

It is observed in [1] that these operators reproduce constant as well as linear functions. Later, some direct approximation results for the iterative combinations of these operators were studied in [14].

A lot of works on q-calculus are available in literature of different branches of mathematics and physics. For systematic study, we refer to the work of Ernst [5], Kim [10, 11], and Kim and Rim [9]. The application of q-calculus in approximation theory was initiated by Phillips [13], who was the first to introduce q-Bernstein polynomials and study their approximation properties. Very recently the q-analogues of the Baskakov operators and their Kantorovich and Durrmeyer variants have been studied in [2, 3] and [7] respectively. We recall some notations and concepts of q-calculus. All of the results can be found in [5] and [8]. In what follows, q is a real number satisfying 0 < q < 1 .

For n N ,
The q-binomial coefficients are given by
[ n k ] q = [ n ] q ! [ k ] q ! [ n k ] q ! , 0 k n .
The q-Beta integral is defined by [12]
Γ q ( t ) = 0 1 1 q x t 1 E q ( q x ) d q x , t > 0 ,
(1.2)
which satisfies the following functional equation:
Γ q ( t + 1 ) = [ t ] q Γ q ( t ) , Γ q ( 1 ) = 1 .
For f C [ 0 , ) q > 0 and each positive integer n, the q-Baskakov operators [2] are defined as
B n , q ( f , x ) = k = 0 [ n + k 1 k ] q q k ( k 1 ) 2 x k ( 1 + x ) q n + k f ( [ k ] q q k 1 [ n ] q ) = k = 0 p n , k q ( x ) f ( [ k ] q q k 1 [ n ] q ) ,
(1.3)
where
( 1 + x ) q n : = { ( 1 + x ) ( 1 + q x ) ( 1 + q n 1 x ) , n = 1 , 2 , , 1 , n = 0 .
Remark 1 The first three moments of the q-Baskakov operators are given by
As the operators D n ( f , x ) have mixed basis functions in summation and integration and have an interesting property of reproducing linear functions, we were motivated to study these operators further. Here we define the q-analogue of the operators as
D n q ( f , x ) = [ n ] q k = 1 p n , k q ( x ) 0 q / ( 1 q n ) q k s n , k 1 q ( t ) f ( t q k ) d q t + p n , 0 q ( x ) f ( 0 ) ,
(1.4)
where x [ 0 , ) and
p n , k q ( x ) = [ n + k 1 k ] q q k ( k 1 ) 2 x k ( 1 + x ) q n + k , s n , k q ( t ) = E q ( [ n ] q t ) ( [ n ] q t ) k [ k ] q ! .

In case q = 1 , the above operators reduce to the operators (1.1). In the present paper, we estimate a local approximation theorem and the rate of convergence of these new operators as well as their weighted approximation properties.

2 Moment estimation

Lemma 1 The following equalities hold:
  1. (i)

    D n q ( 1 , x ) = 1 ,

     
  2. (ii)

    D n q ( t , x ) = x ,

     
  3. (iii)

    D n q ( t 2 , x ) = x 2 + x [ n ] q ( 1 + q + x q ) .

     
Proof The operators D n q are well defined on the function 1 , t , t 2 . Then for every x [ 0 , ) , we obtain
D n q ( 1 , x ) = [ n ] q k = 1 p n , k q ( x ) 0 q / ( 1 q n ) q k ( [ n ] q t ) k 1 [ k 1 ] q ! E q ( [ n ] q t ) d q t + p n , 0 q ( x ) .
Substituting [ n ] q t = q y and using (1.2), we have
D n q ( 1 , x ) = [ n ] q k = 1 p n , k q ( x ) 0 1 / ( 1 q ) q k ( q y ) k 1 [ k 1 ] q ! E q ( q y ) q d q y [ n ] q + p n , 0 q ( x ) = k = 1 p n , k q ( x ) + p n , 0 q ( x ) = B n , q ( 1 , x ) = 1 ,

where B n , q ( f , x ) is the q-Baskakov operator defined by (1.3).

