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Some approximation properties of a kind of q-Gamma-Stancu operators

Abstract

In this paper we propose the Stancu type generalization of a kind of q-Gamma operators. We estimate the moments of these operators and establish two direct and local approximation theorems of the operators. We also obtain the estimates of the rate of convergence and the weighted approximation of the operators. Furthermore, we present a Voronovskaya type asymptotic formula.

MSC:41A10, 41A25, 41A36.

1 Introduction

In 2007, Karsli [1] introduced a kind of new Gamma type operators defined as

L n (f;x)= ( 2 n + 3 ) ! x n + 3 n ! ( n + 2 ) ! 0 t n ( x + t ) 2 n + 4 f(t)dt,x>0.
(1)

He also estimated the rate of convergence of the operators (1) for functions with derivatives of bounded variation on (0,).

In 2009, Karsli et al. [2] gave an estimate of the rate of pointwise convergence of the operators (1) on a Lebesgue point of a bounded variation function f defined on the interval (0,).

In recent years Kim used q calculus to study several results on number theory and the related areas [3, 4]. Now we mention certain definitions based on q-integers, details can be found in [57]. For any fixed real number q>0 and each nonnegative integer n, we denote q-integers by [ n ] q , where

[ n ] q = { 1 q n 1 q , if  q 1 ; n , if  q = 1 .
(2)

Also q-factorial and q-binomial coefficients are defined as follows:

[ n ] q ! = { [ 1 ] q [ 2 ] q [ n ] q , if  n = 1 , 2 , ; 1 , if  n = 0 ; [ n k ] q = [ n ] q ! [ n k ] q ! [ k ] q ! , k = 0 , 1 , , n .

It is obvious that q-binomial coefficients will reduce to the ordinary case when q=1. The q-improper integrals are defined as (see [8, 9])

0 a f(x) d q x=(1q)a n = 0 f ( a q n ) q n ,aR

and

0 / A f(x) d q x=(1q) f ( q n A ) q n A ,A>0,
(3)

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,
(4)

where K(x;t)= 1 x + 1 x t ( 1 + 1 x ) q t ( 1 + x ) q 1 t and ( a + b ) q s = j = 0 s 1 (a+ q j b), s Z + .

In particular, for any positive integers m, n

K(x;n)= q n ( n 1 ) 2 ,K(x;0)=1and B q (m;n)= Γ q ( m ) Γ q ( n ) Γ q ( m + n ) ,
(5)

where Γ q (t) is the q-Gamma function.

For m>0, the q-Gamma function is defined by

Γ q (m)= 0 1 1 q x m 1 E q (qx) d q x,
(6)

where E q (x)= n = 0 q n ( n 1 ) 2 x n [ n ] q ! . Obviously, it satisfies the following functional equations:

Γ q (t+1)= [ t ] q Γ q (t), Γ q (1)=1.

More details of the q-Gamma function and the q-Beta function can be found in [10].

Very recently, Cai and Zeng [11] proposed a kind of q-generalization of Gamma operators and studied their approximation properties. These operators are defined as follows:

G n , q (f;x)= [ 2 n + 3 ] q ! ( q n + 3 2 x ) n + 3 q n ( n + 1 ) 2 [ n ] q ! [ n + 2 ] q ! 0 / A t n ( q n + 3 2 x + t ) q 2 n + 4 f(t) d q t,x>0.
(7)

In approximation theory the Stancu type modification of operators is an interesting research topic. In this paper, we propose a kind of q-Gamma-Stancu operators G n , q α , β (f;x) as follows.

Definition 1 For fC(0,), q(0,1), 0αβ and nN, we can define q-Gamma-Stancu operators G n , q α , β (f;x) as

G n , q α , β (f;x)= [ 2 n + 3 ] q ! ( q n + 3 2 x ) n + 3 q n ( n + 1 ) 2 [ n ] q ! [ n + 2 ] q ! 0 / A t n ( q n + 3 2 x + t ) q 2 n + 4 f ( [ n ] q t + α [ n ] q + β ) d q t.
(8)

The rest of the paper is organized as follows. In Section 2 we present the moments of the operators G n , q α , β (f;x). In Section 3 we present two direct and local approximation results for the operators G n , q α , β (f;x) by means of the first- and second-order modulus of continuity and the second-order central moment of the operators. In Section 4 we study the rate of convergence of the operators G n , q α , β (f;x). In Section 5 we discuss the weighted approximation theorem and Voronovskaya type asymptotic formula of the operators.

