# Rate of Approximation for Modified Lupaş-Jain-Beta Operators

Authors: M. Qasim; Asif Khan; Zaheer Abbas; Princess Raina; Qing-Bo Cai
Journal: Journal of Function Spaces
 (2020)

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## Abstract

The main intent of this paper is to innovate a new construction of modified Lupaş-Jain operators with weights of some Beta basis functions whose construction depends onσ such that σ0=0 and infx∈0,∞σ′x≥1. Primarily, for the sequence of operators, the convergence is discussed for functions belong to weighted spaces. Further, to prove pointwise convergence Voronovskaya type theorem is taken into consideration. Finally, quantitative estimates for the local approximation are discussed.


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## Body

## 1. Introduction

In 1972, Jain [1] with the help of Poisson distribution introduced a famous linear positive operators as follows:
(1)Pmαf;x=∑j=0∞Wαj,mxfjm,x≥0,where m≥1, f defined on 0,∞, and
(2)Wαj,mx=mxmx+jαj−1e−mx+jαj!,0≤α<1,∑j=0∞Wαj,mx=1.If we putα=0 in (1), then it becomes Szász-Mirakyan-type.In 1995, Lupaş [2] introduced a sequence of linear positive operators; later on in 1999, it was modified by Agratini [3] as follows:
(3)Lmf;x=2−mx∑j=0∞mxjj!2jfjm,x≥0,and also discussed the Kantorovich and Durrmeyer variant of operator (1).In 2018, Tunca et al. [4] modified operator (3) in such a way that in the construction, authors take the negative subscript -1 of the Pochhammer symbol into consideration; due to this, the calculations become simpler in a remarkable degree just as
(4)Lmαf;x=2−mx+jα∑j=0∞mxmx+1+jαj−1j!2jfjm,x≥0.In order to approximate Lebesgue integrable functions, the most important modifications are Kantorovich and Durrmeyer integral operators. The Durrmeyer variant of operator (1) is introduced by Tarabie [5] and Mishra and Patel [6] with some beta basis functions.In2011, Cárdenas-Morales et al. [7] defined the Bernstein-type operators by Bmfoσ−1oσ and also presents a better degree of approximation depending on σ. This type of approximation operators generalizes the Korovkin set from e0,e1,e2 to e0,σ,σ2. The Durrmeyer variant of Bmfoσ−1oσ is defined in [8]. In 2014, Aral et al. [9] defined a similar modification of Szász-Mirakyan-type operators by using a suitable function σ.Motivated by the above mentioned work very recently, Bodur [10] introduced a new modification of operator (4) by using a suitable function σ, which satisfies the following properties:σ1σ be a continuously differentiable function on 0,∞σ2σ0=0 and infx∈0,∞σ′x≥1The new formulated operators are defined as(5)Lm,σαf;x=2−mσx+jα∑j=0∞mσxmσx+1+jαj−12jj!foσ−1jm,x≥0,for m≥1, x≥0, and suitable functions f defined on 0,∞.As we know, in order to approximate Lebesgue integrable functions, the most important modifications are Kantorovich and Durrmeyer integral operators. Motivated by the above mentioned Durrmeyer type generalizations of various operators and also from [11–23], in this paper, Durrmeyer-type modification of generalized Lupaş-Jain operators (5) by taking weights of some beta basis function is defined as follows:
(6)Dm,σαf;x=2−mσx+jα∑j=0∞mσxmσx+1+jαj−12jj!∫0∞bm,jϑfoσ−1ϑdϑ,where m∈ℕ and bm,jϑ is defined as
(7)bm,jϑ=1βm+1,jϑj−11+ϑm+j+1,where βm+1,j is the beta basis function and σ is a function satisfying the conditions σ1 and σ2 given above.The rest of the work is organized as follows: in the second section, moments and central moments forDm,σα are calculated. In the third section, we study convergence properties of Dm,σα in the light of weighted space. In the fourth section, we obtain the order of approximation of new constructed operators associated with the weighted modulus of continuity. In the fifth section, we shall prove Voronovskaya-type theorem in quantitative form. These kinds of results are very useful to describe the rate of point-wise convergence. Finally, in the last section, we obtain some local approximation results related to K-functional.


