# Approximation Properties of New Modified Gamma Operators

Authors: Yun-Shun Wu; Wen-Tao Cheng; Wei-Ping Zhou; Lun-Zhi Deng
Journal: Journal of Function Spaces
 (2021)

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## Abstract

This paper is aimed at constructing new modified Gamma operators using the second central moment of the classic Gamma operators. And we will compute the first, second, fourth, and sixth order central moments by the moment computation formulas, and their quantitative properties are researched. Then, the global results are established in certain weighted spaces and the direct results including the Voronovskaya-type asymptotic formula, and point-wise estimates are investigated. Also, weighted approximation of these operators is discussed. Finally, the quantitative Voronovskaya-type asymptotic formula and Grüss Voronovskaya-type approximation are presented.


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## Body

## 1. Introduction

Recently, Karsli et al. [1] constructed and estimated the rate of convergence for functions with derivatives of bounded variation on ℝ+≔0,∞ of new Gamma type operators preserving z2 as (see also [2])
(1)Φlλtz=2l+3!zl+3l!l+2!∫0∞tlz+t2l+4λtdt,z∈ℝ+.In [3], Karsli et al. used analysis methods to obtain the rate of point-wise convergence for the operators (1). In [4], Karsli and Özarslan obtained some direct local and global approximation results for the operators (1). In [5], İzgi studied some direct results in asymptotic approximation about the operators (1). In [6], Krech gave a note about the results of İzgi in [5] and obtained an error estimate for the operators (1). In [7], Krech gave direct approximation theorems for the operators (1) in certain weighted spaces. In [8], Cai and Zeng constructed q-Gamma operators and gave their approximation properties. In [9], Zhao et al. extended the works of Cai and Zeng and considered the stancu generalization of q-Gamma operators. Recently, Cheng et al. constructed p,q-Gamma operators using p,q-Beta function of the second kind and discussed their approximation properties in [10]. In [11], Zhou et al. extended the works of Cheng et al. in [10] and constructed p,q-Gamma-Stancu operators. There are many papers about the research and application of other Gamma-type operators, and we mention some of them [12–17].In this paper, we construct new modified Gamma operators using the second central moment of the operators (1) as follows:Definition 1.
Forl=1,2,⋯ and λ:ℝ+⟶ℝ, we construct new modified Gamma operators by
(2)Ylλtz=∫0∞Klt,zλtdt,z∈ℝ+,where
(3)Klt,z=2l2l+3!!zl+1l!tlz−t2z+t2l+4,t,z∈ℝ+.The paper is organized as follows: In Section1, we introduce the history of Gamma operators and construct new modified Gamma operators using the second central moment. In Section 2, we obtain the basic results by the moment computation formulas. And the first, second, fourth, and sixth order central moment computation formulas and limit equalities are also obtained. In Section 3, we establish the global approximation results for the operators (2) in certain weighted spaces. In Sections 4 and 5, we investigate the direct results including the Voronovskaya-type asymptotic formula and point-wise estimates in three different Lipschitz classes and discuss weighted approximation. In Section 6, we present a quantitative Voronovskaya-type asymptotic formula and a Grüss Voronovskaya-type approximation (for the quantitative Voronovskaya type theorem and Grüss-Voronovskaya theorem for the other operators, see also [18–24]).


## 2. Basic Results

In this section, we present certain auxiliary results which will be used to prove our main theorems for the operators (2).Lemma 2 (see [1]).
For anyl∈ℕ+, p=0,1,2,⋯,l+2, we have
(4)φlp≔Φltpz=l+p!l+2−p!l!l+2!zp,z∈ℝ+.Lemma 3.
If we defineϕlp≔Yltpz, then there holds the following relation
(5)2z2l+2ϕlp=φlp+2−2φlp+1z+φlpz2,where p=0,1,⋯,l, z∈ℝ+.
Then, the following lemma can be obtained immediately:Lemma 4.
For anyl∈ℕ+, z∈ℝ+, we have
(6)ϕl0=1;ϕl1=l+3lz;ϕl2=l+2l+9ll−1z2,forl>1;(7)ϕl3=l+2l+3l+19ll−1l−2z3,forl>2;(8)ϕl4=l+2l+3l+4l+33ll−1l−2l−3z4,forl>3;(9)ϕlp=l−p!l+p!l+2p2+1l!l+1!zp,forl≥p;(10)Alz≔Ylt−zz=3lz;(11)Blz≔Ylt−z2z=6l+24ll−1z2,forl>1;(12)Ylt−z3z=90l+230ll−1l−2z3,forl>2;(13)Ylt−z4z=60l2+23l+48ll−1l−2l−3z4,forl>3;(14)Ylt−z6z=840l3+69l2+506l+768ll−1l−2l−3l−4l−5z6,forl>5;(15)liml⟶∞Alz=3z;(16)liml⟶∞Blz=6z2;(17)liml⟶∞l2Ylt−z4z=60z4;(18)liml⟶∞l3Ylt−z6z=840z6.
By the classical Korovkin theorem, we easily obtain the following lemma:Lemma 5.
For allλ∈CBℝ+ and any finite interval I⊂ℝ+, then the sequence Ylλtz converges to λ uniformly on I, where CBℝ+ denotes the set of all real-valued bounded and continuous functions defined on ℝ+, endowed with the norm λ=supz∈ℝ+λz.