2 Moment estimates

In order to obtain the approximation properties of the operators G n , q α , β (f;x), we need the following lemmas.

Lemma 1 ([11])

For any kN, kn+2 and q(0,1), we have

G n , q ( t k ; x ) = [ n + k ] q ! [ n k + 2 ] q ! [ n ] q ! [ n + 2 ] q ! q 2 k k 2 2 x k .
(9)

Lemma 2 If we define the moments as

T n , m α , β ( x ) = G n , q α , β ( t m ; x ) = [ 2 n + 3 ] ! ( q n + 3 2 x ) n + 3 q n ( n + 1 ) 2 [ n ] q ! [ n + 2 ] q ! 0 / A t n ( q n + 3 2 x + t ) q 2 n + 4 ( [ n ] q t + α [ n ] q + β ) m d q t ,
(10)

then we have

  1. (i)

    T n , 0 α , β (x)= G n , q α , β (1;x)=1,

  2. (ii)

    T n , 1 α , β (x)= G n , q α , β (t;x)= q [ n ] q [ n + 1 ] q ( [ n ] q + β ) [ n + 2 ] q x+ α [ n ] q + β , for n>2,

  3. (iii)

    T n , 2 α , β (x)= G n , q α , β ( t 2 ;x)= ( [ n ] q [ n ] q + β ) 2 x 2 + 2 α q [ n ] q [ n + 1 ] q ( [ n ] q + β ) 2 [ n + 2 ] q x+ ( α [ n ] q + β ) 2 , for n>3.

Proof From Lemma 1, we have T n , 0 α , β (x)= G n , q α , β (1;x)=1. Thus

T n , 1 α , β ( x ) : = [ 2 n + 3 ] q ! ( q n + 3 2 x ) n + 3 q n ( n + 1 ) 2 [ n ] q ! [ n + 2 ] q ! 0 / A t n ( q n + 3 2 x + t ) q 2 n + 4 ( [ n ] q t + α [ n ] q + β ) d q t = [ n ] q [ n ] q + β G n , q ( t ; x ) + α [ n ] q + β G n , q ( 1 ; x ) = q [ n ] q [ n + 1 ] q ( [ n ] q + β ) [ n + 2 ] q x + α [ n ] q + β .

Finally,

T n , 2 α , β ( x ) : = [ 2 n + 3 ] q ! ( q n + 3 2 x ) n + 3 q n ( n + 1 ) 2 [ n ] q ! [ n + 2 ] q ! 0 / A t n ( q n + 3 2 x + t ) q 2 n + 4 ( [ n ] q t + α [ n ] q + β ) 2 d q t = ( [ n ] q [ n ] q + β ) 2 G n , q ( t 2 ; x ) + 2 [ n ] q α ( [ n ] q + β ) 2 G n , q ( t ; x ) + ( α [ n ] q + β ) 2 G n , q ( 1 ; x ) = ( [ n ] q [ n ] q + β ) 2 x 2 + 2 α q [ n ] q [ n + 1 ] q ( [ n ] q + β ) 2 [ n + 2 ] q x + ( α [ n ] q + β ) 2 .

 □

Remark 1 If we put q=1 and α=β=0, we get the moments of the Gamma operators (see [1]):

L n ( t ; x ) = n + 1 n + 2 x , n > 2 , L n ( t 2 ; x ) = x 2 , n > 3 .

