# Approximation Properties ofp,q-Szász-Mirakjan-Durrmeyer Operators

Authors: Zhi-Peng Lin; Wen-Tao Cheng; Xiao-Wei Xu
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In this article, we introduce a new Durrmeyer-type generalization ofp,q-Szász-Mirakjan operators using the p,q-gamma function of the second kind. The moments and central moments are obtained. Then, the Voronovskaja-type asymptotic formula is investigated and point-wise estimates of these operators are studied. Also, some local approximation properties of these operators are investigated by means of modulus of continuity and Peetre K-functional. Finally, the rate of convergence and weighted approximation of these operators are presented.


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## Body

## 1. Introduction

In recent years,p,q-analogues of well-known positive operators have been widely constructed and researched since Mursaleen et al. first introduced p,q-Berstein operators [1] and p,q-Bernstein-Stancu operators [2]. In [3], Acar first introduced p,q-Szász-Mirakjan operators and gave a recurrence relation for the moments of these operators. In [4], Mursaleen et al. proposed a Kantorovich variant of the p,q-analogue of Szász-Mirakjan operators under the nondecreasing condition. In [5], Sharma and Gupta introduced p,q-Szász-Mirakjan-Kantorovich operators and studied their approximation properties. In [6], Acar et al. constructed King’s type p,q-Szász-Mirakjan operators preserving x2 and discussed the order of approximation and weighted approximation properties of these operators. In [7], Aral and Gupta constructed p,q-Szász-Mirakjan-Durrmeyer operators using p,q-gamma function of the first kind and estimated moments and established some direct results of these operators. In [8], Mursaleen et al. introduced a new modification of Szász-Mirakjan operators based on the p,q-calculus and investigated their approximation properties including weighted approximation and Voronovskaya-type theorem. In [9], Mursaleen et al. proposed two different Kantorovich-type p,q-Szász-Mirakjan operators and discussed their error estimated. In [10], Kara and Mahmudov constructed a new p,q-Szász-Mirakjan operators asDefinition 1.
Let0<q<p≤1 and m∈ℕ. For g:0,∞⟶ℝ, p,q-Szász-Mirakjan operators can be defined as
(1)Smp,qg;t=∑k=0∞sm,kp,qtgpm−kkp,qmp,q,where sm,kp,qt=pkk−m/qkk−1kk−1/2mp,qktk/kp,q!ep,q−mp,qpk−m+1q−kt.Meantime, quantitative estimates for the convergence in the polynomial weighted spaces and Voronovskaya theorem for newp,q-Szász-Mirakjan operators (1) were given. All these achievement motivates us to construct the Durrmeyer analogue of the p,q-Szász-Mirakjan operators defined by (1).Definition 2.
Let0<q<p≤1 and m∈ℕ. For g:0,∞⟶ℝ, we construct the p,q-Szász-Mirakjan-Durrmeyer operators as
(2)Dmp,qg;t=mp,q∑k=0∞sm,kp,qt∫0∞q−kpk−msm,kp,qugudp,qu.
Let us recall the basic notations ofp,q-calculus which can be found in [11]. For any fixed real number p>0 and q>0, the p,q-integers mp,q are defined as
(3)mp,q=pm−1+pm−2q+pm−3q2+⋯+pqm−2+qm−1=pm−qmp−q,p≠q≠1,mpm−1,p=q≠1,mq,p=1,m,p=q=1,
wheremq denotes the q-integers and m=0,1,2,⋯. Also p,q-factorial mp,q! is defined as follows:
(4)mp,q!=1p,q2p,q⋯mp,q,m≥1,1,m=0.
Now, we introduce two types ofp,q-analogues of exponential function ep,qt and Ep,qt (see [7]):
(5)ep,qt=∑m=0∞pmm−1/2tmmp,q!,t∈ℝ,p<1andq<1,Ep,qt=∑m=0∞qmm−1/2tmmp,q!,t∈ℝ,p<1andq<1.
Letg be an arbitrary function. The improper p,q-integral of gt on 0,∞ was defined as (see [12])
(6)∫0∞gtdp,qt=p−q∑i=−∞∞qipi+1gqipi+1,0<qp<1.
Thep,q-gamma function of the second kind was defined in [12] as follows:
(7)γp,qz=∫0∞qzz−1/2tz−1ep,q−ptdp,qt,Rz>0.
Meantime, thep,q-gamma function fulfills the following relation:
(8)γp,qz+1=zp,qγp,qz,moreover, for any nonnegative integer m>0, the following relation holds:
(9)γp,qm+1=mp,q!.