## 3. Global Results

In this section, we establish some global results by using certain Lipschitz classes. We first recall some basic definitions. Letr∈ℕ≔0,1,2,⋯ and define the weighted function wr as follows:
(19)w0z≔1andwrz≔11+zrforz∈ℝ+andr∈ℕ\0.Meantime, we consider the following subspaceSrℝ+ of Cℝ+ generated by wr:
(20)Srℝ+≔λ∈Cℝ+:wrλisuniformlycontinuousandboundedonℝ+endowed with the norm λr≔supz∈ℝ+wrzλz for λ∈Srℝ+. For every λ∈Srℝ+, δ>0, and α∈0,2, the usual weighted modulus of continuity, the second-order weighted modulus of smoothness, and the corresponding Lipschitz classes are, respectively, defined as
(21)ωr1λ;δ≔supwrzλy−λz:y−z≤δ,y,z∈ℝ+;ωr2λ;δ≔supt∈0,δλz+2t−2λz+t+λzr;Lipr2α≔λ∈Srℝ+:ωr2λ;δ=Oδαasδ⟶0+.Theorem 6.
Letr∈0,1,⋯,l be fixed. Then, there exists a positive constant Cr such that
(22)Yl1wrr≤Cr.
Furthermore, for allλ∈Srℝ+, we have
(23)Ylλr≤Crλr.
Thus,Yl is a linear positive operator from Srℝ+ to Srℝ+ for any r∈0,1,⋯,l.Proof.
Inequality (22) is obvious for r=0. Assume that l≥r≥1, using (6), we have
(24)wrzYl1wrtz=wrzYl1+trz=wrzYl1z+wrzYltrz=wrz+wrzl−r!l+r!l+2r2+1l!l+1!zr≤Crwrz1+zr=Cr,where Cr=max1,supll−r!l+r!l+2r2+1/l!l+1!, and then we obtain (22). Moreover, for every λ∈Srℝ+ and z∈ℝ+, we have
(25)wrzYlλtz≤wrz∫0∞Klt,zλtdt=wrz∫0∞Klt,zλtwrtwrtdt≤λrwrzYl1wrtz≤Crλr.
Taking the supremum overz∈ℝ+, we obtain (23).Theorem 7.
For any fixedr∈0,1,⋯,l−2, l≥2, there exists a positive constant Cr such that
(26)wrzYlt−z2wrtz≤Crz2l.Proof.
The formula (11) implies (26) for r=0. If r=1, then we obtain
(27)Ylt−z21+tz=Ylt−z2z+Ylt−z2tz=Ylt−z3z+1+zYlt−z2z,which by (11) and (12) yield (26) for r=1. Assuming l−2≥r≥2 and using (11) and (6), we obtain
(28)Ylt−z2wrtz=Ylt−z2z+Yltr+2z−2zYltr+1z+z2Yltrz=6l+24ll−1z2+l−r−2!l+r+2!l+2r+22+1l!l+1!zr+2−2l−r−1!l+r+1!l+2r+12+1l!l+1!zr+2+l−r!l+r!l+2r2+1l!l+1!zr+2=z2l6l+24l−1+l+r+1l+r+2l+2r+22+12l−r−1l+r+1l+2r+12+1+l−r−1l−rl+2r2+1l−r−2!l+r!l−1!l+1!zr≤z2l6l+24l−1+6l2+Cr,1l+Cr,2l−r−2!l+r!l−1!l+1!zr≤Crz2l1+zr,where Cr,1 and Cr,1 are two constants only depending on r. This completes the proof.Now, forr∈0,1,⋯,l, we consider the two spaces Sr1ℝ+≔λ∈Srℝ+:λ′∈Srℝ+ and Sr2ℝ+≔λ∈Srℝ+:λ′′∈Srℝ+, and we have the three following theorems:Theorem 8.
For any fixedr, if λ∈Sr1ℝ+, there exists a positive constant Cr such that
(29)wrzYlλtz−λz≤Crλ′rzlfor all z∈ℝ+ and l≥r+2.Proof.
Letz∈ℝ+. By λt−λz=∫tzλ′udu, t∈ℝ+, Lemma (11), and the linearity of Yl, we obtain
(30)Ylλtz−λz=Yl∫ztλ′uduz.
Using(31)∫tzλ′udu≤λ′r∫zt1wrudu≤λ′r1wrt+1wrzt−z.
