# Approximation Properties ofλ-Gamma Operators Based on q-Integers

Authors: Wen-Tao Cheng; Xiao-Jun Tang
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In the present paper, we will introduceλ-Gamma operators based on q-integers. First, the auxiliary results about the moments are presented, and the central moments of these operators are also estimated. Then, we discuss some local approximation properties of these operators by means of modulus of continuity and Peetre K-functional. And the rate of convergence and weighted approximation for these operators are researched. Furthermore, we investigate the Voronovskaja type theorems including the quantitative q-Voronovskaja type theorem and q-Grüss-Voronovskaja theorem.


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## Body

## 1. Introduction

Gamma operators are very important positive linear operators and have been widely used in probability theory and computational mathematics. Forf∈Cℝ+, n=1,2,3,⋯ where ℝ+=0,∞ and Cℝ+ be the space of all continuous functions f on the interval ℝ+, the Gamma operators were introduced in [1] by
(1)Gnf;x=1n!∫0∞e−ttnfnxtdt,x∈ℝ+.We can learn some properties of Gamma operators and their modified operators in [2–7]. In [8], Qi et al. defined new Gamma operators as follows:
(2)Gn,λf;x=1n!∫0∞e−ttnntλfnxtdt,x∈ℝ+.where f∈Cℝ+, λ∈ℕ=0,1,2,⋯. Obviously, if fλ∈Cℝ+, then Gnf;xλ=Gn,λfλ;x. Meantime, Gn,λ1;x=nλn−λ!/n!≠1 (while λ≠0). In order to preserve the constant, we defined λ-Gamma operators as follows:Definition 1.
Forf∈Cℝ+, λ∈ℕ, n=λ,λ+1,⋯, the λ-Gamma operators are defined by
(3)Gn,λf;x=1n−λ!∫0∞e−ttn−λfnxtdt,x∈ℝ+.Let us recall some useful concepts and notations fromq-calculus, which can be founded in [9–11]. For nonnegative integer i, the q-integer iq and q-factorial iq! are defined by
(4)iq=1+q+⋯+qi−1=1−qi1−q,q≠1,i,q=1,iq!=1q2q⋯iq,i≥1,1,i=0.Further,q-power basis can be defined by
(5)x+yqi=x+yx+qy⋯x+qi−1y,i=1,2,⋯,1,i=0,x−yqi=x−yx−qy⋯x−qi−1y,i=1,2,⋯,1,i=0.Theq-derivative Dqf of a function f can be defined by
(6)Dqfx=fx−fqx1−qx,ifx≠0,and Dqf0=f′0 provided f′0 exists. High-order q-derivatives can be defined by Dq0f=f, Dqi=DqDqi−1f, i=1,2,⋯. The formula for the q-derivative of a product is Dqfxgx=Dqfxgx+Dqgxfqx. We easily know that if a function f is continuous on an interval which does not include 0, then f is continuous q-differentiable.Theq-improper integral of function f can be defined by
(7)∫0∞/1−qftdqt=∑i=−∞∞fqi1−qqi,q∈0,1.Theq-analogue of the classical exponential function ex is
(8)Eqx=∑i=0∞qii−1/2xiiq!=1+1−qxq∞,q∈0,1.Theq-Gamma function is defined by
(9)Γqs=∫0∞/1−qxs−1Eq−qxdqx,s∈ℝ+,and satisfies the functional relation: Γqs+1=sqΓqs, Γq1=1. Moreover, for any nonnegative integer i>0, the relation holds: Γqi+1=iq!.Now, we construct theq-analogue of λ-Gamma operators using q-Gamma function as follows.Definition 2.
Forf:ℝ+→ℝ, q∈0,1, λ∈ℕ, n=λ,λ+1,⋯, the q-analogue of λ-Gamma operators (3) are defined as
(10)Gn,λqf;x=1n−λq!∫0∞/1−qfnqxtEq−qttn−λdqt,x∈ℝ+.The paper is organized as follows: In Section1, we introduce the history of Gamma operators, recall some basic notations about the q-calculus, and construct λ-Gamma operators based on q-integers with q-Gamma function. In Section 2, we obtain the auxiliary results about the moment computation formula. The second- and fourth-order central moments computation formula and other quantitative properties are also presented. In Section 3, we discuss local approximation about the operators by means of modulus of continuity and Peetre K-functional. In Section 4 and Section 5, the rate of convergence and weighted approximation for these operators are researched. In the last section, we firstly prove quantitative q-Voronovskaja type theorems in terms of weighted modulus of continuity, and then the q-Grüss-Voronovskaja theorem in the quantitative mean is also presented (for the quantitative q-Voronovskaja type theorem0 and the q-Grüss-Voronovskaja theorem for the other operators, see also [12, 13]).