Remark 2 Let n>2 be a given natural number. For every q(0,1) we have

A n , q α , β (x)= G n , q α , β (tx;x)= { q [ n ] q [ n + 1 ] q ( [ n ] q + β ) [ n + 2 ] q 1 } x+ α [ n ] q + β ,
(11)
B n , q α , β ( x ) = G n , q α , β ( ( t x ) 2 ; x ) B n , q α , β ( x ) = ( [ n ] q [ n ] q + β ) 2 x 2 + 2 α q [ n ] q [ n + 1 ] q ( [ n ] q + β ) 2 [ n + 2 ] q x + ( α [ n ] q + β ) 2 B n , q α , β ( x ) = 2 x { q [ n ] q [ n + 1 ] q ( [ n ] q + β ) [ n + 2 ] q x + α [ n ] q + β } + x 2 B n , q α , β ( x ) = { ( [ n ] q [ n ] q + β ) 2 2 q [ n ] q [ n + 1 ] q ( [ n ] q + β ) [ n + 2 ] q + 1 } x 2 B n , q α , β ( x ) = + 2 α { q [ n ] q [ n + 1 ] q ( [ n ] q + β ) 2 [ n + 2 ] q 1 [ n ] q + β } x + ( α [ n ] q + β ) 2 ,
(12)
C n , q α , β ( x ) = G n , q α , β ( ( t x ) 4 ; x ) C n , q α , β ( x ) = { ( [ n ] q ) 3 [ n + 3 ] q [ n + 4 ] q [ n 1 ] q ( [ n ] q + β ) 4 q 4 4 q ( [ n ] q ) 2 [ n + 3 ] q q 2 ( [ n ] q + β ) 3 C n , q α , β ( x ) = + 6 ( [ n ] q ) 2 ( [ n ] q + β ) 2 4 q [ n ] q [ n + 1 ] q [ n + 2 ] q ( [ n ] q + β ) + 1 } x 4 C n , q α , β ( x ) = + 4 α [ n ] q + β { q ( [ n ] q ) 2 [ n + 3 ] q q 2 ( [ n ] q + β ) 3 3 ( [ n ] q ) 2 ( [ n ] q + β ) 2 + 3 q [ n + 1 ] q [ n ] q [ n ] q ( [ n ] q + β ) 1 } x 3 C n , q α , β ( x ) = + 6 α 2 ( [ n ] q + β ) 2 { ( [ n ] q ) 2 ( [ n ] q + β ) 2 2 q [ n ] q [ n + 1 ] q [ n + 2 ] q ( [ n ] q + β ) + 1 } x 2 C n , q α , β ( x ) = + 4 α 3 ( [ n ] q + β ) 3 { q [ n ] q [ n + 1 ] q [ n + 2 ] q ( [ n ] q + β ) 1 } x + α 4 ( [ n ] q + β ) 4 .
(13)

3 Local approximation

In this section we establish two direct and the local approximation theorems of the operators G n , q α , β (f;x).

Let C B [0,+) denote the space of all real valued continuous bounded functions f defined on the interval [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+2h)2f(x+h)+f(x)|.

From [12, 13], there exists an absolute constant M 0 >0 such that

K 2 (f;δ) M 0 ω 2 (f; δ ),δ>0.
(14)

Also we set

ω(f;δ):= sup 0 < h δ sup x [ 0 , + ) |f(x+h)f(x)|.

In order to prove the theorems of this section, we need the following lemma.

Lemma 3 Let q(0,1), x(0,+), f C B [0,+). Then, for all g C B 2 [0,+), we have

| G ˆ n , q α , β (g;x)g(x)| ( B n , q α , β ( x ) + D n , q 2 ( x ) ) g ,

where

G ˆ n , q α , β ( f ; x ) = G n , q α , β ( f ; x ) + f ( x ) f ( T n , 1 α , β ( x ) ) , D n , q α , β ( x ) = T n , 1 α , β ( x ) x .
(15)

Proof From (15) and Lemma 2, we have

G ˆ n , q α , β (tx;x)=0.

For x(0,+) and g C B 2 [0,+), using Taylor’s formula,

g(t)g(x)=(tx) g (x)+ x t (tu) g (u)du,

we have

G ˆ n , q α , β ( g ; x ) g ( x ) = G ˆ n , q α , β ( ( t x ) g ( x ) ; x ) + G ˆ n , q α , β ( x t ( t u ) g ( u ) d u ; x ) = g ( x ) G ˆ n , q α , β ( ( t x ) ; x ) + G n , q α , β ( x t ( t u ) g ( u ) d u ; x ) x T n , 1 α , β ( x ) ( T n , 1 α , β ( x ) u ) g ( u ) d u = G n , q α , β ( x t ( t u ) g ( u ) d u ; x ) x T n , 1 α , β ( x ) ( T n , 1 α , β ( x ) u ) g ( u ) d u .