Hence,(32)wrzYlλtz−λz≤λ′rYlt−zz+wrzYlt−zwrtz.
Applying the well-known Cauchy-Schwarz inequality, we can obtain(33)Yl∣t−z∣z≤Ylt−z2z,Yl∣t−z∣wrtz≤Yl1wrtzYlt−z2wrtz.
Combining (22) and (26), we can get the required result.Theorem 9.
For any fixedr, if λ∈Sr1ℝ+, then there exists a positive constant Cr such that
(34)wrzYlλtz−λz≤Crωr1λ;zlfor all z∈ℝ+ and l≥r+2.Proof.
Letz∈ℝ+. We denote the Steklov means of λ by λs, s∈ℝ+:
(35)λsz=1s∫0sλu+zdu,z,s∈ℝ+.
It is obvious that(36)λsz−λz=1s∫0sλu+z−λzdu,λs′z=1sλz+s−λzfor z,s∈ℝ+. Hence, if λ∈Sr1ℝ+, then λs∈Sr2ℝ+ for every fixed s∈ℝ+. Furthermore, we have
(37)λs−λr≤ωr1λ;s,λs′r≤1sωr1λ;s.
By(38)wrzYlλtz−λz≤wrzYlλt−λstz+wrzYlλstz−λsz+wrzλz−λsz.
Using (23) and (37), we have
(39)wrzYlλt−λstz≤Crλ−λsr≤Crωr1λ;sfor any z,s∈ℝ+. From (29) and (37), we have
(40)wrzYlλstz−λsz≤Crλs′rzl≤Cr1sωr1λ;szl.
By (37), we have
(41)wrzλz−λsz≤λ−λsr≤ωr1λ;sfor any z,s∈ℝ+. Finally, we have
(42)wrzYlλtz−λz≤ωr1λ;sCr+1sCrzl+1for any z,s∈ℝ+. Choosing s=z/l, the proof is proved.Theorem 10.
Defining a new operator,(43)Yl∗λtz=Ylλtz−λz+Alz+λz.
For any fixedr, if λ∈Sr2ℝ+, then there exists a positive constant Cr such that
(44)wrzYl∗λtz−λz≤Crλ′′rz2l,for all z∈ℝ+ and l≥r+2.Proof.
Using Taylor’s expansion, we have(45)λt−λz=t−zλ′z+∫ztt−uλ′′udu,z,t∈ℝ+.
ByYl∗t−zz=0 and Yl∗1z=1, we have
(46)Yl∗λtz−λz≤Yl∗λt−λzz≤Yl∗∫ztt−uλ′′uduzYl∫ztt−uλ′′uduz−∫zz+Alzz+Alz−uλ′′udu.
Since(47)∫ztt−uλ′′u≤λ′′rt−z221wrz+1wrt,∫zz+Alzz+Alz−uλ′′udu≤λ′′r2wrzAlz2,we have
(48)wrzYl∗λtz−λz≤λ′′r2Blz+wrzYlt−z2wrtz+λ′′r2Alz2.
Combining Lemma4 and (26), we have
(49)wrzYl∗λtz−λz≤Crλ′′rz2lfor all z∈ℝ+ and l≥r+2. The theorem is completed.Theorem 11.
For any fixedr, if λ∈Sr2ℝ+, then there exists a positive constant Cr such that
(50)wrzYlλtz−λz≤Crωr2λ;zl+ωr1λ;Alzfor all z∈ℝ+ and l≥r+2. In particular, if λ∈Lipr2α for some α∈0,2, then
(51)wrzYlλtz−λz≤Crz2lα/2+ωr1λ;Alzholds.Proof.
Letλ∈Srℝ+, and the Steklov means λ~sz of the second order of λ defined by
(52)λ~sz=4s2∫0s/2∫0s/22λz+u+v−λz+2u+2vdudvfor z,s∈ℝ+. By simple computation, we have
(53)λ−λ~sr≤ωr2λ;s,λ~s′′r≤9s2ωr2λ;s.
Meantime,λ~s∈Sr2ℝ+ while λ∈Srℝ+. Using the following inequality,
(54)Ylλtz−λz≤Yl∗λt−λ~stz+λz−λ~sz+Yl∗λ~stz−λ~sz+λz+Alz−λz.
Combining (23) and (44), we have
(55)wrzYlλtz−λz≤Cr+3λ−λ~sr+Crλ~s′′rz2lwrzλz+Alz−λz≤Crωr2λ;s1+1s2z2l+ωr1λ;Alz.
Hence, choosings=z/l, the first part of the proof is proved. The second part of the proof can be directly observed from the definition of the space Lipr2α.