## 2. Auxiliary Results

In this section, we will give some lemmas and corollaries, which are necessary to obtain the approximation properties of the operatorsGn,λqf;x.Lemma 3.
Forq∈0,1, λ∈ℕ, i∈ℕ, n=λ+i,λ+i+1,⋯, the following formula holds:
(11)Gn,λqti;x=nqin−λ−iq!n−λq!xi.Proof.
According to the properties ofq-Gamma function, we have
(12)Gn,λqti;x=1n−λq!∫0∞/1−qnqxtiEq−qttn−λdqt=nqixin−λq!∫0∞/1−qtn−λ−iEq−qtdqt=nqixiΓqn−λ−i+1n−λq!=nqin−λ−iq!n−λq!xi.
Lemma3 is proved.Corollary 4.
By the lemma given above and some elementary calculations, we can get the results(13)Ax=Gn,λqt−x;x=qn−λλqn−λqxforn≥λ+1,Bx=Gn,λqt−x2;x=qn−λ−1λ+1qn−λ−1q−qn−λλqn−λq+q2n−2λ−1λqλ+1qn−λ−1qn−λqx2forn≥λ+2.Lemma 5.
Letq=qn be a sequence satisfying qn∈0,1, qn→1 and qnn→a∈0,1. Then, for each x∈ℝ+, Anx≔Gn,λqnt−x;x, Bnx≔Gn,λqnt−x2;x,we can obtain
(14)limn→∞nqnAnx=λax,limn→∞nqnBnx=ax2,(15)Gn,λqnt−x3;x=O1nqn2,(16)Gn,λqnt−x4;x=O1nqn2,(17)Gn,λqnt−x6;x=O1nqn3.Proof.
By Lemma3, we can easily get (14). Without loss of generality, we only prove equation (15). Equation (16) and equation (17) can be proved in some way. Set Gn,λqti;x=Cixi, i=1,2,3, and Gn,λqnt−x3;x=C4x3. Using nqn/n−λ−iqn=1+qnn−λ−iλ+iqn/n−λ−iqn, i=0,1,2,⋯,n−λ, we can easily get
(18)C1=1+qnn−λλqnn−λqn,C2=1+qnn−λλqnn−λqn+qnn−λ−1λ+1qnn−λ−1qn+qn2n−2λ−1λqnλ+1qnn−λ−1qnn−λqn,C3=1+qnn−λλqnn−λqn+qnn−λ−1λ+1qnn−λ−1qn+qnn−λ−2λ+2qnn−λ−2qn+qn2n−2λ−3λ+2qnλ+1qnn−λ−2qnn−λ−1qn+qn2n−2λ−2λ+2qnλqnn−λ−2qnn−λqn+qn2n−2λ−1λqnλ+1qnn−λ−1qnn−λqn+o1nqn2.
Combining(19)C4=C3−3C2+3C1−1=qnn−λλqnn−λqn−2qnn−λ−1λ+1qnn−λ−1qn+qnn−λ−2λ+2qnn−λ−2qn+qn2n−2λ−3λ+2qnλ+1qnn−λ−2qnn−λ−1qn+qn2n−2λ−2λ+2qnλqnn−λ−2qnn−λqn−2qn2n−2λ−1λqnλ+1qnn−λ−1qnn−λqn+o1nqn2≔qnn−λλqnn−λqn−2qnn−λ−1λ+1qnn−λ−1qn+qnn−λ−2λ+2qnn−λ−2qn+I=qnn−λ−2∏i=02n−λ−iqnλ+2qnn−λqnn−λ−1qn+qn2λqnn−λ−2qnn−λ−1qn−2qnλ+1qnn−λ−2qnn−λqn+I=qnn−λ−2∏i=02n−λ−iqnn−λqnλ+2qnn−λ−1qn−qnλ+1qnn−λ−2qn−qnn−λ−2qnλ+1qnn−λqn−qnλqnn−λ−1qn+I=qnn−λ−2∏i=02n−λ−iqnn−λqnnqn−qnn−λ−2qnnqn+I=qnn−λ−2nqn1+qnn−λ−1∏i=02n−λ−iqn+I,limn→∞nqn2I=3λ+2a2,we have limn→∞nqn2Gn,λqnt−x3;x=3λ+1a2x3+ax3. This means that equation (15) is obtained. Thus, the proof of Lemma 5 is accomplished.Lemma 6.
Letq=qn be a sequence satisfying qn∈0,1, qn→1 and qnn→a∈0,1. Then, for each x∈ℝ+, the following relations
(20)Gn,λqnt−xqn2;x≤O1nqn,(21)Gn,λqnt−xqn2t−x4;x≤O1nqn3,hold.Proof.
By the definition ofq-power basis, we have t−xqn2=t−xt−qnx=t−x2+1−qnnxt−x=t−x2+x1−qnn/nqnt−x. Thus, we can write t−xqn2≤t−x2+x1−qnn/nqn∣t−x∣. Using the monotonicity of the operators Gn,λqn and the Cauchy-Schwarz inequality, we can get
(22)Gn,λqnt−xqn2;x≤Gn,λqnt−x2;x+x1−qnnnqnGn,λqn∣t−x∣;x≤Gn,λqnt−x2;x+x1−qnnnqnGn,λqnt−x2;x≤O1nqn+O1nqn3/2=O1nqn.
The inequality (21) can be get in the same way. Using the monotonicity of the operators Gn,λqn, (16)and (17), Cauchy-Schwarz inequality, respectively, we can obtain
(23)Gn,λqnt−xqn2t−x4;x≤Gn,λqnt−x6;x+xnqnGn,λqnt−x5;x≤Gn,λqnt−x6;x+xnqnGn,λqnt−x4;xGn,λqnt−x6;x≤O1nqn3+O1nqn7/2=O1nqn3.
Thus, we complete the proof.


