Open Access

Some statistical approximation properties of Kantorovich-type q-Bernstein-Stancu operators

Journal of Inequalities and Applications20142014:10

https://doi.org/10.1186/1029-242X-2014-10

Received: 9 August 2013

Accepted: 9 December 2013

Published: 6 January 2014

Abstract

In this paper two kinds of Kantorovich-type q-Bernstein-Stancu operators are introduced, and the statistical approximation properties of these operators are investigated. Furthermore, by means of modulus of continuity, the rates of statistical convergence of these operators are also studied.

MSC:41A10, 41A25, 41A36.

Keywords

Kantorovich-type q-Bernstein-Stancu operators statistical approximation q-integers modulus of continuity Korovkin-type theorem rate of statistical convergence

1 Introduction

In 1997, Phillips [1] introduced and studied q analogue of Bernstein polynomials. During the last decade, the applications of q-calculus in the approximation theory have become one of the main area of research, q-calculus has been extensively used for constructing various generalizations of many classical approximation processes. It is well known that many q-extensions of the classical objects arising in the approximation theory have been recently introduced and studied (e.g., see [28]). Recently the statistical approximation properties have also been investigated for q-analogue polynomials. For instance, in [9] Kantorovich-type q-Bernstein operators; in [10]q-Baskakov-Kantorovich operators; in [11] Kantorovich-type q-Szász-Mirakjan operators; in [12]q-Bleimann, Butzer and Hahn operators; in [13]q-analogue of MKZ operators were introduced and their statistical approximation properties were studied.

The goal of this paper is to introduce two kinds of new Kantorovich-type q-Bernstein-Stancu operators and to study the statistical approximation properties of these operators with the help of the Korovkin-type approximation theorem. We also estimate the rate of statistical convergence of the mentioned sequences of operators to the appropriate function f, respectively.

Before proceeding further, let us give some basic definitions and notations from q-calculus. Details on q-integers can be found in [14, 15].

Let q > 0 , for each nonnegative integer k, the q-integer [ k ] q and the q-factorial [ k ] q ! are defined by
[ k ] q : = { ( 1 q k ) / ( 1 q ) , q 1 , k , q = 1
and
[ k ] q ! : = { [ k ] q [ k 1 ] q [ 1 ] q , k 1 , 1 , k = 0 ,

respectively.

Then for q > 0 and integers n, k, n k 0 , we have
[ k + 1 ] q = 1 + q [ k ] q and [ k ] q + q k [ n k ] q = [ n ] q .
For the integers n, k, n k 0 , the q-binomial coefficient is defined by
[ n k ] q : = [ n ] q ! [ k ] q ! [ n k ] q ! .
For an arbitrary function f ( x ) , the q-differential is given by
d q f ( x ) = f ( q x ) f ( x ) .
The q-Jackson integral in the interval [ 0 , b ] is defined as
0 b f ( t ) d q t = ( 1 q ) b j = 0 f ( q j b ) q j , 0 < q < 1 ,

provided that sums converge absolutely.

Suppose 0 < a < b . The q-Jackson integral in a generic interval [ a , b ] is defined as
a b f ( t ) d q t = 0 b f ( t ) d q t 0 a f ( t ) d q t , 0 < q < 1 .

2 Construction of the operators

In this part, we first construct the Kantorovich-type q-Bernstein-Stancu operators as follows.

Definition 1 Let f be a q-integrable function on [ 0 , 1 ] , for 0 α β , x [ 0 , 1 ] , n N , 0 < q < 1 , we define the Kantorovich-type q-Bernstein-Stancu operators by
S n , q ( α , β ) ( f ; x ) = ( [ n + 1 ] q + β ) k = 0 n q k p n , k ( q ; x ) [ k ] q + α [ n + 1 ] q + β [ k + 1 ] q + α [ n + 1 ] q + β f ( t ) d q t ,
(1)
where
p n , k ( q ; x ) = [ n k ] q x k s = 0 n k 1 ( 1 q s x ) = [ n k ] q x k ( 1 x ) q n k .

In the following we give some lemmas.