## 2. Basic Results

In this section, we prove some lemmas forDm,σα which are required to prove our main results.Lemma 1.
LetDm,σα be given by (6). Then for each x≥0, m∈ℕ, and 0≤α<1, we have
(i)
Dm,σα1;x=1(ii)
Dm,σασ;x=σx/1−α(iii)
Dm,σασ2;x=1/m−11−α2mσ2x+α2−2α+3/1−ασx(iv)
Dm,σασ3;x=1/m−1m−21−α3m2σ3x+6m+3m1−α2/1−ασ2x+6α+1+61−α2+21−α4/1−α2σx(v)
Dm,σασ4;x=1/m−1m−2m−31−α4m3σ4x+12m2+6m21−α2/1−ασ3x+24mα+36m+36m1−α2+11m1−α4/1−α2σ2x+26α2+68α+24+36α+361−α2+221−α4+61−α6/1−α3σxBy using beta function and Lemma 2.1 in [4], it can be proved. So we omit it. Now, from the linearity of the operators Dm,σα, we can state Lemma 2.Lemma 2.
For operatorsDm,σα, we have the following properties:
(i)
Dm,σασϑ−σx;x=α/1−ασx(ii)
Dm,σασϑ−σx2;x=m/m−11−α2−1+α/1−ασ2x+α2−2α+3/m−11−α3σx(iii)
Dm,σασϑ−σx4;x=1/m−1m−2m−31−α4m3+6mm−2m−31−α2−4m2m−31−α−4m−1m−2m−31−α3+m−1m−2m−31−α4σ4x+6α2−12α+18m−2m−31−α−24m+12m1−α2m−3+12m2+6m21−α2/1−ασ3x+24mα+36m+36m1−α2+11m1−α4/1−α2−24α+1+241−α2+81−α4m−3/1−ασ2x+26α2+68α+24+36α+361−α2+221−α4+61−α6/1−α3σx


## 3. Convergence ofDm,σα

Here, we prove the convergence ofDm,σα by using weight function. Let λx be a function satisfying the conditions σ1 and σ2 given above. Also, let λx=1+σ2x be a weight function and the weighted space is defined as follows:
(8)Bλ0,∞=f:0,∞⟶ℝ∣fx≤Mfλx,x≥0,where Mf is a constant which depends only on f, with the norm
(9)fλ=supx∈0,∞fxλx.Also, we mention some subspaces ofBλ0,∞ as
(10)Cλ0,∞=f∈Bλ0,∞:fiscontinuouson0,∞,Cλ∗0,∞=f∈Cλ0,∞:limx→∞fxλx=Mf=constant,Uλ0,∞=f∈Cλ0,∞:fxλxisuniformlycontinuouson0,∞.It is obvious thatCλ∗0,∞⊂Uλ0,∞⊂Cλ0,∞⊂Bλ0,∞.We have the following results for the weighted Korovkin-type theorems due to Gadjiev [24]Lemma 3.
[24].
The positive linear operatorsTm,m≥1 act from Cλ0,∞ to Bλ0,∞ if and only if the inequality
(11)Tmλ;x≤Mmλx,x≥0holds, where Mm>0 is a constant depending on m.Theorem 4 (see [24]).
Let the sequence of positive linear operatorsTm,m≥1 acting from Cλ0,∞ to Bλ0,∞ and satisfying
(12)limm→∞Tmσr−σrλ=0,r=0,1,2.
Then, for eachf∈Cλ∗0,∞, we have
(13)limm→∞Tmf−fλ=0.Remark 5.
Examining Lemma1 based on the famous Korovkin theorem [25], it is clear that Dm,σαm≥1 does not form an approximation process. Now, in order to obtain convergence properties, we replace the constant α by αm∈0,1 such that limm→∞αm=0.Theorem 6.
Let0≤αm<1 such that limm→∞αm=0, and also, let Dm,σα be the sequence of positive linear operators. Then, for each function f∈Cλ∗0,∞, we have
(14)limm→∞Dm,σαmf−fλ=0.Proof.
From Lemma1, we obtain
(15)Dm,σαm1;x−1λ=0,Dm,σαmσ;x−σλ≤αm1−αm,Dm,σασ2−σ2λ≤m−m−11−αm2m−11−αm2+α2−2αm+3m−11−αm3.
Hence, by Theorem4, we deduce
(16)limm→∞Dm,σαf−fλ=0.