On the other hand, from

| x t (tu) g (u)du|| x t |tu|| g (u)|du| g | x t |tu|du| ( t x ) 2 g

and

| x T n , 1 α , β ( x ) ( T n , 1 α , β ( x ) u ) g (u)du| ( T n , 1 α , β ( x ) x ) 2 g = ( D n , q α , β ( x ) ) 2 g ,

we conclude that

| G ˆ n , q α , β ( g ; x ) g ( x ) | = | G n , q α , β ( x t ( t u ) g ( u ) d u ; x ) x T n , 1 α , β ( x ) ( T n , 1 α , β ( x ) u ) g ( u ) d u | G n , q α , β ( ( t x ) 2 g ; x ) + ( D n , q α , β ( x ) ) 2 g ( B n , q α , β ( x ) + ( D n , q α , β ( x ) ) 2 ) g .

This completes the proof. □

Theorem 1 Let q(0,1), f C B [0,+). Then, for every x(0,+), there exists a constant M 1 >0 such that

| G n , q α , β (f;x)f(x)| M 1 ω 2 ( f ; B n , q α , β ( x ) + ( D n , q α , β ( x ) ) 2 ) +ω ( f ; | D n , q α , β ( x ) | ) .

Proof By (15), we have

| G ˆ n , q α , β (f;x)|| G n , q α , β (f;x)|+2f3f.
(16)

Using Lemma 3, for every g C B 2 [0,+), we obtain

| G n , q α , β ( f ; x ) f ( x ) | | G ˆ n , q α , β ( f ; x ) f ( x ) | + | f ( x ) f ( T n , 1 α , β ( x ) ) | | G ˆ n , q α , β ( f g ; x ) ( f g ) ( x ) | + | f ( x ) f ( T n , 1 α , β ( x ) ) | + | G ˆ n , q α , β ( g ; x ) g ( x ) | 4 f g + | f ( x ) f ( T n , 1 α , β ( x ) ) | + ( B n , q α , β ( x ) + ( D n , q α , β ( x ) ) 2 ) g 4 f g + ω ( f ; | D n , q α , β ( x ) | ) + ( B n , q α , β ( x ) + ( D n , q α , β ( x ) ) 2 ) g .

Now, by taking infimum on the right-hand side for all g C B 2 [0,) and using (14), we get the following result:

| G n , q α , β ( f ; x ) f ( x ) | 4 K 2 ( f ; B n , q α , β ( x ) + ( D n , q α , β ( x ) ) 2 ) + ω ( f ; | D n , q α , β ( x ) | ) 4 M 0 ω 2 ( f ; B n , q α , β ( x ) + ( D n , q α , β ( x ) ) 2 ) + ω ( f ; | D n , q α , β ( x ) | ) = M 1 ω 2 ( f ; B n , q α , β ( x ) + ( D n , q α , β ( x ) ) 2 ) + ω ( f ; | D n , q α , β ( x ) | ) .

This completes the proof. □

Theorem 2 Let 0<γ1 and E be any bounded subset of the interval [0,+). If f C B [0,+) Lip M 3 (γ), then we have

| G n , q α , β (f;x)f(x)| M 3 { ( B n , q α , β ( x ) ) γ 2 + 2 ( d ( x ; E ) ) γ } ,

where M 3 is a constant depending only on α, d(x;E) is the distance between x and E defined as

d(x;E)=inf { | t x | : t E  and  x ( 0 , + ) } .

Proof From the properties of the infimum, there is at least one point t 0 in the closure of E such that

d(x;E)=| t 0 x|.

By the triangle inequality, we have

|f(t)f(x)||f(t)f( t 0 )|+|f( t 0 )f(x)|.

Thus

| G n , q α , β ( f ; x ) f ( x ) | G n , q α , β ( | f ( t ) f ( x ) | ; x ) G n , q α , β ( | f ( t ) f ( t 0 ) | ; x ) + G n , q α , β ( | f ( t 0 ) f ( x ) | ; x ) M 3 { G n , q α , β ( | t t 0 | γ ; x ) + | t 0 x | γ } M 3 { G n , q α , β ( | t x | γ ; x ) + 2 | t 0 x | γ }

holds. Now we choose p 1 = 2 γ and p 2 = 2 2 γ such that 1 p 1 + 1 p 2 =1, then by Hölder inequality we have

| G n , q α , β ( f ; x ) f ( x ) | M 3 { [ G n , q α , β ( | t x | γ p 1 ; x ) ] 1 p 1 [ G n , q α , β ( 1 p 2 ; x ) ] 1 p 2 + 2 | t 0 x | γ } = M 3 { G n , q α , β ( | t x | 2 ; x ) γ 2 + 2 | t 0 x | γ } = M 3 { ( B n , q α , β ( x ) ) γ 2 + 2 ( d ( x ; E ) ) γ } .