Lemma 1 For S n , q ( α , β ) ( t i ; x ) , i = 0 , 1 , 2 , 0 < q < 1 , we have
S n , q ( α , β ) ( 1 ; x ) = 1 ,
(2)
S n , q ( α , β ) ( t ; x ) = [ n ] q [ n + 1 ] q + β x + 1 + 2 α [ 2 ] q ( [ n + 1 ] q + β ) ,
(3)
S n , q ( α , β ) ( t 2 ; x ) = q [ n ] q [ n 1 ] q ( [ n + 1 ] q + β ) 2 x 2 + ( 2 + 3 q + q 2 + 3 α + 3 α q ) [ n ] q [ 3 ] q ( [ n + 1 ] q + β ) 2 x S n , q ( α , β ) ( t 2 ; x ) = + 1 + 3 α + 3 α 2 [ 3 ] q ( [ n + 1 ] q + β ) 2 .
(4)

Proof For i = 0 , since k = 0 n p n , k ( q ; x ) = 1 , [ k ] q + α [ n + 1 ] q + β [ k + 1 ] q + α [ n + 1 ] q + β d q t = q k [ n + 1 ] q + β , so Eq. (2) holds.

For i = 1 ,
S n , q ( α , β ) ( t ; x ) = ( [ n + 1 ] q + β ) k = 0 n q k p n , k ( q ; x ) [ k ] q + α [ n + 1 ] q + β [ k + 1 ] q + α [ n + 1 ] q + β t d q t = k = 0 n p n , k ( q ; x ) 1 + 2 α + [ 2 ] q [ k ] q [ 2 ] q ( [ n + 1 ] q + β ) = [ n ] q [ n + 1 ] q + β x + 1 + 2 α [ 2 ] q ( [ n + 1 ] q + β ) .
For i = 2 ,
S n , q ( α , β ) ( t 2 ; x ) = ( [ n + 1 ] q + β ) k = 0 n q k p n , k ( q ; x ) [ k ] q + α [ n + 1 ] q + β [ k + 1 ] q + α [ n + 1 ] q + β t 2 d q t = [ n ] q [ 3 ] q ( [ n + 1 ] q + β ) 2 × k = 0 n p n , k ( q ; x ) 1 + 3 α + 3 α 2 + ( 1 + 2 q + 3 α [ 2 ] q ) [ k ] q + [ 3 ] q [ k ] q 2 [ n ] q = q [ n ] q [ n 1 ] q ( [ n + 1 ] q + β ) 2 x 2 + ( 2 + 3 q + q 2 + 3 α + 3 α q ) [ n ] q [ 3 ] q ( [ n + 1 ] q + β ) 2 x + 1 + 3 α + 3 α 2 [ 3 ] q ( [ n + 1 ] q + β ) 2 .

 □

Lemma 2 For n N , x [ 0 , 1 ] , 0 < q < 1 , 0 α β , we have
S n , q ( α , β ) ( ( t x ) 2 ; x ) 13 ( 1 + β ) 2 [ n + 1 ] q + β ( 1 4 + 1 [ n + 1 ] q + β ) .
Proof In view of Lemma 1 and max x [ 0 , 1 ] x ( 1 x ) = 1 4 , by a simple computation, we have
S n , q ( α , β ) ( ( t x ) 2 ; x ) = S n , q ( α , β ) ( t 2 ; x ) 2 x S n , q ( α , β ) ( t ; x ) + x 2 S n , q ( α , β ) ( 1 ; x ) = 1 + 3 α + 3 α 2 [ 3 ] q ( [ n + 1 ] q + β ) 2 + x [ 2 ] q [ 3 ] q ( [ n + 1 ] q + β ) 2 × [ [ 2 ] q ( 2 + 3 q + q 2 + 3 α + 3 α q ) [ n ] q 2 ( 1 + 2 α ) [ 3 ] q ( [ n + 1 ] q + β ) ] + q [ n ] q [ n 1 ] q 2 [ n ] q ( [ n + 1 ] q + β ) + ( [ n + 1 ] q + β ) 2 ( [ n + 1 ] q + β ) 2 x 2 13 ( 1 + β ) 2 [ n + 1 ] q + β ( x ( 1 x ) + 1 [ n + 1 ] q + β ) 13 ( 1 + β ) 2 [ n + 1 ] q + β ( 1 4 + 1 [ n + 1 ] q + β ) .

 □

Lemma 3 ([9])

Let 0 a < b , 0 < q < 1 , and let f be a positive function defined on the interval [ 0 , b ] . If f is monotone increasing on [ 0 , b ] , then a b f ( t ) d q t 0 in this interval.