Next, we have
D n q ( t , x ) = [ n ] q k = 1 p n , k q ( x ) 0 q / ( 1 q n ) q k ( [ n ] q t ) k 1 [ k 1 ] q ! E q ( [ n ] q t ) t q k d q t .
Again substituting [ n ] q t = q y and using (1.2), we have
D n q ( t , x ) = [ n ] q k = 1 p n , k q ( x ) 0 1 / ( 1 q ) q k ( q y ) k [ k 1 ] q ! [ n ] q E q ( q y ) q d q y [ n ] q q k = k = 0 p n , k q ( x ) q [ k ] q [ n ] q q k = B n , q ( t , x ) = x .
Finally,
D n q ( t 2 , x ) = [ n ] q k = 0 p n , k q ( x ) 0 q / ( 1 q n ) q k ( [ n ] q t ) k 1 [ k 1 ] q ! E q ( [ n ] q t ) t 2 q 2 k d q t .
Again substituting [ n ] q t = q y , using (1.2) and [ k + 1 ] q = [ k ] q + q k , we have
D n q ( t 2 , x ) = [ n ] q k = 1 p n , k q ( x ) 0 1 / ( 1 q ) q k ( q y ) k + 1 [ k 1 ] q ! [ n ] q 2 E q ( q y ) q 2 k q d q y [ n ] q = k = 1 p n , k q ( x ) [ k + 1 ] q [ k ] q [ n ] q 2 q 2 k 2 = k = 1 p n , k q ( x ) ( [ k ] q + q k ) [ k ] q [ n ] q 2 q 2 k 2 = B n , q ( t 2 , x ) + q [ n ] q B n , q ( t , x ) = x 2 + x [ n ] q ( 1 + q + x q ) .

 □

Remark 2 If we put q = 1 , we get the moments of a new sequence D n ( f , x ) considered in [1] as operators as
Lemma 2 Let q ( 0 , 1 ) , then for x [ 0 , ) we have
D n q ( ( t x ) 2 , x ) = x ( x + q [ 2 ] q ) q [ n ] q .

3 Direct theorems

By C B [ 0 , ) we denote the space of real valued continuous bounded functions f on the interval [ 0 , ) ; the norm- on the space C B [ 0 , ) is given by
f = sup 0 x < | f ( x ) | .
The Peetre’s K-functional is defined by
K 2 ( f , δ ) = inf { f g + δ g : g W 2 } ,
where W 2 = { g C B [ 0 , ) : g , g C B [ 0 , ) } . By [4], pp.177], there exists a positive constant C > 0 such that K 2 ( f , δ ) C ω 2 ( f , δ 1 / 2 ) δ > 0 and the second order modulus of smoothness is given by
ω 2 ( f , δ ) = sup 0 < h δ sup 0 x < | f ( x + 2 h ) 2 f ( x + h ) + f ( x ) | .
Also, for f C B [ 0 , ) a usual modulus of continuity is given by
ω ( f , δ ) = sup 0 < h δ sup 0 x < | f ( x + h ) f ( x ) | .
Theorem 1 Let f C B [ 0 , ) and 0 < q < 1 . Then for all x [ 0 , ) and n N , there exists an absolute constant C > 0 such that
| D n q ( f , x ) f ( x ) | C ω 2 ( f , x ( x + q [ 2 ] q ) q [ n ] q ) .
Proof Let g W 2 and x , t [ 0 , ) . By Taylor’s expansion, we have
g ( t ) = g ( x ) + g ( x ) ( t x ) + x t ( t u ) g ( u ) d u .
Applying Lemma 2, we obtain
D n q ( g , x ) g ( x ) = D n q ( x t ( t u ) g ( u ) d u , x ) .
Obviously, we have | x t ( t u ) g ( u ) d u | ( t x ) 2 g . Therefore,
| D n q ( g , x ) g ( x ) | D n q ( ( t x ) 2 , x ) g = x ( x + q [ 2 ] q ) q [ n ] q g .
Using Lemma 1, we have
| D n q ( f , x ) | [ n ] q k = 1 p n , k q ( x ) 0 q / ( 1 q n ) q k s n , k 1 q ( t ) | f ( t q k ) | d q t + p n , 0 q ( x ) | f ( 0 ) | f .
Thus
| D n q ( f , x ) f ( x ) | | D n q ( f g , x ) ( f g ) ( x ) | + | D n q ( g , x ) g ( x ) | 2 f g + x ( x + q [ 2 ] q ) q [ n ] q g .