## 3. Local Approximation

LetCBℝ+ be the space of all real-valued continuous bounded functions f on ℝ+, endowed with the norm ∥f∥=supx∈ℝ+∣fx∣. Moreover, the Peetre’s K-functional is defined by
(24)K2f;δ=infh∈CB2ℝ+∥f−h∥+δ∥h′′∥,where CB2ℝ+≔h∈CBℝ+:h′,h′′∈CBℝ+. By ([14], p. 177, Theorem 2.4), there exists an absolute constant C>0 such that
(25)K2f;δ≤Cω2f;δ,where δ>0 and the second-order modulus of smoothness is defined by
(26)ω2f;δ=sup0<t≤δsupx∈ℝ+fx+2t−2fx+t+fx,f∈CBℝ+.The usual modulus of smoothness is defined by(27)ωf;δ=sup0<t≤δsupx∈ℝ+fx+t−fx,f∈CBℝ+.Theorem 7.
Letf∈CBℝ+, q∈0,1, λ=1,2,⋯. Then for all x∈ℝ+ and n≥λ+1, there exists an absolute C1=4C such that
(28)Gn,λqf;x−fx≤C1ω2f;A2x+Bx+ωf;∣Ax∣.Proof.
Using Definition2, we easily obtain ∣Gn,λqf;x∣≤∥f∥ for all f∈CBℝ+. Next, we define new operators by
(29)Pn,λqf;x=Gn,λqf;x+fx−fAx+x,x∈ℝ+.
We can getPn,λqt−x;x=Gn,λqt−x;x−Ax=0 and ∣Pn,λqf;x∣≤3∥f∥ for all f∈CBℝ+. For x,t∈ℝ+ and h∈CB2ℝ+, using Taylor’s expansion, we can write
(30)ht=hx+h′xt−x+∫xth′′ut−udu.
Hence,(31)Pn,λqh;x−hx=h′xPn,λqt−x;x+Pn,λq∫xth′′ut−udu;x≤Pn,λq∫xth′′ut−udu;x≤Gn,λq∫xth′′ut−udu;x−∫xAx+xh′′uAx+x−udu≤Gn,λq∫xt∣h′′u∣t−udu;x+∫xAx+x∣h′′u∣Ax+x−udu≤Bx+A2x∥h′′∥.
Further, for allh∈CB2ℝ+, we can write
(32)Gn,λqf;x−fx=Pn,λqf;x+fAx+x−2fx≤Pn,λqf−h;x−f−hx+Pn,λqh;x−hx+∣fAx+x−fx∣≤4∥f−h∥+A2x+Bx∥h′′∥+ωf;∣Ax∣.
Taking infimum over allh and using (25), we can get the desired conclusion.Corollary 8.
Letf∈CBℝ+, q∈0,1. Then for all x∈ℝ+ and n≥1, there exists an absolute C1=4C such that
(33)Gn,0qf;x−fx≤C1ω2f;Bx.Corollary 9.
Letf∈CBℝ+,q=qn be a sequence satisfying qn∈0,1, qn⟶1 and qnn⟶a∈0,1 as n⟶∞, the limit
(34)limn→∞Gn,λqnf;x=fxholds for all x∈ℝ+.