This completes the proof. □

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. Let C x 2 [0,+) denote the subset of all continuous functions belonging to B x 2 [0,+). If f C x 2 [0,+) and lim x + f ( x ) 1 + x 2 exists, we write f C x 2 [0,+). The norm on C x 2 [0,+) is given by f x 2 = sup x [ 0 , + ) | f ( x ) | 1 + x 2 . The modulus of continuity of f on the closed interval [0,a] is defined by

ω a (f;δ)= sup | t x | < δ sup x , t [ 0 , a ] |f(t)f(x)|.

We know that for a function f C x 2 [0,+), the modulus of continuity ω a (f;δ) tends to zero as δ0.

Now we give a rate of convergence theorem for the operators G n , q α , β (f;x).

Theorem 3 Let f C x 2 [0,+), q(0,1) and let ω a + 1 (f;δ) be modulus of the continuity of f on the finite interval [0,a+1][0,+), where a>0. Then for n>3,

| G n , q α , β (f;x)f(x)|5 M f ( 1 + a 2 ) B n , q α , β (x)+ ω a + 1 (f;δ) ( 1 + 1 δ [ B n , q α , β ( x ) ] 1 2 ) .
(17)

Proof For x(0,a] and t>a+1, since tx>1, we have

| f ( t ) f ( x ) | M f ( 2 + x 2 + t 2 ) M f ( 2 + x 2 + t 2 + ( 2 x t ) 2 ) M f ( 2 + 3 x 2 + 2 ( x t ) 2 ) 5 M f ( 1 + a 2 ) ( t x ) 2 .
(18)

For x(0,a] and ta+1, we have

|f(t)f(x)| ω a + 1 ( f ; | t x | ) ( 1 + | t x | δ ) ω a + 1 ( f ; | t x | )
(19)

with δ>0.

From (18) and (19) we get

|f(t)f(x)|5 M f ( 1 + a 2 ) ( t x ) 2 + ( 1 + | t x | δ ) ω a + 1 (f;δ),
(20)

for x(0,a] and t>0. Thus

| G n , q α , β ( f ; x ) f ( x ) | G n , q α , β ( | f ( t ) f ( x ) | ; x ) 5 M f ( 1 + a 2 ) G n , q α , β ( ( t x ) 2 ; x ) + ω a + 1 ( f ; δ ) ( 1 + 1 δ [ G n , q α , β ( ( t x ) 2 ; x ) ] 1 2 ) 5 M f ( 1 + a 2 ) B n , q α , β ( x ) + ω a + 1 ( f ; δ ) ( 1 + 1 δ [ B n , q α , β ( x ) ] 1 2 ) .

The proof is completed. □

As is well known, if f is not uniformly continuous on the interval [0,), then the usual first modulus of continuity ω(f;δ) does not tend to zero as δ0. For every f C x 2 [0,), we would like to take a weighted modulus of continuity Ω(f;δ) which tends to zero as δ0.

Let

Ω(f;δ)= sup 0 < h δ , x 0 | f ( x + h ) f ( x ) | 1 + ( x + h ) 2 ,for every f C x 2 [0,).

The weighted modulus of continuity Ω(f;δ) was defined by Yuksel and Ispir in [14]. It is well known that Ω(f;δ) has the following properties.

Lemma 4 ([14])

Let f C x 2 [0,), then:

  1. (i)

    Ω(f;δ) is a monotone increasing function of δ.

  2. (ii)

    For each f C x 2 [0,), lim δ 0 + Ω(f;δ)=0.

  3. (iii)

    For each mN{0}, Ω(f;mδ)mΩ(f;δ).

  4. (iv)

    For each λ R + , Ω(f;λδ)(1+λ)Ω(f;δ).