Finally, taking the infimum over all g W 2 and using the inequality K 2 ( f , δ ) C ω 2 ( f , δ 1 / 2 ) , δ > 0 , we get the required result. This completes the proof of Theorem 1. □

We consider the following class of functions:

Let H x 2 [ 0 , ) be the set of all functions f defined on [ 0 , ) satisfying the condition | f ( x ) | M f ( 1 + x 2 ) , where M f is a constant depending only on f. By C x 2 [ 0 , ) , we denote the subspace of all continuous functions belonging to H x 2 [ 0 , ) . Also, let C x 2 [ 0 , ) be the subspace of all functions f C x 2 [ 0 , ) , for which lim | x | f ( x ) 1 + x 2 is finite. The norm on C x 2 [ 0 , ) is f x 2 = sup x [ 0 , ) | f ( x ) | 1 + x 2 . We denote the modulus of continuity of f on closed interval [ 0 , a ] , a > 0 as by
ω a ( f , δ ) = sup | t x | δ sup x , t [ 0 , a ] | f ( t ) f ( x ) | .

We observe that for function f C x 2 [ 0 , ) , the modulus of continuity ω a ( f , δ ) tends to zero.

Theorem 2 Let f C x 2 [ 0 , ) , q ( 0 , 1 ) and ω a + 1 ( f , δ ) be its modulus of continuity on the finite interval [ 0 , a + 1 ] [ 0 , ) , where a > 0 . Then for every n > 2 ,
D n q ( f ) f C [ 0 , a ] 6 M f a ( 1 + a 2 ) ( 2 + a ) q [ n ] q + 2 ω ( f , a ( a + q [ 2 ] q ) q [ n ] q ) .
Proof For x [ 0 , a ] and t > a + 1 , since t x > 1 , we have
| f ( t ) f ( x ) | M f ( 2 + x 2 + t 2 ) M f ( 2 + 3 x 2 + 2 ( t x ) 2 ) 6 M f ( 1 + a 2 ) ( t x ) 2 .
(3.1)
For x [ 0 , a ] and t a + 1 , we have
| f ( t ) f ( x ) | ω a + 1 ( f , | t x | ) ( 1 + | t x | δ ) ω a + 1 ( f , δ )
(3.2)

with δ > 0 .

From (3.1) and (3.2) we can write
| f ( t ) f ( x ) | 6 M f ( 1 + a 2 ) ( t x ) 2 + ( 1 + | t x | δ ) ω a + 1 ( f , δ )
(3.3)
for x [ 0 , a ] and t 0 . Thus
| D n q ( f , x ) f ( x ) | D n q ( | f ( t ) f ( x ) | , x ) 6 M f ( 1 + a 2 ) D n q ( ( t x ) 2 , x ) + ω a + 1 ( f , δ ) ( 1 + 1 δ D n q ( ( t x ) 2 , x ) ) 1 2 .
Hence, by using Schwarz inequality and Lemma 2, for every q ( 0 , 1 ) and x [ 0 , a ]
| D n q ( f , x ) f ( x ) | 6 M f ( 1 + a 2 ) x ( q [ 2 ] q + x ) q [ n ] q + ω a + 1 ( f , δ ) ( 1 + 1 δ x ( q [ 2 ] q + x ) q [ n ] q ) 6 M f a ( 1 + a 2 ) ( 2 + a ) q [ n ] q + ω a + 1 ( f , δ ) ( 1 + 1 δ a ( a + q [ 2 ] q ) q [ n ] q ) .