## 4. Direct Results

### 4.1. Voronovskaya-Type Theorem

Theorem 12.
Ifλ∈CBℝ+ and λ′′ exists at a point z∈ℝ+, then
(56)liml⟶∞lYlλtz−λz=3zλ′z+zλ′′z.Proof.
By the Taylor’s expansion formula forλ, we have
(57)λt=λz+λ′zt−z+12λ′′zt−z2+Rt;zt−z2,where
(58)Rt;z=λt−λz−λ′zt−z−1/2λ′′zt−z2t−z2,t≠z;0,t=z.
Applying the L’Hospital’s Rule,(59)limt⟶zRt;z=12limt⟶zλ′t−λ′zt−z−12λ′′z=0.
Thus,R.;z∈CBℝ+. Consequently, we can write
(60)Ylλtz−λz=Alzλ′z+12Blzλ′′z+YlRt;zt−z2z.
By the Cauchy-Schwarz inequality, we have(61)lYlRt;zt−z2z≤YlR2t;zzl2Ylt−z4z.
We observe thatR2z;z=0 and R2t;z∈CBℝ+. Then, it follows in Lemma 5 that
(62)liml⟶∞YlR2t;zz=R2z;z=0.
Hence, from (17), we can obtain
(63)liml⟶∞lYlRt;zt−z2z=0.
Combining (15) and (16), we complete the proof of Theorem 12.Corollary 13.
Ifλ,λ′′∈CBℝ+, then we have
(64)liml⟶∞lYlλtz−λz=3zλ′z+zλ′′z,uniformly with respect to any finite interval I⊂ℝ+.