## 4. Rate of Convergence

As is known, iff is not uniformly continuous on ℝ+, we cannot get ωf;δ→0 as δ→0. To research the rate of convergence of the operators Gn,λqn on ℝ+, we recall the weighted modulus of continuity Ωf;δ(see [15] or [16]). First, we shall consider the following three classes of functions:
(35)B2ℝ+≔f:ℝ+→ℝ;fx≤Cf1+x2,where Cf is a positive constant which depends only on f,
(36)C2ℝ+≔f∈B2ℝ+:fiscontinuous,C20ℝ+≔f∈B2ℝ+:limx→∞fx1+x2isfinite.The spaceC20ℝ+ is a linear normed space endowed with the norm ∥f∥2=supx∈ℝ+fx/1+x2. For any f∈C2ℝ+, Ωf;δ is defined by
(37)Ωf;δ=sup0≤t<δ,x∈ℝ+∣fx+t−fx∣1+t21+x2,if f∈C20ℝ+, then Ωf;δ has the following properties:
(i)
limδ→0+Ωf;δ=0(ii)
Ωf;ρδ≤21+ρ1+δ2Ωf;δ, ρ∈ℝ+In [17–19], the following inequality was introduced and used
(38)∣ft−fx∣≤1+t−x21+x2Ωf;∣t−x∣≤21+∣t−x∣δ1+δ21+t−x21+x2Ωf;δ≤41+δ221+x2Ωf;δ,∣t−x∣<δ,41+δ221+x2t−x4δ4Ωf;δ,∣t−x∣≥δ.Meanwhile, we introduce the modulus of continuity off∈C0,aa>0 by ωaf;δ=sup∣t−x∣≤δsupx,t∈0,a∣ft−fx∣.The following is a theorem of the rate of convergence for the operatorsGn,λq:Theorem 9.
Letf∈C2ℝ+, λ∈ℕ, n=λ+1,λ+2,⋯, a∈ℝ+, we have
(39)∥Gn,λqf;x−f∥C0,a≤4Cf1+a2Ba+2ωa+1f,Ba.Proof.
For anyx∈0,a, t∈a+1,∞, we can easily obtain 1≤t−a2≤t−x2, therefore
(40)∣ft−fx∣≤Cf2+x2+t2≤Cf2+3x2+2t−x2≤4Cf1+a2t−x2.
Ift∈0,a+1, for any δ∈ℝ+, we can obtain
(41)∣ft−fx∣≤ωa+1f;∣t−x∣≤1+∣t−x∣δωa+1f;δ.
Combining (39) with (40), we can get
(42)∣ft−fx∣≤4Cf1+a2t−x2+1+∣t−x∣δωa+1f;δ.
By Cauchy-Schwarz’s inequality and Corollary4, for all x∈0,a, we have
(43)∣Gn,λqf;x−fx∣≤Gn,λq∣ft−fx∣;x≤4Cf1+a2Gn,λqt−x2;x+Gn,λq1+∣t−x∣δ;xωa+1f,δ≤4Cf1+a2Gn,λqt−x2;x+ωa+1f,δ1+1δGn,λqt−x2;x≤4Cf1+a2Bx+ωa+1f,δ1+1δBx≤4Cf1+a2Ba+ωa+1f,δ1+1δBa.
By choosingδ=Ba and taking supremum over all x∈0,a, we can get the desired results.Theorem 10.
q=qn be a sequence satisfying qn∈0,1, qn→1, and qnn→a as n→∞ and f∈C20ℝ+; then, there exists a positive integer N∈ℕ+ such that for all n>N and υ>0, the inequality
(44)supx∈ℝ+Gn,λqnf;x−fx1+x2υ≤64Ωf;1nqn,holds.Proof.
Using (14) and (16), there exists a positive integer N∈ℕ+ such that for all n>N,
(45)Gn,λqnt−x2;x≤94nqn,Gn,λqnt−x4;x≤1.
By Cauchy-Schwarz’s inequality, we can get(46)Gn,λqn∣t−x∣;x≤Gn,λqnt−x2;x≤32nqn,(47)Gn,λqnt−x3;x≤Gn,λqnt−x2;xGn,λqnt−x4;x≤32nqn.
SinceGn,λqn is linear and positive, using (38), (46), and (47), for any δ∈0,1, we can obtain
(48)Gn,λqnf;x−fx≤161+x2Ωf;δ1+Gn,λqn∣t−x∣+t−x3;xδ≤161+x21+3δnqnΩf;δ.
Takingδ=1/nqn, we complete the proof.