Theorem 4 Let f C x 2 [0,) and q= q n (0,1) such that q n 1 and [ n ] q n as n, then there exists a positive constant A such that the inequality

sup x ( 0 , ) | G n , q n α , β ( f ; x ) f ( x ) | ( 1 + x 2 ) 5 2 AΩ ( f ; 1 [ n + 2 ] q n )
(21)

holds.

Proof For t>0, x(0,) and δ>0, by the definition of Ω(f;δ) and Lemma 4, we get

| f ( t ) f ( x ) | ( 1 + ( x + | x t | ) ) 2 Ω ( f ; | t x | ) 2 ( 1 + x 2 ) ( 1 + ( t x ) 2 ) ( 1 + | t x | δ ) Ω ( f ; δ ) .

Since G n , q n α , β is linear and positive, we have

| G n , q n α , β ( f ; x ) f ( x ) | 2 ( 1 + x 2 ) Ω ( f ; δ ) { 1 + G n , q n α , β [ ( t x ) 2 ; x ] + G n , q n α , β [ ( 1 + ( t x ) 2 ) | t x | δ ; x ] } .
(22)

From Remark 2, we have

G n , q n α , β ( ( t x ) 2 ; x ) A 1 1 + x 2 [ n + 2 ] q n ,
(23)

for some positive constant A 1 . To estimate the second term of (22), applying the Cauchy-Schwartz inequality, we have

G n , q n α , β ( ( 1 + ( t x ) 2 ) | t x | δ ; x ) 2 ( G n , q n α , β ( 1 + ( t x ) 4 ; x ) ) 1 2 ( G n , q n α , β ( ( t x ) 2 δ 2 ; x ) ) 1 2 .
(24)

By Remark 2 and (23), there exist two positive constants A 2 , A 3 such that

( G n , q n α , β ( 1 + ( t x ) 4 ; x ) ) 1 2 A 2 ( 1 + x 2 )
(25)

and

( G n , q n α , β ( ( t x ) 2 δ 2 ; x ) ) 1 2 A 3 δ 1 + x 2 [ n + 2 ] q n .
(26)

Now we take A=1+2 A 1 +2 A 2 A 3 and δ= 1 [ n + 2 ] q n , and combining the above estimates, we obtain the inequality (21). □

5 Weighted approximation and Voronovskaya type asymptotic formula

In this section we will discuss the weighted approximation theorem and Voronovskaya type asymptotic formula.

Theorem 5 Let the sequence q={ q n } satisfy 0< q n <1, q n 1 and [ n ] q n as n. Then for f C x 2 [0,), we have

lim n G n , q n α , β ( f ) f x 2 =0.
(27)

Proof Using the Korovkin theorem in [15], we know that it is sufficient to verify the following three equations:

lim n G n , q n α , β ( t k ; x ) x k x 2 =0,k=0,1,2.
(28)

Since G n , q n α , β (1;x)=1, (28) holds true for k=0.

By Lemma 2, for n>1, we have

G n , q n α , β ( t ; x ) x x 2 = sup x [ 0 , ) { | { q n [ n ] q n [ n + 1 ] q n ( [ n ] q n + β ) [ n + 2 ] q n 1 } x + α [ n ] q n + β | × 1 1 + x 2 } | q n [ n ] q n [ n + 1 ] q n ( [ n ] q n + β ) [ n + 2 ] q n 1 | sup x [ 0 , ) x 1 + x 2 + α [ n ] q n + β sup x [ 0 , ) 1 1 + x 2 | q n [ n ] q n [ n + 1 ] q n ( [ n ] q n + β ) [ n + 2 ] q n 1 | + α [ n ] q n + β | q n [ n + 1 ] q n [ n + 2 ] q n 1 | + β q n [ n + 1 ] q n ( [ n ] q n + β ) [ n + 2 ] q n + α [ n ] q n + β | 1 q n 1 | + 1 q n [ n + 2 ] q n + α + β q n [ n ] q n + β .

Thus

lim n G n , q n α , β ( t ; x ) x x 2 =0.