### 4.2. Point-Wise Estimates

In this subsection, we establish three point-wise estimates of the operators (2). First, we obtain the rate of convergence locally by using functions belonging to the Lipschitz class. We denote that λ∈CBℝ+ is in LipMγ,D, γ∈0,1, and D⊂ℝ+ if it satisfies the following condition:
(65)λt−λz≤Mt−zγ,t∈D,z∈ℝ+,where M is a positive constant depending only on γ and λ.Theorem 14.
Ifλ∈CBℝ+⋂LipMγ,D, then for any z∈ℝ+, we have
(66)Ylλtz−λz≤MBlzγ/2+2dz;D,where dz;D=inft−z:t∈D denotes the distance between z and D.Proof.
LetD¯ be the closure of D. Using the properties of infimum, there is at least a point t0∈D¯ such that dz;D=z−t0. By the triangle inequality
(67)λt−λz≤λt−λt0+λz−λt0,we have
(68)Ylλtz−λz≤Ylλt−λt0z+Ylλz−λt0z≤MYlt−t0γz+z−t0γ≤MYlt−zγ+z−t0γz+z−t0γ≤MYlt−zγz+2z−t0γ.
Choosingp=2/γ and q=2/2−γ and using the well-known Hölder inequality, we have
(69)Ylλtz−λz≤MYlt−zpγz1/pYl1qz1/q+2dγz;D≤MYlt−z2zγ/2+2dγz;D≤MBlzγ/2+2dγz;D.
Next, we obtain the local direct estimate of the operators (2), using the Lipcshitz type maximal function of the order γ introduced by Lenze [25] as
(70)ω~γλ;z=supz,t∈ℝ+,t≠zλt−λzt−zγ,γ∈0,1.Theorem 15.
Ifλ∈CBℝ+, then for any z∈ℝ+, we have
(71)Ylλtz−λz≤ω~γλ;zBlzγ/2.Proof.
From equation (70), we have
(72)Ylλtz−λz≤ω~γλ;zYlt−zγz.
Applying the well-known Hölder inequality, we have(73)Ylλtz−λz≤ω~γλ;zYlt−z2zγ/2=ω~γλ;zBlzγ/2.
Finally, we establish point-wise estimate of the operators (2) in the following Lipschitz-type space (see [26]) with two distinct parameters μ1,μ2∈ℝ+:
(74)LipMμ1,μ2γ≔λ∈Cℝ+:λt−λz≤Mt−zγt+μ1z2+μ2z,t,z∈ℝ+,where γ∈0,1, M, is a positive constant depending only on γ,μ1,μ2 and λ.Theorem 16.
Ifλ∈LipMμ1,μ2γ, then for any z∈ℝ+, we have
(75)Ylλtz−λz≤MBlzμ1z2+μ2zγ/2.Proof.
Applying the well-known Hölder inequality withp=2/γ and q=2/2−γ, we have
(76)Ylλtz−λz≤Ylλt−λzz≤YlMt−zγt+μ1z2+μ2zz≤Mμ1z2+μ2zYlt−zγz≤Mμ1z2+μ2zYlt−zpγz1/pYl1qz1/q=MBlzμ1z2+μ2zγ/2.
Thus, the proof is completed.


## 4.1. Voronovskaya-Type Theorem

Theorem 12.
Ifλ∈CBℝ+ and λ′′ exists at a point z∈ℝ+, then
(56)liml⟶∞lYlλtz−λz=3zλ′z+zλ′′z.Proof.
By the Taylor’s expansion formula forλ, we have
(57)λt=λz+λ′zt−z+12λ′′zt−z2+Rt;zt−z2,where
(58)Rt;z=λt−λz−λ′zt−z−1/2λ′′zt−z2t−z2,t≠z;0,t=z.
Applying the L’Hospital’s Rule,(59)limt⟶zRt;z=12limt⟶zλ′t−λ′zt−z−12λ′′z=0.
Thus,R.;z∈CBℝ+. Consequently, we can write
(60)Ylλtz−λz=Alzλ′z+12Blzλ′′z+YlRt;zt−z2z.
By the Cauchy-Schwarz inequality, we have(61)lYlRt;zt−z2z≤YlR2t;zzl2Ylt−z4z.
We observe thatR2z;z=0 and R2t;z∈CBℝ+. Then, it follows in Lemma 5 that
(62)liml⟶∞YlR2t;zz=R2z;z=0.
Hence, from (17), we can obtain
(63)liml⟶∞lYlRt;zt−z2z=0.
Combining (15) and (16), we complete the proof of Theorem 12.Corollary 13.
Ifλ,λ′′∈CBℝ+, then we have
(64)liml⟶∞lYlλtz−λz=3zλ′z+zλ′′z,uniformly with respect to any finite interval I⊂ℝ+.