## 5. Weighted Approximation

In this section, we will discuss the weighted approximation theorems for the operatorsGn,λqn.Theorem 11.
Letq=qn be a sequence satisfying qn∈0,1, qn→1, and qnn→a∈0,1 as n→∞ and f∈C20ℝ+, we have
(49)limn→∞∥Gn,λqnf;x−f∥2=0.Proof.
Using Korovkin’s theorem (see [20]), it is sufficient to verify the following three conditions:
(50)limn→∞∥Gnpn,qntk−xk∥2=0,k=0,1,2.
SinceGnpn,qn1;x=1, (51) holds for k=1. By Lemma 3 and limn→∞nqn/n−λqn=limn→∞nqn/n−λ−1qn=1, we can easily obtain
(51)∥Gn,λqnt;x−x∥2=supx∈ℝ+11+x2Gn,λqnt;x−x=supx∈ℝ+x1+x2nqnn−λqn−1≤nqnn−λqn−1→0,n→∞.∥Gn,λqnt2;x−x2∥2=supx∈ℝ+11+x2Gn,λqnt2;x−x2=supx∈ℝ+x21+x2nqn2n−λqnn−λ−1qn−1≤nqn2n−λqnn−λ−1qn−1→0,n→∞.
We can draw the final conclusion through all the estimates above.Theorem 12.
Letq=qn be a sequence satisfying qn∈0,1, qn→1 and qnn→1 as n→∞ and f∈C20ℝ+. For any f∈C20ℝ+ and υ>0, we have
(52)limn→∞supx∈ℝ+∣Gnpn,qnf;x−fx∣1+x21+υ=0.Proof.
Letx0∈ℝ+ be arbitrary but fixed. Then,
(53)supx∈ℝ+∣Gn,λqnf;x−fx∣1+x21+υ≤supx∈0,x0∣Gn,λqnf;x−fx∣1+x21+υ+supx∈x0,∞∣Gn,λqnf;x−fx∣1+x21+υ≤∥Gn,λqnf;x−f∥C0,x0+∥f∥2supx∈x0,∞∣Gn,λqn1+t2;x∣1+x21+υ+supx∈x0,∞∣fx∣1+x21+υ.
Since∣fx∣≤∥f∥21+x2, we have supx∈x0,∞∣fx∣/1+x21+υ≤∥f∥2/1+x02υ. Let ε>0 be arbitrary, we can choose x0 to be so large that
(54)∥f∥21+x02υ<ε.
In view of Corollary 9, whilex∈x0,∞, we obtain
(55)∥f∥2limn→∞∣Gn,λqn1+t2;x∣1+x21+υ=∥f∥21+x21+x21+υ=∥f∥21+x2υ≤∥f∥21+x02υ<ε.
Using Theorem9, we can see that the first term of the inequality (53) implies that
(56)∥Gn,λqnf;x−f∥C0,x0<ε,asn→∞.
Combining (53)–(56), we get the desired result.


## 6. Voronovskaja Type Theorems

As is known, Voronovskaja type theorems of many positive operators are widely researched and discussed (see [21–28]). In this section, we will discuss the quantitative q-Voronovskaja theorem and q-Grüss-Voronovskaja theorem.


### 6.1. Quantitativeq-Voronovskaja Theorem

In this subsection, we will obtain the Quantitativeq-Voronovskaja theorem and Voronovskaja type asymptotic formula for the operators Gn,λqn.Theorem 13.
Letq=qn be a sequence satisfying qn∈0,1, qn→1 and qnn→a∈0,1 as n→∞ and f∈C20ℝ+ satisfy Dqn2f∈C20ℝ+. Then, the inequality
(57)nqnGn,λqnf;x−fx−DqnfxAnx−nqnBnx+1−qnnAnxx2qn!Dqn2fx≤O1ΩDqn2f;1nqn,holds for any x∈ℝ+.Proof.
Using theq-Taylor expansion formula (58), we have
(58)ft=fx+Dqnfxt−x+Dqn2fξ2qn!t−xqn2=fx+Dqnfxt−x+Dqn2fx2qn!t−xqn2+R2t,x;qn,where ξ is a number between t and x and
(59)R2t,x;qn=Dqn2fξ−Dqn2fx2qn!t−xqn2.
Applying the operatorsGn,λqn to both sides of (58) and using t−xqn2=t−x2+1−qnn/nqnt−xx, we have
(60)Gn,λqnf;x−fx−DqnfxAnx−Dqn2fx2qn!Gn,λqnt−xqn2;x=Gn,λqnf;x−fx−DqnfxAnx−Bnx+1−qnnAnx/nqnx2qn!Dqn2fx≤Gn,λqn∣R2t,x;qn∣;x.
Multiplying the above inequality bynqn, we have
(61)nqnGn,λqnf;x−fx−DqnfxAnx−nqnBnx+1−qnnAnxx2qn!Dqn2fx≤nqnGn,λqn∣R2t,x;qn∣;x.
Furthermore,(62)Dqn2fξ−Dqn2fx2qn!≤12qn!ΩDqn2f;∣ξ−x∣1+ξ−x21+x2≤12qn!ΩDqn2f;∣t−x∣1+t−x21+x2≤22qn!1+∣t−x∣δ1+t−x21+δ21+x2ΩDqn2f;δ≤161+x21+t−x4δ4ΩDqn2f;δ,for all δ∈0,1. Hence,
(63)∣R2t,x;qn∣≤161+x2∣t−xqn2∣+∣t−xqn2∣t−x4δ4ΩDqn2f;δ.
Using (20), (21), for any x∈ℝ+, we can write
(64)Gn,λqn∣R2t,x;qn∣;x≤161+x2Gn,λqn∣t−xqn2∣;x+Gn,λqn∣t−xqn2∣t−x4;xδ4ΩDqn2f;δ≤O1nqn+1δ4O1nqn3ΩDqn2f;δ.
If we chooseδ=1/nqn, we can easily get
(65)nqnGn,λqnR2t,x;qn;x≤O1ΩDqn2f;1nqn,which completes the proof of Theorem 13.Corollary 14.
Letqn be a sequence satisfying qn∈0,1, qn→1 and qnn→a∈0,1 as n→∞ and f∈C20ℝ+ satisfy f′′∈C20ℝ+. Then, we can obtain
(66)limn→∞nqnGn,λqnf;x−fx=λaxf′x+12ax2f′′x.


