Open Access

Approximation properties of the modification of q-Stancu-Beta operators which preserve x 2

Journal of Inequalities and Applications20142014:505

https://doi.org/10.1186/1029-242X-2014-505

Received: 5 April 2014

Accepted: 2 December 2014

Published: 12 December 2014

Abstract

In this paper, we introduce a new kind of modification of q-Stancu-Beta operators which preserve x 2 based on the concept of q-integer. We investigate the moments and central moments of the operators by computation, obtain a local approximation theorem, and get the pointwise convergence rate theorem and also a weighted approximation theorem.

MSC:41A10, 41A25, 41A36.

Keywords

q-integerq-Stancu-Beta operatorsweighted approximation

1 Introduction

In 2012, Aral and Gupta [1] introduced the q analog of Stancu-Beta operators as
L n , q ( f ; x ) = K ( A ; [ n ] q x ) B q ( [ n ] q x ; [ n ] q + 1 ) 0 / A u [ n ] q x 1 ( 1 + u ) q [ n ] q x + [ n ] q + 1 f ( q [ n ] q x u ) d q u ,
(1)

for every n N , q ( 0 , 1 ) , x [ 0 , ) . They estimated moments, established direct result in terms of modulus of continuity and present an asymptotic formula.

Since the types of operators which preserve x 2 are important in approximation theory, in this paper, we will introduce a modification of q-Stancu-Beta operators which will be defined in (5). The advantage of these new operators is that they reproduce not only constant functions but also x 2 .

Firstly, we recall some concepts of q-calculus. All of the results can be found in [2]. For any fixed real number 0 < q 1 and each nonnegative integer k, we denote q-integers by [ k ] q , where
[ k ] q = { 1 q k 1 q , q 1 ; k , q = 1 .
Also the q-factorial and q-binomial coefficients are defined as follows:
[ k ] q ! = { [ k ] q [ k 1 ] q [ 1 ] q , k = 1 , 2 , ; 1 , k = 0
and
[ n k ] q = [ n ] q ! [ k ] q ! [ n k ] q ! ( n k 0 ) .
The q-improper integrals are defined as
0 / A f ( x ) d q x = ( 1 q ) f ( q n A ) q n A , A > 0 ,
(2)

provided the sums converge absolutely.

The q-Beta integral is defined as
B q ( t ; s ) = K ( A ; t ) 0 / A x t 1 ( 1 + x ) q t + s d q x ,
(3)

where K ( x ; t ) = 1 x + 1 x t ( 1 + 1 x ) q t ( 1 + x ) q 1 t , and ( 1 + x ) q τ = ( 1 + x ) ( 1 + q x ) ( 1 + q 2 x ) ( 1 + q τ x ) ( 1 + q τ + 1 x ) ( 1 + q τ + 2 x ) , τ > 0 ( τ = t + s ).

In particular for any positive integer n,
K ( x ; n ) = q n ( n 1 ) 2 , K ( x ; 0 ) = 1 and B q ( t ; s ) = Γ q ( t ) Γ q ( s ) Γ q ( t + s ) .
(4)
For f C [ 0 , ) , q ( 0 , 1 ) , and n N , we introduce the new modification of q-Stancu-Beta operators L n , q ( f , x ) as
L n , q ( f ; x ) = K ( A ; [ n ] q v n ( x ) ) B q ( [ n ] q v n ( x ) ; [ n ] q + 1 ) 0 / A u [ n ] q v n ( x ) 1 ( 1 + u ) q [ n ] q v n ( x ) + [ n ] q + 1 f ( q [ n ] q v n ( x ) u ) d q u ,
(5)
where
v n ( x ) = q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 1 2 [ n ] q .
(6)

2 Some preliminary results

In this section we give the following lemmas, which we need to prove our theorems.

Lemma 1 (see [[1], Lemma 1])

The following equalities hold:
L n , q ( 1 ; x ) = 1 , L n , q ( t ; x ) = x , L n , q ( t 2 ; x ) = ( [ n ] q x + 1 ) x q ( [ n ] q 1 ) .
Lemma 2 Let q ( 0 , 1 ) , x [ 0 , ) , we have
L n , q ( 1 ; x ) = 1 , L n , q ( t ; x ) = q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 1 2 [ n ] q , L n , q ( t 2 ; x ) = x 2 .
(7)
Proof From Lemma 1, we get L n , q ( 1 ; x ) = 1 and L n , q ( t ; x ) = q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 1 2 [ n ] q easily. Finally, we have
L n , q ( t 2 ; x ) = ( [ n ] q v n ( x ) + 1 ) v n ( x ) q ( [ n ] q 1 ) = [ n ] q q [ n ] q q ( 1 4 [ n ] q 2 + q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 1 [ n ] q q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 ) + 1 q [ n ] q q ( q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 1 2 [ n ] q ) = x 2 .