## 4.2. Point-Wise Estimates

In this subsection, we establish three point-wise estimates of the operators (2). First, we obtain the rate of convergence locally by using functions belonging to the Lipschitz class. We denote that λ∈CBℝ+ is in LipMγ,D, γ∈0,1, and D⊂ℝ+ if it satisfies the following condition:
(65)λt−λz≤Mt−zγ,t∈D,z∈ℝ+,where M is a positive constant depending only on γ and λ.Theorem 14.
Ifλ∈CBℝ+⋂LipMγ,D, then for any z∈ℝ+, we have
(66)Ylλtz−λz≤MBlzγ/2+2dz;D,where dz;D=inft−z:t∈D denotes the distance between z and D.Proof.
LetD¯ be the closure of D. Using the properties of infimum, there is at least a point t0∈D¯ such that dz;D=z−t0. By the triangle inequality
(67)λt−λz≤λt−λt0+λz−λt0,we have
(68)Ylλtz−λz≤Ylλt−λt0z+Ylλz−λt0z≤MYlt−t0γz+z−t0γ≤MYlt−zγ+z−t0γz+z−t0γ≤MYlt−zγz+2z−t0γ.
Choosingp=2/γ and q=2/2−γ and using the well-known Hölder inequality, we have
(69)Ylλtz−λz≤MYlt−zpγz1/pYl1qz1/q+2dγz;D≤MYlt−z2zγ/2+2dγz;D≤MBlzγ/2+2dγz;D.
Next, we obtain the local direct estimate of the operators (2), using the Lipcshitz type maximal function of the order γ introduced by Lenze [25] as
(70)ω~γλ;z=supz,t∈ℝ+,t≠zλt−λzt−zγ,γ∈0,1.Theorem 15.
Ifλ∈CBℝ+, then for any z∈ℝ+, we have
(71)Ylλtz−λz≤ω~γλ;zBlzγ/2.Proof.
From equation (70), we have
(72)Ylλtz−λz≤ω~γλ;zYlt−zγz.
Applying the well-known Hölder inequality, we have(73)Ylλtz−λz≤ω~γλ;zYlt−z2zγ/2=ω~γλ;zBlzγ/2.
Finally, we establish point-wise estimate of the operators (2) in the following Lipschitz-type space (see [26]) with two distinct parameters μ1,μ2∈ℝ+:
(74)LipMμ1,μ2γ≔λ∈Cℝ+:λt−λz≤Mt−zγt+μ1z2+μ2z,t,z∈ℝ+,where γ∈0,1, M, is a positive constant depending only on γ,μ1,μ2 and λ.Theorem 16.
Ifλ∈LipMμ1,μ2γ, then for any z∈ℝ+, we have
(75)Ylλtz−λz≤MBlzμ1z2+μ2zγ/2.Proof.
Applying the well-known Hölder inequality withp=2/γ and q=2/2−γ, we have
(76)Ylλtz−λz≤Ylλt−λzz≤YlMt−zγt+μ1z2+μ2zz≤Mμ1z2+μ2zYlt−zγz≤Mμ1z2+μ2zYlt−zpγz1/pYl1qz1/q=MBlzμ1z2+μ2zγ/2.
Thus, the proof is completed.


## 5. Weighted Approximation

LetB2ℝ+ be the set of all functions λ defined on ℝ+ satisfying the condition ∣λz∣≤Mλ1+z2 with an absolute constant Mλ>0 which depends only on λ. C2ℝ+ denotes the subspace of all continuous functions λ∈B2ℝ+ with the norm λ2=supz∈ℝ+λz/1+z2. By C20ℝ+, we denote the subspace of all functions f∈C2ℝ+ for which limz⟶+∞λz/1+z2 is finite.Theorem 17.
Ifλ∈C20ℝ+ and κ>0, we have
(77)liml⟶∞supz∈ℝ+Ylλtz−λz1+z21+κ=0.Proof.
Letz0∈ℝ+ be arbitrary but fixed.
(78)supz∈ℝ+Ylλtz−λz1+z21+κ≤supz∈0,z0Ylλtz−λz1+z21+κ+supz∈z0,∞Ylλtz−λz1+z21+κ≤Ylλtz−λ0,z0+λ2supz∈z0,∞Yl1+t2z1+z21+κ+supz∈z0,∞λz1+z21+κ≔I1+I2+I3.
Applyingλz≤λ21+z2, we have
(79)I3=supz∈z0,∞λz1+z21+κ≤supz∈z0,∞λ21+z21+z21+κ≤λ21+z02κ.
Letε>0. Since liml⟶∞supz∈z0,∞Yl1+t2z/1+z2=1, there exists L1∈ℕ, such that for all l>L1,
(80)λ2Yl1+t2z1+z21+κ≤λ21+z21+κ1+z2+ε3λ2≤λ21+z2κ+ε3.
Hence,(81)λ2supz∈z0,∞Yl1+t2z1+z21+κ≤λ21+z02κ+ε3,∀l≥L1.
Thus,(82)I2+I3<2λ21+z02κ+ε3,∀l≥L1.
Next, for sufficiently largez0 such that λ2/1+z02κ<ε/6, then I2+I3<2ε/3,∀l≥L1. Applying Lemma 5, there exists L2∈ℕ, such that for all l>L2,
(83)Ylλtz−λ0,z0<ε3.
LetL=maxL1,L2. Combining (80) (82), and (83), we have
(84)supz∈ℝ+Ylλtz−λz1+z21+κ<ε,∀l≥L.
Hence, the proof of Theorem17 is completed.Theorem 18.
Ifλ∈C20ℝ+, then we have
(85)liml⟶∞Ylλtz−λ2=0.Proof.
Applying the Korovkin theorem [27], it is sufficient to show the following three conditions:
(86)liml⟶∞Yltpz−zp2=0,p=0,1,2.
SinceYl1z=1, the condition (86) holds for p=0. From Lemma (11), we have
(87)Yltz−z2=supz∈ℝ+11+z2l+3lz−z≤3l.
Thus,liml⟶∞Yltz−z2=0. Finally, we have
(88)Ylt2z−z22=supz∈ℝ+11+z2l+2l+9ll−1z2−z2≤12l+19ll−1,which implies that liml⟶∞Ylt2z−z22=0.