### 6.2.q-Grüss-Voronovskaja Theorem

In this subsection, we will obtain theq-Grüss-Voronovskaja theorem and its quantitative version for the operators Gn,λqn.Theorem 15.
Letq=qn be a sequence satisfying qn∈0,1, qn→1 and qnn→a∈0,1 as n→∞ and f,g∈C20ℝ+ satisfy Dqn2f,Dqn2g,Dqn2fg∈C20ℝ+. Then, the following inequality
(67)nqnGn,λqnfg;x−Gn,λqnf;xGn,λqng;x−DqnfqnxDqngx+Dqngqnx2qn!Gn,λqnt−xqn2;x≤O1ΩDqn2fg;1nqn+O1∥f∥2+O1nqn∥Dqnf∥2+∥Dqn2f∥2ΩDqn2g;1nqn+O1∥g∥2+O1nqn∥Dqng∥2+∥Dqn2g∥2ΩDqn2f;1nqn+O1nqn∥Dqnf∥2+∥Dqn2f∥2∥Dqng∥2+∥Dqn2g∥2+O1nqnΩDqn2f;1nqnΩDqn2g;1nqn,holds for any x∈ℝ+.Proof.
Using the equalities(68)Dqnfxgx=Dqnfxgx+fqnxDqngx,Dqn2fxgx=Dqn2fxgx+DqnfqnxDqngx+fqnxDqn2gx+DqnfqnxDqngqnx,by simple computations, for x∈ℝ+ and n=λ+1,⋯, we can obtain
(69)Gn,λqnfg;x−Gn,λqnf;xGn,λqng;x=Gn,λqnfg;x−fxgx−Gn,λqnt−x;xDqnfxgx−Gn,λqnt−xqn2;x2qn!Dqn2fxgx−gxGn,λqnf;x−fx−Gn,λqnt−x;xDqnfx−Gn,λqnt−xqn2;x2qn!Dqn2fx−Gn,λqnf;xGn,λqng;x−gx−Gn,λqnt−x;xDqngx−Gn,λqnt−xqn2;x2qn!Dqn2gx+DqnfqnxDqngx+Dqngqnx2qn!Gn,λqnt−xqn2;x−Dqn2gxGn,λqnf;x−fxGn,λqnt−xqn2;x2qn!+Dqn2gxDqnfxGn,λqnt−xqn2;x2qn!qn−1x−Gn,λqnt−x;xGn,λqnf;x−fqnxDqngx.
Hence, we can write(70)Gn,λqnfg;x−Gn,λqnf;xGn,λqng;x−DqnfqnxDqngx+Dqngqnx2qn!Gn,λqnt−xqn2;x=Gn,λqnfg;x−fxgx−Gn,λqnt−x;xDqnfxgx−Gn,λqnt−xqn2;x2qn!Dqn2fxgx−gxGn,λqnf;x−fx−Gn,λqnt−x;xDqnfx−Gn,λqnt−xqn2;x2qn!Dqn2fx−fxGn,λqng;x−gx−Gn,λqnt−x;xDqngx−Gn,λqnt−xqn2;x2qn!Dqn2gx+fx−Gn,λqnf;xGn,λqng;x−gx+Gn,λqnt−x;xqn−1xDqnfxDqngx+Gn,λqnt−xqn2;x2qn!qn−1xDqnfxDqn2gx≔I1+I2+I3+I4+I5+I6.
By Theorem 13, for any fixedx∈ℝ+, we can easily have the following estimates
(71)nqn∣I1∣≤O1ΩDqn2fg;1nqn,(72)nqn∣I2∣≤∣gx∣O1ΩDqn2f;1nqn≤∥g∥2O1ΩDqn2f;1nqn,(73)nqn∣I3∣≤∣fx∣O1ΩDqn2g;1nqn≤∥f∥2O1ΩDqn2g;1nqn.
Using (14), (20), and ∣qn−1∣=qnn−1/nqn≤O1/nqn, we have
(74)nqn∣I5∣≤O1nqn∥Dqnf∥2∥Dqng∥2,nqn∣I6∣≤O1nqn∥Dqnf∥2∥Dqn2g∥2.
Using (14), (20), and Theorem 13, we can get
(75)Gn,λqnf;x−fx≤O1nqn∥Dqnf∥2+∥Dqn2f∥2+O1nqnΩDqn2f;1nqn,hence, we can know
(76)nqn∣I4∣≤O1nqn∥Dqnf∥2+∥Dqn2f∥2∥Dqng∥2+∥Dqn2g∥2+O1nqnΩDqn2f;1nqn∥Dqng∥2+∥Dqn2g∥2+O1nqnΩDqn2g;1nqn∥Dqnf∥2+∥Dqn2f∥2+O1nqnΩDqn2f;1nqnΩDqn2g;1nqn.
Combining (71)–(76), we complete the proof of Theorem 15.Corollary 16.
Letq=qn be a sequence satisfying qn∈0,1, qn→1 and qnn→a∈0,1 as n→∞ and f,g∈C20ℝ+ satisfy f′′,g′′,fg′′∈C20ℝ+. Then, the following limit equality
(77)limn→∞nqnGn,λqnfg;x−Gn,λqnf;xGn,λqng;x=af′xg′xx2,holds for any x∈ℝ+.