It is clear that the operator S n , q ( α , β ) ( f ; x ) is a linear and positive operator for any monotone increasing function f [ 0 , 1 ] .

Remark 1 To guarantee the positivity of S n , q ( α , β ) ( f ; x ) , f must be a monotone increasing function on the interval [ 0 , 1 ] . But for the function f this condition is strong. In order to solve the problems, a special type of q-integral, which is the Riemann-type q-integral, is defined by Marinković et al. [16].

Definition 2 ([16])

Let 0 < a < b , 0 < q < 1 . The Riemann-type q-integral is defined as
R q ( f ; a , b ) = a b f ( t ) d q R t = ( 1 q ) ( b a ) j = 0 f ( a + ( b a ) q j ) q j ,
(5)

provided the sums converge absolutely.

We now redefine S n , q ( α , β ) ( f ; x ) by putting the Riemann-type q-integral into the operators instead of the general q-integral as
S ˜ n , q ( α , β ) ( f ; x ) = ( [ n + 1 ] q + β ) k = 0 n q k p n , k ( q ; x ) [ k ] q + α [ n + 1 ] q + β [ k + 1 ] q + α [ n + 1 ] q + β f ( t ) d q R t ,
(6)

where f is a Riemann-type q-integrable function on [ 0 , 1 ] .

Let us give some lemmas as follows.

Lemma 4 Let 0 < q < 1 , 0 α β , then { S ˜ n , q ( α , β ) ( f ; x ) } is a linear and positive operator.

Proof The proof is clear, so we omit it. □

Lemma 5 For S ˜ n , q ( α , β ) ( t i ; x ) , i = 0 , 1 , 2 , 0 < q < 1 , 0 α β , we have
S ˜ n , q ( α , β ) ( 1 ; x ) = 1 ,
(7)
S ˜ n , q ( α , β ) ( t ; x ) = 2 q [ n ] q [ 2 ] q ( [ n + 1 ] q + β ) x + 1 + [ 2 ] q α [ 2 ] q ( [ n + 1 ] q + β ) ,
(8)
S ˜ n , q ( α , β ) ( t 2 ; x ) = q [ n ] q [ n 1 ] q ( [ n + 1 ] q + β ) 2 ( 1 + 2 ( q 1 ) [ 2 ] q + ( q 1 ) 2 [ 3 ] q ) x 2 S ˜ n , q ( α , β ) ( t 2 ; x ) = + [ n ] q ( [ n + 1 ] q + β ) 2 ( 1 + 2 α + 2 ( q 1 ) ( 1 + α ) [ 2 ] q + 2 [ 2 ] q S ˜ n , q ( α , β ) ( t 2 ; x ) = + 2 ( q 1 ) [ 3 ] q + ( q 1 ) 2 [ 3 ] q ) x + 1 ( [ n + 1 ] q + β ) 2 ( 1 [ 3 ] q + 2 α [ 2 ] q + α 2 ) .
(9)
Proof By Definition 2, we have
[ k ] q + α [ n + 1 ] q + β [ k + 1 ] q + α [ n + 1 ] q + β d q R t = q k [ n + 1 ] q + β ,
(10)
[ k ] q + α [ n + 1 ] q + β [ k + 1 ] q + α [ n + 1 ] q + β t d q R t = 1 ( [ n + 1 ] q + β ) 2 { q k ( [ k ] q + α ) + q 2 k [ 2 ] q } ,
(11)
[ k ] q + α [ n + 1 ] q + β [ k + 1 ] q + α [ n + 1 ] q + β t 2 d q R t = 1 ( [ n + 1 ] q + β ) 3 { q k ( [ k ] q + α ) 2 + 2 q 2 k [ 2 ] q ( [ k ] q + α ) + q 3 k [ 3 ] q } .
(12)
Hence, by using the equality k = 0 n p n , k ( q ; x ) = 1 and Eq. (10), we get
S ˜ n , q ( α , β ) ( 1 ; x ) = 1 .
By using Eq. (11) and the equality q k = ( q 1 ) [ k ] q + 1 , we have
S ˜ n , q ( α , β ) ( t ; x ) = 1 [ n + 1 ] q + β k = 0 n p n , k ( q ; x ) { [ k ] q + α + ( q 1 ) [ k ] q + 1 [ 2 ] q } = 2 q [ n ] q [ 2 ] q ( [ n + 1 ] q + β ) x + 1 + [ 2 ] q α [ 2 ] q ( [ n + 1 ] q + β ) .
By using Eq. (12) and the equality q k = ( q 1 ) [ k ] q + 1 , we have
S ˜ n , q ( α , β ) ( t 2 ; x ) = 1 ( [ n + 1 ] q + β ) 2 k = 0 n p n , k ( q ; x ) { ( [ k ] q + α ) 2 + 2 q k [ 2 ] q ( [ k ] q + α ) + q 2 k [ 3 ] q } = [ n ] q ( [ n + 1 ] q + β ) 2 k = 0 n p n , k ( q ; x ) [ n ] q { 1 [ 3 ] q + 2 α [ 2 ] q + α 2 + ( 2 α + 2 ( q 1 ) α [ 2 ] q + 2 [ 2 ] q + 2 ( q 1 ) [ 3 ] q ) [ k ] q + ( 1 + 2 ( q 1 ) [ 2 ] q + ( q 1 ) 2 [ 3 ] q ) [ k ] q 2 } = q [ n ] q [ n 1 ] q ( [ n + 1 ] q + β ) 2 ( 1 + 2 ( q 1 ) [ 2 ] q + ( q 1 ) 2 [ 3 ] q ) x 2 + [ n ] q ( [ n + 1 ] q + β ) 2 ( 1 + 2 α + 2 ( q 1 ) ( 1 + α ) [ 2 ] q + 2 [ 2 ] q + 2 ( q 1 ) [ 3 ] q + ( q 1 ) 2 [ 3 ] q ) x + 1 ( [ n + 1 ] q + β ) 2 ( 1 [ 3 ] q + 2 α [ 2 ] q + α 2 ) .