## 6. Some Voronovskaya-Type Approximation Theorem

As is known, ifλ∈Cℝ+ is not uniform, the limit limδ⟶0+ωλ;δ=0 may be not true. In [28], Yüksel and Ispir defined the following weighted modulus of continuity:
(89)Ωλ;δ=supz∈ℝ+,0<h≤δλz+h−λz1+z21+h2forλ∈C20ℝ+and proved the properties of monotone increasing about Ωλ;δ as δ>0, limδ⟶0+Ωλ;δ=0, and the inequality
(90)Ωλ;τδ≤21+τ1+δ2Ωλ;δ,τ>0.For anyλ∈C20ℝ+, it follows from (89) and (90) that
(91)λt−λz≤1+t−z21+z2Ωλ;t−z≤21+t−zδ1+δ2Ωλ;δ1+t−z21+z2.In the next theorem, we obtain the degree of approximation ofλ by the operators (2) in the weighted space of continuous functions C20ℝ+ in terms of the weighted modulus of smoothness Ωλ;δ,δ>0.


### 6.1. Quantitative Voronovskaya-Type Theorem

Theorem 19.
Ifλ∈C20ℝ+ satisfies λ′,λ′′∈C20ℝ+, then for sufficiently large l and any z∈ℝ+,
(92)lYlλtz−λz−λ′zAlz−λ′′z2!Blz≤O1Ωλ′′;1l.Proof.
By Taylors’ expansion formula forλ, we have
(93)λt=λz+λ′zt−z+λ′′y2!t−z2=λz+λ′zt−z+λ′′z2!t−z2+R1t,z,where y−z≤t−z and hence
(94)R1t,z=λ′′y−λ′′z2!t−z2.
Applying the inequality (91) of the weighted modulus of continuity, we have
(95)λ′′y−λ′′z≤1+y−z21+z2Ωλ′′;y−z≤1+t−z21+z2Ωλ′′;t−z≤21+t−zδ1+δ2Ωλ′′;δ1+t−z21+z2≤41+δ221+z2Ωλ′′;δ,t−z≤δ,41+δ221+z2Ωλ′′;δt−z4δ4,t−z>δ,≤41+δ221+z2Ωλ′′;δ1+t−z4δ4.
Combining (94) and (95) and choosing δ∈0,1, we have
(96)R1t,z≤21+δ221+z2Ωλ′′;δ1+t−z4δ4t−z2.
Using the operator (2) and Lemma 4 on both sides of (94), we have
(97)Ylλtz−λz−λ′zAlz−λ′′z2!Blz≤YlR1t,zz.
Applying (16), (18), and (96), we have
(98)YlR1t,zz≤21+δ221+z2Ωλ′′;δYlt−z2+t−z6δ4z≤21+δ221+z2Ωλ′′;δBlz+1δ4Ylt−z6z≤21+δ221+z2Ωλ′′;δO1l+1δ4O1l3.
Choosingδ=1/l, we have
(99)lYlR1t,zz≤O1Ωλ′′;1l.
Combining (97)-(99), we complete the proof of Theorem 19.