## 2. Auxiliary Results

In order to discuss the approximation properties of the operatorsDmp,qg;t, we need the following lemmas.Lemma 3 (see ([10], Lemma 4)).
For0<q<p≤1, m∈ℕ, and t∈0,∞, we have
(10)Smp,q1;t=1,Smp,qu;t=t,Smp,qu2;t=t2+pm−1mp,qt,Smp,qu3;t=t3+2p+qpm−2mp,qt2+p2m−2mp,q2t,Smp,qu4;t=t4+3p2+2qp+q2pm−3mp,qt3+3p2+3qp+q2p2m−4mp,q2t2+p3m−3mp,q3t.Lemma 4.
For0<q<p≤1, r=p/q, m∈ℕ, and t∈0,∞, we have
(11)Dmp,q1;t=1,Dmp,qu;t=t+pmq−1mp,q,Dmp,qu2;t=t2+pm2p,q2p−1q−2mp,qt+p2mq−32p,qmp,q2,Dmp,qu3;t=t3+q−1g1r+2p−1+p−2qpmmp,qt2+p−2+g2rq−2+g1rp−1q−1p2mmp,q2t+p3mq−3g3rmp,q3,Dmp,qu4;t=t4+3p−1+2qp−2+q2p−3+q−1h1rpmmp,qt3+3+2r+1h1rp−2+3p−3q+p−4q2+q−2h2rp2mmp,q2t2+p−3+p−2q−1h1r+p−1q−2h2r+q−3h3rp3mmp,q3t+q−4h4rp4mmp,q4,where g1r=∑i=13ir, g2r=∑1≤i<j≤3irjr, g3r=2r3r, h1r=∑i=14ir, h2r=∑1≤i<j≤4irjr, h3r=∑1≤i<j<k≤4irjrkr, and h4r=2r3r4r.Proof.
Using thep,q-gamma function of the second kind, we can obtain
(12)Dmp,q1;t=mp,q∑k=0∞sm,kp,qt∫0∞q−kpk−msm,kp,qudp,qu=mp,q∑k=0∞sm,kp,qt∫0∞q−kk+1/2pk−mk+1mp,qukkp,q!ep,q−mp,qpk−m+1q−kudp,qu=mp,q∑k=0∞sm,kp,qt∫0∞q−kk+1/2pk−mk+1mp,qukkp,q!ep,q−pmp,qpk−mq−kudp,qu=∑k=0∞sm,kp,qt∫0∞qkk+1/2mp,qpk−mq−kukkp,q!ep,q−pmp,qpk−mq−kudp,qmp,qpk−mq−ku=∑k=0∞sm,kp,qtγp,qk+1kp,q!=∑k=0∞sm,kp,qt=1.
and next usingk+1p,q=pk+qkp,q, we have
(13)Dmp,qu;t=mp,q∑k=0∞sm,kp,qt∫0∞q−kk+1/2pk−mk+1mp,qukkp,q!ep,q−mp,qpk−m+1q−kuudp,qu=∑k=0∞sm,kp,qtpm−kmp,qkp,q!∫0∞q−kk+1/2+kk+2mp,qpk−mq−kuk+1ep,q−pmp,qpk−mq−ku×dp,qmp,qpk−mq−ku=∑k=0∞sm,kp,qtpm−kq−1mp,qkp,q!∫0∞qk+1k+2/2sk+1ep,q−psdp,qs=∑k=0∞sm,kp,qtpm−kq−1mp,qkp,q!γp,qk+2=∑k=0∞sm,kp,qtpm−kq−1k+1p,qmp,q=∑k=0∞sm,kp,qtpm−kq−1pk+qkp,qmp,q=∑k=0∞sm,kp,qtpm−kkp,qmp,q+∑k=0∞sm,kp,qtpmq−1mp,q=Smp,qu;t+Smp,q1;t=t+pmq−1mp,q.
Using the equalityk+1p,qk+2p,q=q3kp,q2+q2q+ppkkp,q+2p,qp2k and Lemma 3, we have
(14)Dmp,qu2;t=mp,q∑k=0∞sm,kp,qt∫0∞q−kpk−msm,kp,quu2dp,qu=mp,q∑k=0∞sm,kp,qt∫0∞q−kk+1/2pk−mk+1mp,qukkp,q!ep,q−mp,qpk−m+1q−kuu2dp,qu=∑k=0∞sm,kp,qtp2m−2kmp,q2kp,q!∫0∞q−kk+1/2+kk+3mp,qpk−mq−kuk+2ep,q−pmp,qpk−mq−ku×dp,qmp,qpk−mq−ku=∑k=0∞sm,kp,qtp2m−2kq−3mp,q2kp,q!∫0∞qk+2k+3/2sk+2ep,q−psdp,qs=∑k=0∞sm,kp,qtp2m−2kq−3mp,q2kp,q!γp,qk+3=∑k=0∞sm,kp,qtp2m−2kq−3k+1p,qk+2p,qmp,q2=∑k=0∞sm,kp,qtpm−kkp,qmp,q2+pmq−2p+2qmp,q∑k=0∞sm,kp,qtpm−kkp,qmp,q+p2mq−32p,qmp,q2∑k=0∞sm,kp,qt=Smp,qu2;t+pmq−2p+2qmp,qSmp,qu;t+p2mq−32p,qmp,q2Smp,q1;t=t2+p−1+2q−1+pq−2pmmp,qt+p2mq−32p,qmp,q2=t2+pm2p,q2p−1q−2mp,qt+p2mq−32p,qmp,q2.
Usingmp,q=qm−1mr, we have
