# Approximation Properties of a New Gamma Operator

Authors: Jieyu Huang; Qiulan Qi
Journal: Journal of Mathematics
 (2022)

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## Abstract

This paper deals with a kind of modification of the classical Gamma operators defined on the semiaxis which holds fixed functions 1 ande−μxμ>0. We study the uniform approximation effect and the direct results. We also investigate the weighted A-statistical convergence. Finally, the Voronovskaja type asymptotic formula is given.


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## Body

## 1. Introduction

Forn∈N=1,2,…, the Gamma operators are given by [1].(1)Gnf;x=1n!∫0∞e−ττnfxnτdτ,x∈0,+∞.These operators play important role in many fields inside applied mathematics [1–11]. A considerable amount of literature has been published on approximation processes. Various operators were conveniently modified, which preserve the test functions that emerged in this field and a lot of advancements have been made regarding this subject to acquire better approximation [10–17]. For some recent studies on linear positive operators preserving exponential functions, we refer the readers to the literature [2, 5, 10, 11, 13, 14, 16, 17]. Following the idea of King [12], further developed in references [2, 5, 10–17], we introduce the operators which preserve constants and e−μxμ>0. We study their uniform convergence properties, asymptotic formula, and the weighted-statistical convergence results.The improved Gamma operators are defined by(2)Gsμf;x=1Γs+1∫0∞e−ττsfx2ταsdτ,where s is a positive real parameter,(3)α=μx2seμx/s+1−1,μ>0.Remark 1.
By calculating, we havelims⟶+∞α=x.Remark 2.
CB0,+∞ denotes the space of all continuous functions bounded on 0,+∞. C∗0,+∞≔f∈C0,+∞:limx⟶+∞fxexists and is finite.Remark 3.
In this paper, the norm of the function is defined as a uniform norm:f∞≔supx∈0,+∞fx.


## 2. Auxiliary Results

According to the definition of the operatorsGsμf;x, we have as follows:Lemma 1.
Ifμ>0, x∈0,+∞, then(4)Gsμ1;x=1,Gsμe−μt;x=e−μx,Gsμe−2μt;x=1−2μx2sα+2μx2s+1.Lemma 2.
Ifμ>0, x∈0,+∞, then(5)Gsμt;x=s+1x2sα,Gsμt2;x=s+1s+2x4α2s2.Meanwhile, letΦs,kx≔Gsμt−xk;x, k=1,2,4, we have(6)lims⟶∞sΦs,1x=12μx2,lims⟶∞sΦs,2x=x2,lims⟶∞s2Φs,4x=3x4.Proof.
From the definition of the operators, one has(7)Gsμt;x=1Γs+1∫0∞e−ττsx2ταsdτ=s+1x2sα,Gsμt2;x=1Γs+1∫0∞e−ττsx2ταs2dτ=s+1s+2x4α2s2,Gsμt3;x=1Γs+1∫0∞e−ττsx2ταs3dτ=s+1s+2s+3x6α3s3,Gsμt4;x=1Γs+1∫0∞e−ττsx2ταs4dτ=s+1s+2s+3s+4x8α4s4,using the software Mathematica, we have the estimate of sΦs,1x,sΦs,2x, and s2Φs,4x.Lemma 3.
For anyλ∈R,μ>0, and x∈0,+∞, we have(8)lims⟶+∞Gsμe−2λt;x=e−2λx.Proof.
(9)lims⟶+∞Gsμe−2λt;x=lims⟶+∞1−2λx2sα+2λx2s+1=e−2λx.
By definition of the operatorsGsμf;x, one hasLemma 4.
For anyfx∈CB0,+∞, then Gsμf∞≤f∞.Lemma 5.
Forfx∈C∗0,+∞, one has Gsμf;x∈C∗0,+∞.Proof.
From the condition:fx∈C∗0,+∞, we know that limx⟶+∞fx exists and is finite. For s,τ>0, 1/Γs+1e−ττs≤1. Without loss of generality, let fx≠0, and limx⟶+∞fx=0. For every ε>0, there exists a constant M>0, such that fx≤ε/2 for x>M. Let A=ε/2f∞, if x≥sαM/A1/2, noting that 1/Γs+1∫0∞e−ττsdτ=1, we have(10)Gsμf;x≤1Γs+1∫0Ae−ττsfx2ταsdτ+1Γs+1∫A+∞e−ττsfx2ταsdτ≤f∞A+ε2=ε.
Forx≥sαM/A1/2, Gsμf;x<ε, i.e. limx⟶+∞Gsμf;x=0. Therefore, Gsμf;x∈C∗0,+∞.Lemma 6.
[5, 9] The sequence of operators An:C∗0,+∞⟶C∗0,+∞ converges to fx as n⟶+∞ uniformly in 0,+∞ for all f∈C∗0,+∞, if and only if(11)limn⟶+∞Ane−kt;x=e−kx,k=0,1,2.