Lemma 2 is proved. □

Remark 1 Let n N and x [ 0 , ) , then for every q ( 0 , 1 ) , by Lemma 2, we have
L n , q ( 1 + t 2 ; x ) = 1 + x 2 .
(8)
Lemma 3 For every q ( 0 , 1 ) and x [ 0 , ) , we have
L n , q ( ( t x ) 2 ; x ) = 2 x 2 2 x q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 + x [ n ] q .
(9)

Proof Since L n , q ( ( t x ) 2 ; x ) = L n , q ( t 2 ; x ) 2 x L n , q ( t ; x ) + x 2 and from Lemma 2, we get Lemma 3 easily. □

Remark 2 Let the sequence q = { q n } satisfy that q n ( 0 , 1 ) and q n 1 as n , then for any fixed x [ 0 , ) , by Lemma 3, we have
lim n L n , q n ( ( t x ) 2 ; x ) = 0 .
(10)

3 Local approximation

In this section we establish direct local approximation theorem in connection with the operators L n , q ( f ; x ) .

We denote the space of all real valued continuous bounded functions f defined on the interval [ 0 , ) by C B [ 0 , ) . The norm on the space C B [ 0 , ) is given by f = sup { | f ( x ) | : x [ 0 , ) } .

Further let us consider Peetre’s K-functional:
K 2 ( f ; δ ) = inf g W 2 { f g + δ g } ,

where δ > 0 and W 2 = { g C B [ 0 , ) : g , g C B [ 0 , ) } .

For f C B [ 0 , ) , the modulus of continuity of second order is defined by
ω 2 ( f ; δ ) = sup 0 < h δ sup x [ 0 , ) | f ( x + 2 h ) 2 f ( x + h ) + f ( x ) | ,
by [[3], p.177] there exists an absolute constant C > 0 such that
K 2 ( f ; δ ) C ω 2 ( f ; δ ) , δ > 0 .
(11)
For f C B [ 0 , ) , the modulus of continuity is defined by
ω ( f ; δ ) = sup 0 < h δ sup x [ 0 , ) | f ( x + h ) f ( x ) | .

Our first result is a direct local approximation theorem for the operators L n , q ( f ; x ) .

Theorem 1 For q ( 0 , 1 ) , x [ 0 , ) , n N , and f C B [ 0 , ) , we have
| L n , q ( f , x ) f ( x ) | C ω 2 ( f ; 2 x 2 2 x q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 + x [ n ] q + ( x + 1 2 [ n ] q q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 ) 2 ) + ω ( f ; x + 1 2 [ n ] q q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 ) .
(12)
Proof For x ( 0 , ] , we define the auxiliary operators L n , q ¯ ( f ; x )
L n , q ¯ ( f ; x ) = L n , q ( f ; x ) f ( q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 1 2 [ n ] q ) + f ( x ) .
(13)
Obviously, we have
L n , q ¯ ( t x ; x ) = 0 .
(14)
Let g W 2 , by Taylor’s expansion, we have
g ( t ) = g ( x ) + g ( x ) ( t x ) + x t ( t u ) g ( u ) d u , x , t [ 0 , ) .
Using (14), we get
L n , q ¯ ( g ; x ) = g ( x ) + L n , q ¯ ( x t ( t u ) g ( u ) d u ; x ) ,
hence, by Lemma 3, we have
| L n , q ¯ ( g ; x ) g ( x ) | = | L n , q ( x t ( t u ) g ( u ) d u ; x ) | + | q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 1 2 [ n ] q x [ u ( q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 1 2 [ n ] q ) ] g ( u ) d u | L n , q ( | x t ( t u ) | g ( u ) | d u | ; x ) + q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 1 2 [ n ] q x | u ( q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 1 2 [ n ] q ) | | g ( u ) | d u [ 2 x 2 2 x q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 + x [ n ] q + ( x + 1 2 [ n ] q q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 ) 2 ] g .
On the other hand, using (13) and Lemma 2, we have
| L n , q ¯ ( f ; x ) | | L n , q ( f ; x ) | + 2 f f L n , q ( 1 ; x ) + 2 f 3 f .
(15)
Thus,
| L n , q ( f ; x ) f ( x ) | | L n , q ¯ ( f g ; x ) ( f g ) ( x ) | + | L n , q ¯ ( g ; x ) g ( x ) | + | f ( q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 1 2 [ n ] q ) f ( x ) | 4 f g + | f ( q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 1 2 [ n ] q ) f ( x ) | + [ 2 x 2 2 x q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 + x [ n ] q + ( x + 1 2 [ n ] q q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 ) 2 ] g .
Hence taking the infimum on the right-hand side over all g W 2 , we get
| L n , q ( f ; x ) f ( x ) | 4 K 2 ( f ; 2 x 2 2 x q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 + x [ n ] q + ( x + 1 2 [ n ] q q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 ) 2 ) + ω ( f ; x + 1 2 [ n ] q q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 ) .
By (11), for every q ( 0 , 1 ) , we have
| L n , q ( f , x ) f ( x ) | C ω 2 ( f ; 2 x 2 2 x q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 + x [ n ] q + ( x + 1 2 [ n ] q q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 ) 2 ) + ω ( f ; x + 1 2 [ n ] q q [ n ] q q [ n ] q x 2 + 1 4 [ n ] q 2 ) .