### 6.2. Grüss Voronovskaya-Type Theorem

Theorem 20.
Ifλ,μ∈C20ℝ+ satisfy λμ,λ′,μ′,λμ′,λ′′,μ′′ and λμ′′∈C20ℝ+. Then, for any z∈ℝ+,
(100)liml⟶∞lYlλ·μtz−Ylλtz·Ylμtz=6λ′zμ′zz2.Proof.
Using the equalities(101)λ·μz=λz·μz,λ·μ′z=λ′z·μz+λz·μ′z,λ·μ′′z=λ′′z·μz+2λ′z·μ′z+λz·μ′′z,by simple computations, for any z∈ℝ+, we have
(102)Ylλ·μtz−Ylλtz·Ylμtz=Ylλ·μtz−λ·μz−λ·μ′zAlz−λ·μ′′z2!Blz−μzYlλtz−λz−λ′zAlz−λ′′z2!Blz−YlλtzYlμtz−μz−μ′zAlz−μ′′z2!Blz+12!Blzλz·μ′′z+2λ′z·μ′z−μ′′z·Ylλtz+Alzλz·μ′z−μ′z·Ylλtz.
By using (16), Lemma 5, and Theorem 19, we have
(103)liml⟶∞lYlλ·μtz−Ylλtz·Ylμtz=6λ′zμ′zz2,which proves our theorem.


## 6.1. Quantitative Voronovskaya-Type Theorem

Theorem 19.
Ifλ∈C20ℝ+ satisfies λ′,λ′′∈C20ℝ+, then for sufficiently large l and any z∈ℝ+,
(92)lYlλtz−λz−λ′zAlz−λ′′z2!Blz≤O1Ωλ′′;1l.Proof.
By Taylors’ expansion formula forλ, we have
(93)λt=λz+λ′zt−z+λ′′y2!t−z2=λz+λ′zt−z+λ′′z2!t−z2+R1t,z,where y−z≤t−z and hence
(94)R1t,z=λ′′y−λ′′z2!t−z2.
Applying the inequality (91) of the weighted modulus of continuity, we have
(95)λ′′y−λ′′z≤1+y−z21+z2Ωλ′′;y−z≤1+t−z21+z2Ωλ′′;t−z≤21+t−zδ1+δ2Ωλ′′;δ1+t−z21+z2≤41+δ221+z2Ωλ′′;δ,t−z≤δ,41+δ221+z2Ωλ′′;δt−z4δ4,t−z>δ,≤41+δ221+z2Ωλ′′;δ1+t−z4δ4.
Combining (94) and (95) and choosing δ∈0,1, we have
(96)R1t,z≤21+δ221+z2Ωλ′′;δ1+t−z4δ4t−z2.
Using the operator (2) and Lemma 4 on both sides of (94), we have
(97)Ylλtz−λz−λ′zAlz−λ′′z2!Blz≤YlR1t,zz.
Applying (16), (18), and (96), we have
(98)YlR1t,zz≤21+δ221+z2Ωλ′′;δYlt−z2+t−z6δ4z≤21+δ221+z2Ωλ′′;δBlz+1δ4Ylt−z6z≤21+δ221+z2Ωλ′′;δO1l+1δ4O1l3.
Choosingδ=1/l, we have
(99)lYlR1t,zz≤O1Ωλ′′;1l.
Combining (97)-(99), we complete the proof of Theorem 19.


## 6.2. Grüss Voronovskaya-Type Theorem

Theorem 20.
Ifλ,μ∈C20ℝ+ satisfy λμ,λ′,μ′,λμ′,λ′′,μ′′ and λμ′′∈C20ℝ+. Then, for any z∈ℝ+,
(100)liml⟶∞lYlλ·μtz−Ylλtz·Ylμtz=6λ′zμ′zz2.Proof.
Using the equalities(101)λ·μz=λz·μz,λ·μ′z=λ′z·μz+λz·μ′z,λ·μ′′z=λ′′z·μz+2λ′z·μ′z+λz·μ′′z,by simple computations, for any z∈ℝ+, we have
(102)Ylλ·μtz−Ylλtz·Ylμtz=Ylλ·μtz−λ·μz−λ·μ′zAlz−λ·μ′′z2!Blz−μzYlλtz−λz−λ′zAlz−λ′′z2!Blz−YlλtzYlμtz−μz−μ′zAlz−μ′′z2!Blz+12!Blzλz·μ′′z+2λ′z·μ′z−μ′′z·Ylλtz+Alzλz·μ′z−μ′z·Ylλtz.
By using (16), Lemma 5, and Theorem 19, we have
(103)liml⟶∞lYlλ·μtz−Ylλtz·Ylμtz=6λ′zμ′zz2,which proves our theorem.



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*Source: 6696979-2021-05-15.xml*