## 3. Properties of Approximation

In this section, we will give the main approximation properties of the operatorsGsμf;x.Theorem 1.
Letμ>0, f∈C∗0,+∞, then the sequence of the operators Gsμf;x converges to fx uniformly in the interval 0,+∞.Proof.
Due to Lemma6, we only need to check the relations:(12)supx>0Gsμe−kt;x−e−kx,k=0,1,2.
Firstly, using Lemma1, we have,(13)supx>0Gsμ1;x−1=0.
Secondly, using Mathematica, we can write the expansiona1+ea2x−1/x:(14)a1+ea2x−1/x=e−lna1+ea2x/x=e−ln1+a1/x−a2/1+a1−a1a22x/2!1+a12−a1−1a1a23x2/3!1+a13−a1a24−4a12a24+a13a24x3/4!1+a14+Ox4.
Leta1=μ−1, a2=xμ, x=1/n+1, from the relation (14), we have(15)Gsμe−t;x−e−x=μμ+eμx/n+1−1n+1−e−x=μn+1e−lnμn+1−x−μ−1x2/2n+1−μ−2μ−1x3/6n+12+On−3−e−x=e−xe−μ−1x2/2n+1−μ−2μ−1x3/6n+12+On−3−1=e−x−μ−1x22n+1−μ−2μ−1x33!n+12−μ−1μ2−6μ+6x44!n+13+On−4.
Then,(16)Gsμe−t;x−e−x≤e−xμ−1x22n+1+e−xμ−2μ−1x33!n+12+e−xμ−1⋅μ2−6μ+6x44!n+13+On−4≤4μ−12e21n+1+1n+12⋅μ−2⋅μ−1333!e3+1n+13⋅μ−1⋅μ2−6μ+6444!e4+On−4,lims⟶+∞supx>0Gsμe−t;x−e−x=0.
Leta1=μ/2−1, a2=xμ, x=1/n+1, from the relation (14), we have(17)Gsμe−2t;x−e−2x=μμ+2eμx/n+1−2n+1−e−2x=μ2n+1e−n+1lnμ/2−2x−μ−2x2/n+1−μ−4μ−2x3/3n+12+On−3−e−2x=e−2xe−μ−2x2/n+1−μ−4μ−2x3/3n+12+On−3−1=e−2x−μ−2x2n+1−μ−4μ−2x33n+12−μ−2μ2−12μ+24x412n+13+On−4.
Therefore,(18)Gsμe−2t;x−e−2x≤μ−2e21n+1+1n+12μ−4⋅μ−23223e3+1n+13μ−2⋅μ2−12μ+242412e4+On−4.
One has(19)lims⟶+∞supx>0Gsμe−2t;x−e−2x=0.
From the relations (13), (16), and (19), the proof of Theorem 1 is complete.Definition 1.
The modulus of continuityω∗f;δ is defined by Holhos [18]. For any δ>0 and f∈C∗0,+∞,(20)ω∗f;δ=supx,t≥0,e−x−e−t≤δfx−ft.Definition 2.
The classical moduli of continuity are defined by [7, 8]:(21)ωf;δ=supx,t≥0,t−x≤δfx−ft.
The relationship between the two moduli defined above is [18] ω∗f;δ=ωf∗;δ, where(22)f∗x=f−lnx,x≠0;limt⟶∞ft,x=0.
From Holhos ([18], theorem 3), we can get the following results.Theorem 2.