 □

Lemma 6 For n N , x [ 0 , 1 ] , 0 < q < 1 , 0 α β , we have
S ˜ n , q ( α , β ) ( ( t x ) 2 ; x ) 2 ( 1 q ) 2 + 1 [ n + 1 ] q + β ( 3 + ( 1 + α ) 2 + ( 1 + β ) 2 [ n + 1 ] q + β ) .
Proof In view of Lemma 5, by a simple computation, we have
S ˜ n , q ( α , β ) ( ( t x ) 2 ; x ) = S ˜ n , q ( α , β ) ( t 2 ; x ) 2 x S ˜ n , q ( α , β ) ( t ; x ) + x 2 S ˜ n , q ( α , β ) ( 1 ; x ) = [ q [ n ] q [ n 1 ] q ( [ n + 1 ] q + β ) 2 ( 1 + 2 ( q 1 ) [ 2 ] q + ( q 1 ) 2 [ 3 ] q ) 4 q [ n ] q [ 2 ] q ( [ n + 1 ] q + β ) + 1 ] x 2 + [ n ] q ( [ n + 1 ] q + β ) 2 ( 1 + 2 α + 2 ( q 1 ) ( 1 + α ) [ 2 ] q + 2 [ 2 ] q + 2 ( q 1 ) [ 3 ] q + ( q 1 ) 2 [ 3 ] q ) x 2 ( 1 + [ 2 ] q α ) [ 2 ] q ( [ n + 1 ] q + β ) x + 1 ( [ n + 1 ] q + β ) 2 ( 1 [ 3 ] q + 2 α [ 2 ] q + α 2 ) ( 2 ( 1 q ) 2 + ( 1 + β ) 2 ( [ n + 1 ] q + β ) 2 ) x 2 + 3 [ n + 1 ] q + β x + ( 1 + α ) 2 ( [ n + 1 ] q + β ) 2 2 ( 1 q ) 2 + 1 [ n + 1 ] q + β ( 3 + ( 1 + α ) 2 + ( 1 + β ) 2 [ n + 1 ] q + β ) .

 □

3 Statistical approximation of Korovkin type

Now, let us recall the concept of statistical convergence which was introduced by Fast [17].

Let set K N and K n = { k n : k K } , the natural density of K is defined by δ ( K ) : = lim n 1 n | K n | if the limit exists (see [18]), where | K n | denotes the cardinality of the set K n .

A sequence x = { x k } is called statistically convergent to a number L if for every ε > 0 , δ { k N : | x k L | ε } = 0 . This convergence is denoted as st - lim k x k = L .

Note that any convergent sequence is statistically convergent, but not conversely. Details can be found in [19].

In approximation theory, the concept of statistical convergence was used by Gadjiev and Orhan [20]. They proved the following Bohman-Korovkin type approximation theorem for statistical convergence.