By taking δ = a ( q [ 2 ] q + a ) q [ n ] q we get the assertion of our theorem. □

4 Higher order moments and an asymptotic formula

Lemma 3 ([6])

Let 0 < q < 1 , we have
B n , q ( t 3 , x ) = 1 [ n ] q x + 1 + 2 q q 2 [ n + 1 ] q [ n ] q 2 x 2 + 1 q 3 [ n + 1 ] q [ n + 2 ] q [ n ] q 2 x 3 , B n , q ( t 4 , x ) = 1 [ n ] q 3 x + 1 q 3 ( 1 + 3 q + 3 q 2 ) [ n + 1 ] q [ n ] q 3 x 2 + 1 q 5 [ 2 ] q ( 1 + 3 q + 5 q 2 + 3 q 3 ) [ n + 1 ] q [ n + 2 ] q [ n ] q 3 x 3 + 1 q 6 [ 2 ] q [ 3 ] q [ 4 ] q ( 1 + 3 q + 5 q 2 + 6 q 3 + 5 q 4 + 3 q 5 + q 6 ) × [ n + 1 ] q [ n + 2 ] q [ n + 3 ] q [ n ] q 3 x 4 .

Now, we present higher order moments for the operators (1.4).

Lemma 4 Let 0 < q < 1 , we have

The proof of Lemma 4 can be obtained by using Lemma 3.

We consider the following classes of functions:

Theorem 3 Let q n ( 0 , 1 ) , then the sequence { D n q n ( f ) } converges to f uniformly on [ 0 , A ] for each f C 2 [ 0 , ) if and only if lim n q n = 1 .

Theorem 4 Assume that q n ( 0 , 1 ) , q n 1 and q n n a as n . For any f C 2 [ 0 , ) such that f , f C 2 [ 0 , ) the following equality holds
lim n [ n ] q n ( D n q n ( f ; x ) f ( x ) ) = ( x 2 + 2 x ) f ( x )

uniformly on any [ 0 , A ] , A > 0 .

Proof Let f , f , f C 2 [ 0 , ) and x [ 0 , ) be fixed. By using Taylor’s formula, we may write
f ( t ) = f ( x ) + f ( x ) ( t x ) + 1 2 f ( x ) ( t x ) 2 + r ( t ; x ) ( t x ) 2 ,
(4.1)
where r ( t ; x ) is the Peano form of the remainder, r ( ; x ) C 2 [ 0 , ) and lim t x r ( t ; x ) = 0 . Applying D n q n to (4.1), we obtain
[ n ] q n ( D n q n ( f ; x ) f ( x ) ) = 1 2 f ( x ) [ n ] q n D n q n ( ( t x ) 2 ; x ) + [ n ] q n D n q n ( r ( t ; x ) ( t x ) 2 ; x ) .
By the Cauchy-Schwarz inequality, we have
D n q n ( r ( t ; x ) ( t x ) 2 ; x ) D n q n ( r 2 ( t ; x ) ; x ) D n q n ( ( t x ) 4 ; x ) .
(4.2)
Observe that r 2 ( x ; x ) = 0 and r 2 ( ; x ) C 2 [ 0 , ) . Then it follows from Theorem 3 and Lemma 4, that
lim n D n q n ( r 2 ( t ; x ) ; x ) = r 2 ( x ; x ) = 0
(4.3)
uniformly with respect to x [ 0 , A ] . Now from (4.2), (4.3) and Remark 2, we get immediately
lim n [ n ] q n D n q n ( r ( t ; x ) ( t x ) 2 ; x ) = 0 .
Then, we get the following

 □

Declarations

Acknowledgements

The work was done while the first author visited Division of General Education-mathematics, Kwangwoon University, Seoul, South Korea for collaborative research during June 15-25, 2010.

Authors’ Affiliations

(1)
School of Applied Sciences, Netaji Subhas Institute of Technology, New Delhi, India
(2)
Department of Mathematics, Kwangwoon University, Seoul, S. Korea
(3)
Division of General Education-Mathematics, Kwangwoon University, Seoul, S. Korea

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© Gupta et al.; licensee Springer 2012

This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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