TheAs:C∗0,+∞⟶C∗0,+∞ are positive linear operators and let(23)As1−1∞=αs,Ase−t−e−x∞=βs,Ase−2t−e−2x∞=γs.If all theαs,βs,andγs vanish at infinity, then for any f∈C∗0,+∞, we have the following conclusion: Asf−f∞≤f∞αs+2+αsω∗f;αs+2βs+γs.Corollary 1.
For anyf∈C∗0,+∞, by Theorem 2, we have(24)Gsμf−f∞≤2ω∗f;2βs+γs.Remark 4.
Forμ=1,2, then βs=0,γs=0, respectively. As a consequence of Theorem 2, we have(25)Gs1f−f∞≤2ω∗f;γs,Gs2f−f∞≤2ω∗f;2βs.Theorem 3.
Iff″∈CB0,+∞, then(26)lims⟶+∞2sGsμf;x−fx=μx2f′x+x2f″x.Proof.
Let’s use Taylor’s formula forf, we get(27)ft=fx+f′xt−x+f″xt−x22+ht,xt−x2,where ht,x=1/2f″ξ−f″x and ξ is between x and t. Next, we apply the operators Gsμ to both sides of the equality,(28)Gsμf;x−fx=f′xΦs,1x+f″x2Φs,2x+Gsμht,xt−x2;x.
By Lemma2, we can get(29)lims⟶+∞sΦs,1x=12μx2,lims⟶+∞sΦs,2x=x2,Φs,4x=Os−2.
Letδ>0, from above definition of ωf;δ, we know that(30)ft−fx≤1+t−xδωf;δ.
Hence, we obtain(31)ht,x≤121+t−xδωf″;δ.
Using the linearity ofGsμ, we get(32)Gsμht,xt−x2;x≤ωf″;δGsμt−x2;x+1δωf″;δGsμt−x3;x.
Choosingδ=1/s, lims⟶+∞sGsμht,xt−x2;x=0.
So, we have the desired result.Definition 3.
Letf∈C0,+∞,M>0 and γ∈0,1. We said f∈LipMγ,if the inequality ft−fx≤Mt−sγ holds.Remark 5.
Forf∈LipMγ, 0<γ≤1, by Hölder inequality and Theorem 3, we have(33)Gsμf−f∞≤CΦs,2γ/2x.Theorem 4.
Letf″∈CB0,+∞. If there exists s0>0 such that(34)fx≤Gsμf;x≤Gsf;x,fors≥s0,x∈0,+∞,then,(35)f″x≥−μf′x≥0.On the other hand, for a given pointx∈0,+∞, if inequality (35) holds, then there exists s0>0 such that fx≤Gsμf;x≤Gsf;x, for each s≥s0.Proof.
By the inequality (34), we have(36)0≤2sGsμf;x−fx≤2sGsf;x−fx.
By Theorem3, we get 0≤μf′x+f″x≤f″x.
We assume that the inequality (35) holds, then 0≤μf′x+f″x≤f″x.
The proof of the theorem is complete.Remark 6.
Theorem4 shows that the new Gamma operators have superior properties.