## 6.1. Quantitativeq-Voronovskaja Theorem

In this subsection, we will obtain the Quantitativeq-Voronovskaja theorem and Voronovskaja type asymptotic formula for the operators Gn,λqn.Theorem 13.
Letq=qn be a sequence satisfying qn∈0,1, qn→1 and qnn→a∈0,1 as n→∞ and f∈C20ℝ+ satisfy Dqn2f∈C20ℝ+. Then, the inequality
(57)nqnGn,λqnf;x−fx−DqnfxAnx−nqnBnx+1−qnnAnxx2qn!Dqn2fx≤O1ΩDqn2f;1nqn,holds for any x∈ℝ+.Proof.
Using theq-Taylor expansion formula (58), we have
(58)ft=fx+Dqnfxt−x+Dqn2fξ2qn!t−xqn2=fx+Dqnfxt−x+Dqn2fx2qn!t−xqn2+R2t,x;qn,where ξ is a number between t and x and
(59)R2t,x;qn=Dqn2fξ−Dqn2fx2qn!t−xqn2.
Applying the operatorsGn,λqn to both sides of (58) and using t−xqn2=t−x2+1−qnn/nqnt−xx, we have
(60)Gn,λqnf;x−fx−DqnfxAnx−Dqn2fx2qn!Gn,λqnt−xqn2;x=Gn,λqnf;x−fx−DqnfxAnx−Bnx+1−qnnAnx/nqnx2qn!Dqn2fx≤Gn,λqn∣R2t,x;qn∣;x.
Multiplying the above inequality bynqn, we have
(61)nqnGn,λqnf;x−fx−DqnfxAnx−nqnBnx+1−qnnAnxx2qn!Dqn2fx≤nqnGn,λqn∣R2t,x;qn∣;x.
Furthermore,(62)Dqn2fξ−Dqn2fx2qn!≤12qn!ΩDqn2f;∣ξ−x∣1+ξ−x21+x2≤12qn!ΩDqn2f;∣t−x∣1+t−x21+x2≤22qn!1+∣t−x∣δ1+t−x21+δ21+x2ΩDqn2f;δ≤161+x21+t−x4δ4ΩDqn2f;δ,for all δ∈0,1. Hence,
(63)∣R2t,x;qn∣≤161+x2∣t−xqn2∣+∣t−xqn2∣t−x4δ4ΩDqn2f;δ.
Using (20), (21), for any x∈ℝ+, we can write
(64)Gn,λqn∣R2t,x;qn∣;x≤161+x2Gn,λqn∣t−xqn2∣;x+Gn,λqn∣t−xqn2∣t−x4;xδ4ΩDqn2f;δ≤O1nqn+1δ4O1nqn3ΩDqn2f;δ.
If we chooseδ=1/nqn, we can easily get
(65)nqnGn,λqnR2t,x;qn;x≤O1ΩDqn2f;1nqn,which completes the proof of Theorem 13.Corollary 14.
Letqn be a sequence satisfying qn∈0,1, qn→1 and qnn→a∈0,1 as n→∞ and f∈C20ℝ+ satisfy f′′∈C20ℝ+. Then, we can obtain
(66)limn→∞nqnGn,λqnf;x−fx=λaxf′x+12ax2f′′x.