Theorem 1 ([20])

If the sequence of linear positive operators A n : C [ a , b ] C [ a , b ] satisfies the conditions
st - lim n A n ( e υ ; ) e υ C [ a , b ] = 0
for e υ ( t ) = t υ , υ = 0 , 1 , 2 . Then, for any f C [ a , b ] ,
st - lim n A n ( f ; ) f C [ a , b ] = 0 .
Theorem 2 Let q = { q n } , 0 < q n < 1 , be a sequence satisfying the following condition
st - lim n q n = 1 , st - lim n q n n = a ( a < 1 ) and st - lim n 1 [ n ] q n = 0 ,
(13)
then for any monotone increasing function f C [ 0 , 1 ] , we have
st - lim n S n , q n ( α , β ) ( f ; ) f C [ 0 , 1 ] = 0 .

Proof From Theorem 1, it is enough to prove that st - lim n S n , q n ( α , β ) ( e υ ; ) e υ C [ 0 , 1 ] = 0 for e υ ( t ) = t υ , υ = 0 , 1 , 2 .

From Eq. (2), we can easily get
st - lim n S n , q n ( α , β ) ( e 0 ; ) e 0 C [ 0 , 1 ] = 0 .
(14)
From Eq. (3), we have
S n , q n ( α , β ) ( e 1 ; x ) e 1 ( x ) = ( [ n ] q n [ n + 1 ] q n + β 1 ) x + 1 + 2 α [ 2 ] q n ( [ n + 1 ] q n + β ) .
In view of [ n + 1 ] q n = 1 + q n [ n ] q n , for β > 0 we have
S n , q n ( α , β ) ( e 1 ; ) e 1 C [ 0 , 1 ] ( 1 q n ) + ( 1 + β + 1 + 2 α [ 2 ] q n ) 1 [ n ] q n .
(15)
Now, for every given ε > 0 , let us define the following sets:
U = { k : S n , q k ( α , β ) ( e 1 ; ) e 1 C [ 0 , 1 ] ε } , U 1 = { k : 1 q k ε 2 } , U 2 = { k : ( 1 + β + 1 + 2 α [ 2 ] q k ) 1 [ k ] q k ε 2 } .
From inequality (15), one can see that U U 1 U 2 , so we have
δ { k n : S n , q k ( α , β ) ( e 1 ; ) e 1 C [ 0 , 1 ] ε } δ { k n : 1 q k ε 2 } + δ { k n : ( 1 + β + 1 + 2 α [ 2 ] q k ) 1 [ k ] q k ε 2 } .

By condition (13), it is clear that st - lim n ( 1 q n ) = 0 , st - lim n ( 1 + β + 1 + 2 α [ 2 ] q n ) 1 [ n ] q n = 0 .

So we have
st - lim n S n , q n ( α , β ) ( e 1 ; ) e 1 C [ 0 , 1 ] = 0 .
(16)
In view of Eq. (4) and the equality [ n + 1 ] q n = 1 + q n [ n ] q n , by a simple computation, for β > 0 we have
S n , q n ( α , β ) ( e 2 ; ) e 2 C [ 0 , 1 ] 11 + 9 α + 3 α 2 + 4 β + β 2 [ n ] q n .
Now, for every given ε > 0 , let us define the following sets:
T = { k : S n , q k ( α , β ) ( e 2 ; ) e 2 C [ 0 , 1 ] ε } , T 1 = { k : 11 + 9 α + 3 α 2 + 4 β + β 2 [ k ] q k ε } .
It is clear that T T 1 , so we get
δ { k n : S n , q k ( α , β ) ( e 2 ; ) e 2 C [ 0 , 1 ] ε } δ { k n : 11 + 9 α + 3 α 2 + 4 β + β 2 [ k ] q k ε } .
By condition (13), we have
st - lim n 11 + 9 α + 3 α 2 + 4 β + β 2 [ n ] q n = 0 ,
so, we can get
st - lim n S n , q n ( α , β ) ( e 2 ; ) e 2 C [ 0 , 1 ] = 0 .
(17)

In view of Eqs. (14), (16) and (17), the proof is complete. □

Theorem 3 Let q = { q n } , 0 < q n < 1 , be a sequence satisfying condition (13), then for all f C [ 0 , 1 ] , we have st - lim n S ˜ n , q n ( α , β ) ( f ; ) f C [ 0 , 1 ] = 0 .