(15)k+1p,qk+2p,qk+3p,q=qkp,q+pkq2kp,q+2p,qpkq3kp,q+3p,qpk=q6kp,q3+q33p,q+q42p,q+q5kp,q2pk+q2p,q3p,q+q23p,q+q32p,qkp,qp2k+2p,q3p,qp3k=q6kp,q3+q53r+2r+1kp,q2pk+q43r2r+3r+2rkp,qp2k+q33r2rp3k≔q6kp,q3+q5g1rkp,q2pk+q4g2rkp,qp2k+q3g3rp3k.
Thus,(16)Dmp,qu3;t=mp,q∑k=0∞sm,kp,qt∫0∞q−kpk−msm,kp,quu3dp,qu=mp,q∑k=0∞sm,kp,qt∫0∞q−kk+1/2pk−mk+1mp,qukkp,q!ep,q−mp,qpk−m+1q−kuu3dp,qu=∑k=0∞sm,kp,qtp3m−3kmp,q3kp,q!∫0∞q−kk+1/2+kk+4mp,qpk−mq−kuk+3ep,q−pmp,qpk−mq−ku×dp,qmp,qpk−mq−ku=∑k=0∞sm,kp,qtp3m−3kq−6mp,q3kp,q!∫0∞qk+3k+4/2sk+3ep,q−psdp,qs=∑k=0∞sm,kp,qtp3m−3kq−6mp,q3kp,q!γp,qk+4=∑k=0∞sm,kp,qtp3m−3kq−6k+1p,qk+2p,qk+3p,qmp,q3=∑k=0∞sm,kp,qtpm−kkp,qmp,q3+pmq−1g1rmp,q∑k=0∞sm,kp,qtpm−kkp,qmp,q2+p2mq−2g2rmp,q2∑k=0∞sm,kp,qtpm−kkp,qmp,q+p3mq−3g3rmp,q3∑k=0∞sm,kp,qt=Smp,qu3;t+pmq−1g1rmp,qSmp,qu2;t+p2mq−2g2rmp,q2Smp,qu;t+p3mq−3g3rmp,q3Smp,q1;t=t3+q−1g1r+2p−1+p−2qpmmp,qt2+p−2+g2rq−2+g1rp−1q−1p2mmp,q2t+p3mq−3g3rmp,q3.
Similarly, using∏i=14k+ip,q=q10kp,q4+h1rq9kp,q3pk+h2rq8kp,q2p2k+h3rq7kp,qp3k+h4rq6p4k, we can obtain
(17)Dmp,qu4;t=mp,q∑k=0∞sm,kp,qt∫0∞q−kpk−msm,kp,quu4dp,qu=mp,q∑k=0∞sm,kp,qt∫0∞q−kk+1/2pk−mk+1mp,qukkp,q!ep,q−mp,qpk−m+1q−kuu4dp,qu=∑k=0∞sm,kp,qtp4m−4kmp,q3kp,q!∫0∞q−kk+1/2+kk+5mp,qpk−mq−kuk+4ep,q−pmp,qpk−mq−ku×dp,qmp,qpk−mq−ku=∑k=0∞sm,kp,qtp4m−4kq−10mp,q4kp,q!∫0∞qk+4k+5/2sk+4ep,q−psdp,qs=∑k=0∞sm,kp,qtp4m−4kq−10mp,q4kp,q!γp,qk+5=∑k=0∞sm,kp,qtp4m−4kq−10k+1p,qk+2p,qk+3p,qk+4p,qmp,q4=∑k=0∞sm,kp,qtpm−kkp,qmp,q4+pmq−1h1rmp,q∑k=0∞sm,kp,qtpm−kkp,qmp,q3+p2mq−2h2rmp,q2∑k=0∞sm,kp,qtpm−kkp,qmp,q2+p3mq−3h3rmp,q3∑k=0∞sm,kp,qtpm−kkp,qmp,q+p3mq−4h4rmp,q4∑k=0∞sm,kp,qt=Smp,qu4;t+pmq−1h1rmp,qSmp,qu3;t+p2mq−2h2rmp,q2Smp,qu2;t+p3mq−3h3rmp,q3Smp,qu;t+p3mq−4h4rmp,q4Smp,q1;t=t4+3p−1+2qp−2+q2p−3+q−1h1rpmmp,qt3+3+2r+1h1rp−2+3p−3q+p−4q2+q−2h2rp2mmp,q2t2+p−3+p−2q−1h1r+p−1q−2h2r+q−3h3rp3mmp,q3t+q−4h4rp4mmp,q4.Lemma 5.
Using Lemma4, we immediately have the following explicit formulas for the central moments:
(18)Amp,qt≔Dmp,qu−t;t=q−1pmmp,q,Bmp,qt≔Dmp,qu−t2;t=p−1+pq−2pmmp,qt+q−32p,qp2mmp,q2.Lemma 6.
The sequencespm, qm satisfy 0<qm<pm≤1 such that pm⟶1, qm⟶1 and pmm⟶β∈0,1, mpm,qm⟶∞ as m⟶∞, rm=pm/qm, then
(19)limm⟶∞mpm,qmAmpm,qmt=β,(20)limm⟶∞mpm,qmBmpm,qmt=2βt,(21)limm⟶∞mpm,qmDmpm,qmu−t4;t=0.Proof.
Applying Lemma5, we can easily obtain (19) and (20). While m⟶∞, we can rewrite
(22)Dmpm,qm1;t=1Dmpm,qmu;tt+βmpm,qm;Dmpm,qmu2;t~t2+4βmpm,qmt,Dmpm,qmu3;t~t3+9βmpm,qmt2;Dmpm,qmu4;t~t4+16βmpm,qmt3.
UsingDmpm,qmu−t4;t=Dmpm,qmu4;t−4Dmpm,qmu3;tt+6Dmpm,qmu2;tt2−4Dmpm,qmu;tt3+Dmpm,qm1;tt4 and 16−9×4+4×6−4×1=0, we can get (21).