## 6.2.q-Grüss-Voronovskaja Theorem

In this subsection, we will obtain theq-Grüss-Voronovskaja theorem and its quantitative version for the operators Gn,λqn.Theorem 15.
Letq=qn be a sequence satisfying qn∈0,1, qn→1 and qnn→a∈0,1 as n→∞ and f,g∈C20ℝ+ satisfy Dqn2f,Dqn2g,Dqn2fg∈C20ℝ+. Then, the following inequality
(67)nqnGn,λqnfg;x−Gn,λqnf;xGn,λqng;x−DqnfqnxDqngx+Dqngqnx2qn!Gn,λqnt−xqn2;x≤O1ΩDqn2fg;1nqn+O1∥f∥2+O1nqn∥Dqnf∥2+∥Dqn2f∥2ΩDqn2g;1nqn+O1∥g∥2+O1nqn∥Dqng∥2+∥Dqn2g∥2ΩDqn2f;1nqn+O1nqn∥Dqnf∥2+∥Dqn2f∥2∥Dqng∥2+∥Dqn2g∥2+O1nqnΩDqn2f;1nqnΩDqn2g;1nqn,holds for any x∈ℝ+.Proof.
Using the equalities(68)Dqnfxgx=Dqnfxgx+fqnxDqngx,Dqn2fxgx=Dqn2fxgx+DqnfqnxDqngx+fqnxDqn2gx+DqnfqnxDqngqnx,by simple computations, for x∈ℝ+ and n=λ+1,⋯, we can obtain
(69)Gn,λqnfg;x−Gn,λqnf;xGn,λqng;x=Gn,λqnfg;x−fxgx−Gn,λqnt−x;xDqnfxgx−Gn,λqnt−xqn2;x2qn!Dqn2fxgx−gxGn,λqnf;x−fx−Gn,λqnt−x;xDqnfx−Gn,λqnt−xqn2;x2qn!Dqn2fx−Gn,λqnf;xGn,λqng;x−gx−Gn,λqnt−x;xDqngx−Gn,λqnt−xqn2;x2qn!Dqn2gx+DqnfqnxDqngx+Dqngqnx2qn!Gn,λqnt−xqn2;x−Dqn2gxGn,λqnf;x−fxGn,λqnt−xqn2;x2qn!+Dqn2gxDqnfxGn,λqnt−xqn2;x2qn!qn−1x−Gn,λqnt−x;xGn,λqnf;x−fqnxDqngx.
Hence, we can write(70)Gn,λqnfg;x−Gn,λqnf;xGn,λqng;x−DqnfqnxDqngx+Dqngqnx2qn!Gn,λqnt−xqn2;x=Gn,λqnfg;x−fxgx−Gn,λqnt−x;xDqnfxgx−Gn,λqnt−xqn2;x2qn!Dqn2fxgx−gxGn,λqnf;x−fx−Gn,λqnt−x;xDqnfx−Gn,λqnt−xqn2;x2qn!Dqn2fx−fxGn,λqng;x−gx−Gn,λqnt−x;xDqngx−Gn,λqnt−xqn2;x2qn!Dqn2gx+fx−Gn,λqnf;xGn,λqng;x−gx+Gn,λqnt−x;xqn−1xDqnfxDqngx+Gn,λqnt−xqn2;x2qn!qn−1xDqnfxDqn2gx≔I1+I2+I3+I4+I5+I6.
By Theorem 13, for any fixedx∈ℝ+, we can easily have the following estimates
(71)nqn∣I1∣≤O1ΩDqn2fg;1nqn,(72)nqn∣I2∣≤∣gx∣O1ΩDqn2f;1nqn≤∥g∥2O1ΩDqn2f;1nqn,(73)nqn∣I3∣≤∣fx∣O1ΩDqn2g;1nqn≤∥f∥2O1ΩDqn2g;1nqn.
Using (14), (20), and ∣qn−1∣=qnn−1/nqn≤O1/nqn, we have
(74)nqn∣I5∣≤O1nqn∥Dqnf∥2∥Dqng∥2,nqn∣I6∣≤O1nqn∥Dqnf∥2∥Dqn2g∥2.
Using (14), (20), and Theorem 13, we can get
(75)Gn,λqnf;x−fx≤O1nqn∥Dqnf∥2+∥Dqn2f∥2+O1nqnΩDqn2f;1nqn,hence, we can know
(76)nqn∣I4∣≤O1nqn∥Dqnf∥2+∥Dqn2f∥2∥Dqng∥2+∥Dqn2g∥2+O1nqnΩDqn2f;1nqn∥Dqng∥2+∥Dqn2g∥2+O1nqnΩDqn2g;1nqn∥Dqnf∥2+∥Dqn2f∥2+O1nqnΩDqn2f;1nqnΩDqn2g;1nqn.
Combining (71)–(76), we complete the proof of Theorem 15.Corollary 16.
Letq=qn be a sequence satisfying qn∈0,1, qn→1 and qnn→a∈0,1 as n→∞ and f,g∈C20ℝ+ satisfy f′′,g′′,fg′′∈C20ℝ+. Then, the following limit equality
(77)limn→∞nqnGn,λqnfg;x−Gn,λqnf;xGn,λqng;x=af′xg′xx2,holds for any x∈ℝ+.



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*Source: 5710510-2020-10-21.xml*