Proof From Theorem 1, it is enough to prove that st - lim n S ˜ n , q n ( α , β ) ( e υ ; ) e υ C [ 0 , 1 ] = 0 , for e υ ( t ) = t υ , υ = 0 , 1 , 2 .

From Eq. (7), we can easily get
st - lim n S ˜ n , q n ( α , β ) ( e 0 ; ) e 0 C [ 0 , 1 ] = 0 .
(18)
From Eq. (8), we have
S ˜ n , q n ( α , β ) ( e 1 ; x ) e 1 ( x ) = ( 2 q n [ n ] q n [ 2 ] q n ( [ n + 1 ] q n + β ) 1 ) x + 1 + [ 2 ] q n α [ 2 ] q n ( [ n + 1 ] q n + β ) .
In view of [ n + 1 ] q n = 1 + q n [ n ] q n , for β > 0 we have
S ˜ n , q n ( α , β ) ( e 1 ; ) e 1 C [ 0 , 1 ] ( 1 q n ) + [ ( 1 + q n ) ( 1 + β ) + 1 + [ 2 ] q n α ] 1 [ n ] q n .
(19)
Now, for every given ε > 0 , let us define the following sets:
U ˜ = { k : S ˜ n , q k ( α , β ) ( e 1 ; ) e 1 C [ 0 , 1 ] ε } , U ˜ 1 = { k : 1 q k ε 2 } , U ˜ 2 = { k : [ ( 1 + q k ) ( 1 + β ) + 1 + [ 2 ] q k α ] 1 [ k ] q k ε 2 } .
From inequality (19), one can see that U ˜ U ˜ 1 U ˜ 2 , so we have
δ { k n : S ˜ n , q k ( α , β ) ( e 1 ; ) e 1 C [ 0 , 1 ] ε } δ { k n : 1 q k ε 2 } + δ { k n : [ ( 1 + q k ) ( 1 + β ) + 1 + [ 2 ] q k α ] 1 [ k ] q k ε 2 } .
By condition (13), it is clear that
st - lim n ( 1 q n ) = 0 , st - lim n [ ( 1 + q n ) ( 1 + β ) + 1 + [ 2 ] q n α ] 1 [ n ] q n = 0 .
So we have
st - lim n S ˜ n , q n ( α , β ) ( e 1 ; ) e 1 C [ 0 , 1 ] = 0 .
(20)
In view of Eq. (9) and the equality [ n + 1 ] q n = 1 + q n [ n ] q n , for β > 0 and 0 < q n < 1 , by a simple computation, we have
S ˜ n , q n ( α , β ) ( e 2 ; ) e 2 C [ 0 , 1 ] 6 ( 1 q n ) + 11 + 6 α + α 2 + 6 ( 1 + β ) ( 3 + β ) [ n ] q n .
Now, for every given ε > 0 , let us define the following sets:
T ˜ = { k : S ˜ n , q k ( α , β ) ( e 2 ; ) e 2 C [ 0 , 1 ] ε } , T ˜ 1 = { k : 6 ( 1 q k ) ε 2 } , T ˜ 2 = { k : 11 + 6 α + α 2 + 6 ( 1 + β ) ( 3 + β ) [ k ] q k ε 2 } .
It is clear that T ˜ T ˜ 1 T ˜ 2 , so we get
δ { k n : S ˜ n , q k ( α , β ) ( e 2 ; ) e 2 C [ 0 , 1 ] ε } δ { k n : 6 ( 1 q k ) ε 2 } + δ { k n : 11 + 6 α + α 2 + 6 ( 1 + β ) ( 3 + β ) [ k ] q k ε 2 } .
By condition (13), we have
st - lim n 6 ( 1 q n ) = 0 , st - lim n 11 + 6 α + α 2 + 6 ( 1 + β ) ( 3 + β ) [ n ] q n = 0 .
So, we can get
st - lim n S ˜ n , q n ( α , β ) ( e 2 ; ) e 2 C [ 0 , 1 ] = 0 .
(21)

In view of Eqs. (18), (20) and (21), the proof is complete. □

4 Rates of statistical convergence

Let f C [ 0 , 1 ] , for any δ > 0 , the usual modulus of continuity for f is defined as ω ( f ; δ ) = sup 0 < h δ sup x , x + h [ 0 , 1 ] | f ( x + h ) f ( x ) | .