Similarly, for n>2, we have

G n , q n α , β ( t 2 ; x ) x 2 x 2 = sup x [ 0 , ) { | ( [ n ] q n [ n ] q n + β ) 2 x 2 + 2 α q n [ n ] q n [ n + 1 ] q n ( [ n ] q n + β ) 2 [ n + 2 ] q n x + ( α [ n ] q n + β ) 2 x 2 | × 1 1 + x 2 } β 2 ( [ n ] q n + β ) 2 + 2 β [ n ] q n + β + α 2 ( [ n ] q n + β ) 2 + 2 α q n [ n ] q n [ n + 1 ] q n ( [ n ] q n + β ) 2 [ n + 2 ] q n α 2 + β 2 ( [ n ] q n + β ) 2 + 2 α q n + 2 β [ n ] q n + β ,

which implies that

lim n G n , q n α , β ( t 2 ; x ) x 2 x 2 =0.

The proof is completed. □

Finally, we give a Voronovskaya type asymptotic formula for G n , q n α , β (f;x) by means of the second and the fourth central moments.

Theorem 6 Let f be a bounded and integrable function on the interval (0,) and { q n } n = 1 be a sequence such that 0< q n <1 and q n 1 as n. Suppose that the second derivative f (x) exists at a point x(0,), then we have

lim n [ n + 2 ] q n ( G n , q n α , β ( f ; x ) f ( x ) ) = ( α ( 1 + β ) x ) f (x)+ x 2 f (x).
(29)

Proof By the Taylor formula we have

f(t)f(x)=(tx) f (x)+ 1 2 f (x) ( t x ) 2 +r(t,x) ( t x ) 2 ,

where r(t,x) is bounded and lim t x r(t,x)=0. By applying the operator G n , q α , β (f;x) to the above equation we obtain

G n , q n α , β ( f ; x ) f ( x ) = f ( x ) G n , q n α , β ( ( t x ) ; x ) + 1 2 f ( x ) G n , q n α , β ( ( t x ) 2 ; x ) + G n , q n α , β ( r ( t , x ) ( t x ) 2 ; x ) = f ( x ) A n , q n α , β ( x ) + 1 2 f ( x ) B n , q n α , β ( x ) + G n , q n α , β ( r ( t , x ) ( t x ) 2 ; x ) .

By direct calculation, we obtain

[ n + 2 ] q n A n , q n α , β ( x ) = { q n [ n ] q n [ n + 1 ] q n ( [ n ] q n + β ) [ n + 2 ] q n } x + α [ n + 2 ] q n [ n ] q n + β = { q n [ n + 1 ] q n [ n + 2 ] q n } x { q n β [ n + 1 ] q n [ n ] q n + β } x + α [ n + 2 ] q n [ n ] q n + β = ( q n q n ) [ n + 1 ] q n x { 1 + q n β [ n + 1 ] q n [ n ] q n + β } x + α [ n + 2 ] q n [ n ] q n + β = q n ( 1 q n n + 1 ) 1 + q n x { 1 + q n β [ n + 1 ] q n [ n ] q n + β } x + α [ n + 2 ] q n [ n ] q n + β α ( 1 + β ) x ( n ) .

Similarly,

[ n + 2 ] q n [ ( [ n ] q n [ n ] q n + β ) 2 2 q n [ n ] q n [ n ] q n ( [ n ] q n + β ) ( [ n + 2 ] q n ) + 1 ] = 2 { [ n + 2 ] q n q n [ n ] q n [ n + 1 ] q n [ n ] q n + β β [ n + 2 ] q n [ n ] q n + β } = 2 { [ n + 2 ] q n q n [ n + 1 ] q n + q n β [ n + 1 ] q n [ n ] q n + β β [ n + 2 ] q n [ n ] q n + β } = 2 { 1 + ( q n q n ) [ n + 1 ] q n + q n β [ n + 1 ] q n [ n ] q n + β β [ n + 2 ] q n [ n ] q n + β } 2 ( n ) .

That means

[ n + 2 ] q n B n , q n α , β (x)2 x 2 (n).
(30)

On the other hand, by simple calculation we obtain

[ n + 2 ] q n ( [ n + 3 ] q n [ n + 4 ] q n [ n ] q n [ n 1 ] q n 1 ) 8(n),
(31)
[ n + 2 ] q n ( [ n + 3 ] q n [ n ] q n 1 ) 3(n).
(32)

Thus from Remark 2, we have

[ n + 2 ] q n G n , q n α , β ( ( t x ) 4 ; x ) 0(n).
(33)

Since r(t,x) is bounded and lim t x r(t,x)=0, then for any given ϵ>0, there exists a δ>0 such that

|r(t,x)|ϵ+ M δ 2 ( t x ) 2 ( t , x ( 0 , ) , M  is a positive constant ) .

