Qing-Bo Cai

School of Mathematics and Computer Sciences,Quanzhou Normal University,Quanzhou 362000,Fujian,P.R.China

Properties of Convergence for a Class of Generalized q-Gamma Operators

Qing-Bo Cai∗

School of Mathematics and Computer Sciences,Quanzhou Normal University,Quanzhou 362000,Fujian,P.R.China

.In this paper,a generalization ofq-Gamma operators based on the concept ofq-integer is introduced.We investigate the moments and central moments of the operators by computation,obtain a local approximation theorem and get the point wise convergence rate theorem and also obtain a weighted approximation theorem.Finally,a Voronovskaya type asymptotic formula was given.

AMS subject classifications:41A10,41A25,41A36

Chinese Library Classifications:O174.4

q-integer,q-Gamma operators,weighted approximation,modulus of continuity.

1 Introduction

It is well known that the Gamma operators are given by

In 2005,Zeng[9]obtained the approximation properties ofGndefined above,supposingfsatisfies exponential growth condition.He studied the approximation properties to the locally bounded functions and the absolutely continuous functions and obtained some good properties.

Since the application ofq-calculus in approximation theory is an active field,many researchers have performed studies in it,we mention some of them[3,5–8],these motivate us to introduce theqanalogue of this kind of Gamma operators.

Firstly,we recall some concepts ofq-calculus.All of the results can be found in[4].For any fixed real number 0＜q≤1 and each nonnegative integerk,we denoteq-integers by[k]q,where

Alsoq-factorial andq-binomial coefficients are defined as follows:

and

Theq-improper integrals are defined as

provided the sums converge absolutely.

Theq-exponential functionEq(x)is given as

where(1−x)∞q= ∏∞j=0(1−qjx).

Theq-Gamma integral is defined as

which satisfies the following functional equations:Γq(t+1)=[t]qΓq(t), Γq(1)=1.

Forf∈C[0,∞),q∈(0,1)andn∈N,we introduce a generalization ofq-Gamma operatorsGn,q(f,x)as

Obviously,Gn,q(f;x)are positive linear operators.It is observed that forq→1−,Gn,1−(f;x)become the Gamma operators defined in(1.1).

2 Some preliminary results

In this section,we give the following lemmas,which are need to prove our theorems:Lemma 2.1.For q∈(0,1),x∈[0,∞)and k=0,1,...,we have

Proof.From(1.3)and(1.4),we have

Lemma 2.1 is proved.

Lemma 2.2.For q∈(0,1),x∈[0,∞),we have

Proof.From Lemma 2.1,we get(2.2)easily.Next,

Finally,

Lemma 2.2 is proved.

Remark 2.1.Letn∈N andx∈[0,∞),then for everyq∈(0,1),by Lemma 2.2,we have

Lemma 2.3.For every q∈(0,1)and x∈[0,∞),we have

Remark 2.2.Let the sequenceq={qn}satisfiesqn∈(0,1)andqn→1 asn→∞,then for any fixedx∈[0,∞),by Lemma 2.3,we have

3 Local approximation

In this section,we establish direct and local approximation theorems in connection with the operatorsLn,q(f;x).

We denote the space of all real valued continuous bounded functionsfdefined on the interval[0,∞)byCB[0,∞).The norm||·||on the spaceCB[0,∞)is given by||f||=sup{|f(x)|:x∈[0,∞)}.

Further let us consider Peetre’sK−functional:

where δ＞0 andW2={g∈CB[0,∞):g′,g′′∈CB[0,∞)}.

Forf∈CB[0,∞),the modulus of continuity of second order is defined by

by[1,p.177],there exists an absolute constantC＞0 such that

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

Theorem 3.1.For q∈(0,1),x∈[0,∞),n∈Nand f∈CB[0,∞),we have

where C is a positive constant.

Proof.Letg∈W2,by Taylor’s expansion,we have

Using(2.5),we get

Hence,we have

On the other hand,using Lemma 2.2,we have

Now(3.3)and(3.4)imply

Hence taking infimum on the right hand side over allg∈W2,we get

By(3.1),for everyq∈(0,1),we have

This completes the proof of Theorem 3.1.

4 Rate of convergence

LetBx2[0,∞)be the set of all functionsfdefined on[0,∞),satisfying the condition|f(x)|≤Mf(1+x2),whereMfis a constant depending only onf.We denote the subspace of all continuous functions belonging toBx2[0,∞)byCx2[0,∞).Also,let[0,∞)be the subspace of all functionsf∈Cx2[0,∞),for whichis finite.The norm onisWe denote the usual modulus of continuity offon the closed interval[0,a],(a＞0)by

Obviously,for the functionf∈Cx2[0,∞),the modulus of continuity ωa(f,δ)tends to zero.

Theorem 4.1.Let f∈Cx2[0,∞),q∈ (0,1)andωa+1(f,δ)be the modulus of continuity on the finite interval[0,a+1]⊂[0,∞),where a＞0.Then we have

Proof.Forx∈[0,a]andt＞a+1,we havet−x≥t−a＞1.Hence(t−x)2＞1.Thus 2+3x2+2(t−x)2≤(2+3x2)(t−x)2+2(t−x)2=(4+3x2)(t−x)2≤(4+3a2)(t−x)2≤4(1+a2)(t−x)2.

Hence,we obtain

Forx∈[0,a]andt≤a+1,we have

From(4.2)and(4.3),we get

Forx∈[0,a]andt≥0,by Schwarz’s inequality and Lemma 2.3,we have

5 Weighted approximation and Voronovskaya type asymptotic formula

Now we will discuss the weighted approximation theorems.

Theorem 5.1.Let the sequence q={qn}satisfies0＜qn＜1and qn→1as n→∞,for f∈C∗x2[0,∞),we have

Proof.By using the Korovkin theorem in[2],we see that it is sufficient to verify the following three conditions

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

Theorem 5.2.Let the sequence q={qn}satisfies0＜qn＜1and qn→1as n→∞.Then for

Proof.Letx∈[0,∞)be fixed.By the Taylor formula,we may write

wherer(t;x)is the Peano form of the remainder,r(t;x)∈Cx2[0,∞),using L’Hopital’s rule,we have

ApplyingGn,q(f;x)to(5.4),we obtain

By the Cauchy-Schwarz inequality,we have

Sincer2(x;x)=0,then it follows from Theorem 5.1 that

we get the desired result.

This work is supported by the Educational Office of Fujian Province of China(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.

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20 July 2014;Accepted 4 December 2014

∗Corresponding author.Email address:qbcai@126.com(Q.-B.Cai)