For f C [ 0 , 1 ] and any t , x [ 0 , 1 ] , we have | f ( t ) f ( x ) | ω ( f , | t x | ) , so for any δ > 0 , we get
ω ( f , | t x | ) { ω ( f , δ ) , | t x | < δ , ω ( f , ( t x ) 2 δ ) , | t x | δ .
In the light of ω ( f ; λ δ ) ( 1 + λ ) ω ( f ; δ ) for λ > 0 , it is clear that we have
| f ( t ) f ( x ) | ( 1 + δ 2 ( t x ) 2 ) ω ( f , δ )
(22)

for any t , x [ 0 , 1 ] and δ > 0 .

Next we will give the rates of convergence of both S n , q n ( α , β ) ( f ; x ) and S ˜ n , q n ( α , β ) ( f ; x ) by means of modulus of continuity.

Theorem 4 Let q = { q n } , 0 < q n < 1 be a sequence satisfying condition (13), then for any monotone increasing function f C [ 0 , 1 ] , we have
S n , q n ( α , β ) ( f ; ) f C [ 0 , 1 ] 2 ω ( f ; δ n ) ,
where
δ n = { 13 ( 1 + β ) 2 [ n + 1 ] q n + β ( 1 4 + 1 [ n + 1 ] q n + β ) } 1 / 2 .
(23)
Proof Using the linearity and positivity of the operator S n , q ( α , β ) ( f ; x ) , by Lemma 2 and inequality (22), we get
| S n , q ( α , β ) ( f ; x ) f ( x ) | S n , q ( α , β ) ( | f ( t ) f ( x ) | ; x ) ( 1 + δ 2 S n , q ( α , β ) ( ( t x ) 2 ; x ) ) ω ( f , δ ) [ 1 + δ 2 13 ( 1 + β ) 2 [ n + 1 ] q + β ( 1 4 + 1 [ n + 1 ] q + β ) ] ω ( f , δ ) .

Take q = { q n } , 0 < q n < 1 , to be a sequence satisfying condition (13) and choose δ = δ n in (23), we have | S n , q n ( α , β ) ( f ; x ) f ( x ) | 2 ω ( f ; δ n ) , which implies the proof is complete. □

Theorem 5 Let q = { q n } , 0 < q n < 1 , be a sequence satisfying condition (13), then for any f C [ 0 , 1 ] , we have S ˜ n , q n ( α , β ) ( f ; ) f C [ 0 , 1 ] 2 ω ( f ; δ n ) , where
δ n = { 2 ( 1 q ) 2 + 1 [ n + 1 ] q n + β ( 3 + ( 1 + α ) 2 + ( 1 + β ) 2 [ n + 1 ] q n + β ) } 1 / 2 .
(24)
Proof Using the linearity and positivity of the operator S ˜ n , q ( α , β ) ( f ; x ) , by Lemma 6 and inequality (22), we get
| S ˜ n , q ( α , β ) ( f ; x ) f ( x ) | S ˜ n , q ( α , β ) ( | f ( t ) f ( x ) | ; x ) ( 1 + δ 2 S ˜ n , q ( α , β ) ( ( t x ) 2 ; x ) ) ω ( f , δ ) { 1 + δ 2 [ 2 ( 1 q ) 2 + 1 [ n + 1 ] q + β ( 3 + ( 1 + α ) 2 + ( 1 + β ) 2 [ n + 1 ] q + β ) ] } ω ( f , δ ) .

Take q = { q n } , 0 < q n < 1 , to be a sequence satisfying condition (13) and choose δ = δ n in (24), we have | S ˜ n , q n ( α , β ) ( f ; x ) f ( x ) | 2 ω ( f ; δ n ) , which implies the proof is complete. □

Declarations

Acknowledgements

The authors are most grateful to the editor and anonymous referee for careful reading of the manuscript and valuable suggestions which helped in improving an earlier version of this paper. This work is supported by the National Natural Science Foundation of China (Grant No. 61170324), the Class A Science and Technology Project of Education Department of Fujian Province of China (Grant No. JA12324), and the Natural Science Foundation of Fujian Province of China (Grant No. 2013J01017).