## 4. Properties of Statistical Approximation

The concept of statistical convergence, which was first introduced by Fast [19], is a generalization of ordinary convergence. Several extensions of statistical approximation have appeared in the literature [20–24] and references therein. In this section, we will give some weighted-statistical approximation properties of the operators Gnμf;x. First, we will recall the following definitions, and notations can be found in ref. [22, 24].Definition 4.
[22] Let E⊆N≔1,2,…, En=k≤n:k∈E. The natural density of E is denoted by(37)δE=limn⟶∞Enn,here En is the cardinality of En. A sequence x=xk is said to be statistically convergent to L, if, for every ε>0, δk∈N:xk−L≥ε=0. In symbol, we write S−limxn=L.Definition 5.
[24] A given non-negative infinite summability matrix A=an,k is said to be regular if limAxn=lim∑k=1∞an,kxk=L, whenever limxk=L. Then, the sequence x=xk is said to be A-statistically convergent to L, denoted by SA−limxn=L, provided that for each ε>0, lim∑k:xk−L≥εan,k=0.Definition 6.
[24] Let A=an,k be a non-negative regular summability matrix, q=qk be a sequence of non-negative numbers such that q0>0 and Qn=∑k=0nqk⟶∞ as n⟶∞. Then, x=xk is said to be weighted A-statistically convergent to L, if, for any ε>0, lim1/Qn∑k=0nqk∑m:qmxm−L≥εak,m=0. In this case, we write SAN¯−limxn=L.Definition 7.
[24] Let A=an,k be a non-negative regular summability matrix, q=qk be a sequence of non-negative numbers such that q0>0 and Qn=∑k=0nqk⟶∞ as n⟶∞, and un be a positive non-increasing sequence. Then, x=xk is weighted A-statistically convergent to L with the rate oun, if for each ε>0,(38)limn⟶∞1unQn∑k=0nqk∑m:qmxm−L≥εak,m=0.
This relation is denoted bySAN¯−oun−limxn=L.Theorem 5.
LetA=an,k be a non-negative regular summability matrix. For any n∈N and f∈C∗0,+∞, we have(39)SAN¯−limGnμf−f∞=0.Proof.
Forfx∈C∗0,+∞, there is a constant C>0, such that fx≤C. Therefore, ft−fx≤2C, 0<t,x<+∞. For any ε>0, there is a δε>0 such that ft−fx<ε, ∀e−t−e−x<δ. Consider Dδ of the form Dδ=x,t∈0,+∞:e−t−e−x<δ, we obtain(40)ft−fx≤ft−fxDδ+ft−fx0,+∞−Dδ=ε+2Cδ2Ω,where Ω=e−t−e−x2. For m∈N, by a direct computation, we write(41)GmμΩ;x=Gmμe−2t;x−e−2x−2e−xGmμe−t;x−e−x+e−2xGmμ1;x−1.
The termGmμf;x−fx can be written as(42)Gmμf;x−fx≤εGmμ1;x+2Cδ2GmμΩ;x+fxGmμ1;x−1≤2Cδ2Gmμe−2t;x−e−2x+4Cδ2Gmμe−t;x−e−x.
For a givenε′>0, such that 0<ε<ε′. If we define the following sets,(43)E=m∈N:qmGmμf;x−fx≥ε′,E1=m∈N:qmGmμe−2t;x−e−2x≥ε′−ε8Cδ2,E2=m∈N:qmGmμe−t;x−e−x≥ε′−ε8Cδ2,we see that E⊂E1∪E2,(44)1Qn∑k=0nqk∑m∈Eak,m≤1Qn∑k=0nqk∑m∈E1∪E2ak,m.
Taking the limitn⟶∞ and noting the conditions, we obtain(45)SAN¯−limGnμf−f∞=0.Remark 7.
Here, we use the Korovkin test functions1,e−x,e−2x. We can also use the usual test functions 1,x,x2.Theorem 6.
LetA=an,k be a non-negative regular summability matrix. If the following condition yields,(46)SAN¯−oun−limωf;hn=0,on0,+∞.Lethn=Gnμt−x2;x∞1/2, then for f∈CB0,∞, we have(47)SAN¯−oun−limGnμf−f∞=0.Proof.
Forfx∈CB0,+∞,m∈N, one has(48)Gmμf;x−fx≤ωf;ξGmμt−xξ+1;x≤ωf;ξ+ωf;ξ1ξGmμt−x2;x1/2.
Letξ≔hm, taking supermum over x∈0,+∞ on both sides, we obtain(49)Gmμf;x−fx≤ωf;hm+ωf;hm1hmGmμt−x2;x1/2=2ωf;hm.
Defining the following sets for a givenε>0,(50)S=m∈N:qmGmμf;x−fx≥ε,S1=m∈N:qmωf;hm≥ε2,it is easy to see that S⊂S1, and(51)1unQn∑k=0nqk∑m∈Sak,m≤1unQn∑k=0nqk∑m∈S1ak,m.
Hence,(52)SAN¯−oun−limGnμf−f∞=0.



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