Thus

[ n + 2 ] q n | G n , q n α , β ( r ( t , x ) ( t x ) 2 ; x ) | ϵ [ n + 2 ] q n G n , q n α , β ( ( t x ) 2 ; x ) + M δ 2 [ n + 2 ] q n G n , q n α , β ( ( t x ) 4 ; x ) 0 ( n ) .

The proof is completed. □

References

  1. Karsli H: Rate of convergence of a new Gamma type operators for functions with derivatives of bounded variation. Math. Comput. Model. 2007,45(5-6):617-624. 10.1016/j.mcm.2006.08.001

    MathSciNet  Article  MATH  Google Scholar 

  2. Karsli H, Gupta V, Izgi A: Rate of point wise convergence of a new kind of gamma operators for functions of bounded variation. Appl. Math. Lett. 2009, 22: 505-510. 10.1016/j.aml.2006.12.015

    MathSciNet  Article  MATH  Google Scholar 

  3. Kim T: q -Generalized Euler numbers and polynomials. Russ. J. Math. Phys. 2006,13(3):293-298. 10.1134/S1061920806030058

    MathSciNet  Article  MATH  Google Scholar 

  4. Kim T: Some identities on the q -integral representation of the product of several q -Bernstein-type polynomial. Abstr. Appl. Anal. 2011., 2011: Article ID 634675

    Google Scholar 

  5. Gasper G, Rahman M Encyclopedia of Mathematics and Its Application 35. In Basic Hypergeometrik Series. Cambridge University Press, Cambridge; 1990.

    Google Scholar 

  6. Kac VG, Cheung P Universitext. In Quantum Calculus. Springer, New York; 2002.

    Chapter  Google Scholar 

  7. Aral A, Gupta V, Agarwal RP: Applications of q-Calculus in Operator Theory. XII. Springer, New York; 2013.

    Book  MATH  Google Scholar 

  8. Jackson FH: On a q -definite integrals. Q. J. Pure Appl. Math. 1910, 41: 193-203.

    MATH  Google Scholar 

  9. Koornwinder TH: q -Special functions, a tutorial. Contemp. Math. 134. In Deformation Theory and Quantum Groups with Applications to Mathematical Physics. Edited by: Gerstenhaber M, Stasheff J. Am. Math. Soc., Providence; 1992.

    Google Scholar 

  10. De Sole A, Kac VG: On integral representations of q -gamma and q -beta functions. Atti Accad. Naz. Lincei, Rend. Lincei, Mat. Appl. 2005,9(16):11-29.

    MathSciNet  MATH  Google Scholar 

  11. Cai Q, Zeng XM: On the convergence of a kind of q -gamma operators. J. Inequal. Appl. 2013., 2013: Article ID 105

    Google Scholar 

  12. Gupta V, Agarwal RP: Convergence Estimates in Approximation Theory. VIII. Springer, New York; 2014.

    Book  MATH  Google Scholar 

  13. DeVore RA, Lorentz GG: Constructive Approximation. Springer, Berlin; 1993.

    Book  MATH  Google Scholar 

  14. Yuksel I, Ispir N: Weighted approximation by a certain family of summation integral-type operators. Comput. Math. Appl. 2006,52(10-11):1463-1470. 10.1016/j.camwa.2006.08.031

    MathSciNet  Article  MATH  Google Scholar 

  15. Gadjiev AD: Theorems of the type of P.P. Korovkin type theorems. Mat. Zametki 1976,20(5):781-786. (English translation: Math. Notes 20(5-6), 996-998 (1976))

    MathSciNet  Google Scholar 

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Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 61170324 and Grant No. 61100105). The authors thank the associate editor and the referees for their important comments and suggestions, which improved the quality of the paper.

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Zhao, C., Cheng, WT. & Zeng, XM. Some approximation properties of a kind of q-Gamma-Stancu operators. J Inequal Appl 2014, 94 (2014). https://doi.org/10.1186/1029-242X-2014-94

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Keywords

  • q-Gamma-Stancu operators
  • approximation theorem
  • weighted approximation
  • rate of convergence