Authors’ Affiliations

(1)
Department of Mathematics and Computer Science, Wuyi University
(2)
Department of Mathematics, Xiamen University

References

  1. Phillips GM: Bernstein polynomials based on the q -integers. Ann. Numer. Math. 1997, 4: 511-518.MathSciNetGoogle Scholar
  2. Aral A, Doǧru O: Bleimann, Butzer, and Hahn operators based on the q -integers. J. Inequal. Appl. 2007., 2007: Article ID 079410 10.1155/2007/79410Google Scholar
  3. Gupta V, Wang HP: The rate of convergence of q -Durrmeyer operators for 0 < q < 1 . Math. Methods Appl. Sci. 2008,31(16):1946-1955. 10.1002/mma.1012MathSciNetView ArticleGoogle Scholar
  4. Doǧru O, Gupta V: Korovkin-type approximation properties of bivariate q -Meyer-König and Zeller operators. Calcolo 2012,43(1):51-63.Google Scholar
  5. Aral A, Gupta V, Agarwal RP: Applications of q-Calculus in Operator Theory. Springer, New York; 2013.View ArticleGoogle Scholar
  6. Verma DK, Agrawal PN: Approximation by Baskakov-Durrmeyer-Stancu operators based on q -integers. Lobachevskii J. Math. 2013,43(2):187-196.MathSciNetView ArticleGoogle Scholar
  7. Ren MY, Zeng XM: Approximation by a kind of complex modified q -Durrmeyer type operators in compact disks. J. Inequal. Appl. 2012., 2012: Article ID 212 10.1186/1029-242X-2012-212Google Scholar
  8. Ren MY, Zeng XM: Approximation of the summation-integral-type q -Szász-Mirakjan operators. Abstr. Appl. Anal. 2012., 2012: Article ID 614810 10.1155/2012/614810Google Scholar
  9. Dalmanoǧlu Ö, Doǧru O: On statistical approximation properties of Kantorovich type q -Bernstein operators. Math. Comput. Model. 2010,52(5-6):760-771. 10.1016/j.mcm.2010.05.005View ArticleGoogle Scholar
  10. Gupta V, Radu C: Statistical approximation properties of q -Baskokov-Kantorovich operators. Cent. Eur. J. Math. 2009,7(4):809-818. 10.2478/s11533-009-0055-yMathSciNetView ArticleGoogle Scholar
  11. Örkcü M, Doǧru O: Weighted statistical approximation by Kantorovich type q -Szász-Mirakjan operators. Appl. Math. Comput. 2011,217(20):7913-7919. 10.1016/j.amc.2011.03.009MathSciNetView ArticleGoogle Scholar
  12. Ersan S, Doǧru O: Statistical approximation properties of q -Bleimann, Butzer and Hahn operators. Math. Comput. Model. 2009,49(7-8):1595-1606. 10.1016/j.mcm.2008.10.008View ArticleGoogle Scholar
  13. Aktuglu H, Ozarslan MA, Duman O: Matrix summability methods on the approximation of multivariate q -MKZ operators. Bull. Malays. Math. Sci. Soc. 2011,34(3):465-474.MathSciNetGoogle Scholar
  14. Kac VG, Cheung P Universitext. In Quantum Calculus. Springer, New York; 2002.View ArticleGoogle Scholar
  15. Gasper G, Rahman M Encyclopedia of Mathematics and Its Applications 35. In Basic Hypergeometric Series. Cambridge University Press, Cambridge; 1990.Google Scholar
  16. Marinković S, Rajković P, Stanković M: The inequalities for some types of q -integrals. Comput. Math. Appl. 2008,56(10):2490-2498. 10.1016/j.camwa.2008.05.035MathSciNetView ArticleGoogle Scholar
  17. Fast H: Sur la convergence statistique. Colloq. Math. 1951,2(3-4):241-244.MathSciNetGoogle Scholar
  18. Niven I, Zuckerman HS, Montgomery H: An Introduction to the Theory Numbers. 5th edition. Wiley, New York; 1991.Google Scholar
  19. Doǧru O: On statistical approximation properties of Stancu type bivariate generalization of q -Balás-Szabados operators. In Seminar on Numerical Analysis and Approximation Theory. Univ. Babeş-Bolya, Cluj-Napoca; 2006:179-194.Google Scholar
  20. Gadjiev AD, Orhan C: Some approximation theorems via statistical convergence. Rocky Mt. J. Math. 2002,32(1):129-138. 10.1216/rmjm/1030539612MathSciNetView ArticleGoogle Scholar

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© Ren and Zeng; licensee Springer. 2014

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