This completes the proof of Theorem 1. □

4 Rate of convergence

Let B 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. We denote the subspace of all continuous functions belonging to B x 2 [ 0 , ) by C 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 usual modulus of continuity of f on the closed interval [ 0 , a ] ( a > 0 ) by
ω a ( f , δ ) = sup | t x | δ sup x , t [ 0 , a ] | f ( t ) f ( x ) | .

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

Theorem 2 Let f C x 2 [ 0 , ) , q ( 0 , 1 ) and ω a + 1 ( f , δ ) be the modulus of continuity on the finite interval [ 0 , a + 1 ] [ 0 , ) , where a > 0 . Then we have
L n , q ( f ) f C [ 0 , a ] 4 M f ( 1 + a 2 ) ( 2 a 2 2 a q [ n ] q q [ n ] q a 2 + 1 4 [ n ] q 2 + a [ n ] q ) + 2 ω a + 1 ( f ; 2 a 2 2 a q [ n ] q q [ n ] q a 2 + 1 4 [ n ] q 2 + a [ n ] q ) .
(16)
Proof For x [ 0 , a ] and t > a + 1 , we have t x t a > 1 . Hence ( t x ) 2 > 1 . Thus 2 + 3 x 2 + 2 ( t x ) 2 ( 2 + 3 x 2 ) ( t x ) 2 + 2 ( t x ) 2 = ( 4 + 3 x 2 ) ( t x ) 2 ( 4 + 3 a 2 ) ( t x ) 2 4 ( 1 + a 2 ) ( t x ) 2 . Hence, we obtain
| f ( t ) f ( x ) | 4 M f ( 1 + a 2 ) ( t x ) 2 .
(17)
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 ) , δ > 0 .
(18)
From (17) and (18), we get
| f ( t ) f ( x ) | 4 M f ( 1 + a 2 ) ( t x ) 2 + ( 1 + | t x | δ ) ω a + 1 ( f ; δ ) .
(19)
For x [ 0 , a ] and t 0 , by Schwarz’s inequality, Lemma 2, and Lemma 3, we have
| L n , q ( f ; x ) f ( x ) | L n , q ( | f ( t ) f ( x ) | ; x ) 4 M f ( 1 + a 2 ) L n , q ( ( t x ) 2 ; x ) + ω a + 1 ( f ; δ ) ( 1 + 1 δ L n , q ( ( t x ) 2 ; x ) ) 4 M f ( 1 + a 2 ) ( 2 a 2 2 a q [ n ] q q [ n ] q a 2 + 1 4 [ n ] q + a [ n ] q ) + ω a + 1 ( f , δ ) ( 1 + 1 δ 2 a 2 2 a q [ n ] q q [ n ] q a 2 + 1 4 [ n ] q + a [ n ] q ) .

By taking δ = 2 a 2 2 a q [ n ] q q [ n ] q a 2 + 1 4 [ n ] q + a [ n ] q , we get the assertion of Theorem 2. □

5 Weighted approximation

Now we will discuss the weighted approximation theorems.

Theorem 3 Let the sequence { q n } satisfy 0 < q n < 1 and q n 1 as n , for f C x 2 [ 0 , ) , we have
lim n L n , q n ( f ) f x 2 = 0 .
(20)
Proof By using the Korovkin theorem in [4], we see that it is sufficient to verify the following three conditions:
lim n L n , q n ( t v ; x ) x v x 2 , v = 0 , 1 , 2 .
(21)

Since L n , q n ( 1 ; x ) = 1 and L n , q n ( t 2 ; x ) = x 2 (see Lemma 2), (21) holds true for v = 0 and v = 2 .

Finally, for v = 1 , we have
L n , q n ( t ; x ) x x 2 = sup x [ 0 , ) | L n , q n ( t ; x ) x | 1 + x 2 ( 1 q [ n ] q q [ n ] q ) sup x [ 0 , ) x 1 + x 2 + 1 2 [ n ] q sup x [ 0 , ) 1 1 + x 2 1 q [ n ] q q [ n ] q + 1 2 [ n ] q ,

since lim n q n = 1 , we get lim n ( 1 q [ n ] q q [ n ] q ) = 0 and lim n 1 2 [ n ] q = 0 , so the second condition of (21) holds for v = 1 as n , then the proof of Theorem 3 is completed. □

Declarations

Acknowledgements

The author thanks the editor and referee(s) for several important comments and suggestions, which improved the quality of the paper. This work is supported by the Educational Office of Fujian Province of China (Grant No. JA13269), the Startup Project of Doctor Scientific Research of Quanzhou Normal University, Fujian Provincial Key Laboratory of Data Intensive Computing and Key Laboratory of Intelligent Computing and Information Processing, Fujian Province University.

Authors’ Affiliations

(1)
School of Mathematics and Computer Science, Quanzhou Normal University

References

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© Cai; licensee Springer. 2014

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