## 4. Rate of Convergence

In this part, we would like to determine the rate of convergence forDm,σα by weighted modulus of continuity ωσf;δ which was introduced by Holhos [26] in 2008, as follows:
(17)ωσ;δ=supx,ϑ∈0,∞,σϑ−σx≤δfϑ−fxλϑ+λx,δ>0,where f∈Cλ0,∞, with the following properties:
(i)
ωσf;0=0(ii)
ωσf;λ≥0, λ≥0, for f∈Cλ0,∞(iii)
limλ→0ωσf;λ=0,foreachf∈Uλ0,∞Theorem 7 (see [26]).
LetTm:Cλ0,∞→Bλ0,∞ be a sequence of positive linear operators with
(18)Tmσ0−σ0λ0=am,Tmσ−σλ1/2=bm,Tmσ2−σ2λ=cm,Tmσ3−σ3λ3/2=dm,where the sequences am, bm, cm, and dm converge to zero as m⟶∞. Then
(19)Tmf−fλ3/2≤7+4am+2cmωσf;λm+fλam,for all f∈Cλ0,∞, where
(20)λm=2am+2bm+cm1+am+am+3bm+3cm+dm.Theorem 8.
Let0≤α<1 such that limm→∞αm=0, and also, let Dm,σα be the sequence of positive linear operators. Then for all f∈Cλ0,∞, we have
(21)Dm,σαmf−fλ3/2≤7+2m−2m−11−αm2m−11−αm2+2αm2−4αm+6m−11−αm3ωσf;δm,where
(22)δm=22αm1−αm+m−m−11−αm2m−21−αm2+αm2−2αm+3m−11−αm3+3αm1−αm+3m−3m−11−αm2m−11−αm2+3αm2−6αm+6m−11−αm3+m2−m−1m−21−αm3m−1m−21−αm3+6m2−3m1−αm2m−1m−21−αm4+21−αm4+61−αm2+6αm+1m−1m−21−αm5.Proof.
We should calculate the sequencesam, bm, cm, and dm, in order to apply Theorem 7. In light of Lemma 1, clearly, we have
(23)am=Dm,σαmσ0−σ0λ0=0,bm=Dm,σαmσ−σλ1/2≤αm1−αm,cm=Dm,σαmσ2−σ2λ≤m−m−11−αm2m−11−αm2+αm2−2αm+3m−11−αm3.
Finally,(24)dm=Dm,σαmσ3−σ3λ3/2≤m2−m−1m−21−αm3m−1m−21−αm3+6m−3m1−αm2m−1m−21−αm4+21−αm4+61−αm2+6αm+1m−1m−21−αm5.
Thus, Theorem7 is satisfied. Hence, we have the desired result.Remark 9.
Forlimλ→0ωσf;λ=0 in Theorem 8, we obtain
(25)limm→∞Dm,σαmf−fλ3/2=0,forf∈Uλ0,∞.


## 5. Pointwise Convergence ofDm,σα

In this section, we shall analyze pointwise convergence ofDm,σα by obtaining the Voronovskaya theorem in a quantitative form by using the same technique in [7].Theorem 10.
Let0≤α<1 such that limm→∞αm=0, and also, let f∈Cλ0,∞, x∈0,∞ and suppose that foσ−1′ and foσ−1′′ exist at σx. If foσ−1′′ is bounded on 0,∞, then, we have
(26)limm→∞mDm,σαf;x−fx=3σxfoσ−1″σx2.Proof.
By using Taylor expansion offoσ−1 at σx∈0,∞, we have
(27)fϑ=foσ−1σϑ=foσ−1σx+foσ−1′σxσϑ−σx+foσ−1′′σxσϑ−σx22+λxϑσϑ−σx2,where
(28)λxϑ=foσ−1′′σϑ−foσ−1′′σx2.
Therefore, (28) together with the assumption on f ensures that
(29)λxϑ≤K,forallϑ∈0,∞and is convergent to zero as ϑ→x. Now applying the operators (6) to the equality (27), we obtain
(30)Dm,σαf;x−fx=foσ−1′σxDm,σασϑ−σx;x+foσ−1′′σxDm,σασϑ−σx2;x2+Dm,σαλxϑσϑ−σx2;x.
From Lemma2, we get
(31)limm→∞mDm,σασϑ−σx;x=0,(32)limm→∞mDm,σασϑ−σx2;x≤3σx.
By estimating the equality (30), we will get the proof.
Since from (28), for every ε>0, limϑ→xλxϑ=0. Let δ>0 such that λxϑ<ε for every ϑ≥0. By Cauchy-Schwartz inequality, we get
(33)limm→∞mDm,σαλxϑσϑ−σx2;x≤εlimm→∞mDm,σασϑ−σx2;x+Kδ2limm→∞Dm,σασϑ−σx4;x.
Since(34)limm→∞mDm,σασϑ−σx4;x=0,we obtain
(35)limm→∞mDm,σα∣λxϑ∣σϑ−σx2;x=0.
Thus, by taking into account equations (31), (32), and (35) to equation (30), the proof is completed.Remark 11.
If we chooseαn=1/m+1a with a>1, then, one can easily see that limm→∞αm=0 and mlimm→∞αm=0.


## 6. Local Approximation

In this section, for the operatorsDm,σα, we shall present local approximation theorems. Let CB0,∞ denote the space of real-valued continuous and bounded functions f defined on the interval 0,∞. The norm ∥·∥ on the space CB0,∞ is defined by
(36)f=sup0≤x<∞fx.K-functional is defined as
(37)K2f,δ=infs∈W2f−s+δf′′,where δ>0 and W2=s∈CB0,∞:s′,s′′∈CB[0,∞). By Devore and Lorentz ([27], p. 177, Theorem 6.4), there exists an absolute constant C>0 such that
(38)Kf,δ≤Cω2f,δ.The second order modulus of smoothness is as follows:(39)ω2f,δ=sup0<h≤δsupx∈0,∞fx+2h−2fx+h+fx,where f∈CB0,∞. The usual modulus of continuity of f∈CB0,∞ is defined by
(40)ωf,δ=sup0<h≤δsupx∈0,∞fx+h−fx.Theorem 12.
Let0≤α<1 such that limm→∞αm=0 and for all f∈CB0,∞. Also, let σ be a function satisfying the conditions σ1 and σ2, and σ′′ is finite. Then, there exists an absolute constant C>0 such that
(41)Dm,σαf;x−fx≤CKf,δmx,where
(42)δmx=mm−11−α2−1+α1−ασ2x+α2−2α+3m−11−α3σx.Proof.
Lets∈W2 and x,ϑ∈0,∞. By Taylor’s formula, we have
(43)sϑ=sx+soσ−1′σxσϑ−σx+∫σxσϑσϑ−vsoσ−1′′vdv.
By using the equality,(44)soσ−1′′σx=s′′xσ′x2−s′′xσ′′xσ′x3.
Now, puttingv=σx in the last term in equality (43), we get
(45)∫σxσϑσϑ−vsoσ−1′′vdv=∫xϑσϑ−σxs′′xσ′x−s′xσ′′vσ′x2dx=∫σxσϑσϑ−vs′′σ−1vσ′σ−1v2dv−∫σxσϑσϑ−vs′σ−1vσ′′σ−1vσ′σ−1v3dv.
By applying operator (6) to the both sides of equality (43) and from Lemma 1, we deduce
(46)Dm,σαs;x=sx+Dm,σα∫σxσϑσϑ−vs′′σ−1vσ′σ−1v2dv;x−Dm,σα∫σxσϑσϑ−vs′σ−1vσ′′σ−1vσ′σ−1v3dv;x.
As we know,σ is strictly increasing on 0,∞, and with condition (σ2), we get
(47)Dm,σαs;x−sx≤Mm,2σxs′′+s′σ′′,where
(48)Mm,2σx=Dm,σασϑ−σx2;x.
Also,(49)∣Dαm,σ∣≤f.
Hence, we have(50)Dm,σαf;x−fx≤Dm,σαf−s;x+Dm,σαs;x−sx+sx−fx≤2∥f−s∥+mm−11−α2−1+α1−ασ2x+α2−2α+3m−11−α3σxs′′+s′σ′′,if C=max2,σ′′; then
(51)Dm,σαf;x−fx≤C2∥f−s∥+mm−11−α2−1+α1−ασ2x+α2−2α+3m−11−α3σx∥s′′∥W2.
Taking infimum over alls∈W2, we obtain
(52)Dm,σαf;x−fx≤CKf,δmx.
Theorem12 is proved.



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