## 3. Voronovskaja-Type Theorem

Theorem 7.
Letpm, qm be the sequences defined in Lemma 6 and g∈CB0,∞. Supposing that g′′t exists at a point t∈0,∞, then, we can obtain
(23)limm⟶∞mpm,qmDmpm,qmg;t−gt=βgt+tg′t,
whereCB0,∞ denotes the set of all real-valued bounded and continuous functions defined on 0,∞, endowed with the norm g=supt∈0,∞gt.Proof.
By the Taylor’s expansion theorem of functiong∈CB0,∞, we can obtain
(24)gu−gt=g′tu−t+12g′′tu−x2+Θu,tu−t2,
whereu,t∈0,∞, Θu,t is bounded and limu⟶tΘu,t=0. Applying the operator Dmpm,qm to the equality above, we can obtain
(25)Dmpm,qmg;t−gt=g′tAmpm,qmt+12g′′tBmpm,qmt+Dmpm,qmΘu,tu−t2;t
Sincelimu⟶tΘu,t=0, then for all ε>0, there exists δ>0 such that u−t<δ and it will imply Θu,t<ε for all fixed t∈0,∞ as m sufficiently large. While if u−t≥δ, then Θu,t≤C/δ2u−t2, where C>0 is a constant. Using Lemma 6(26)mpm,qmDmpm,qmΘu,tu−t2;t≤εmpm,qmDmpm,qmu−t2;t+Cδ2mpm,qmDmpm,qmu−t4;t⟶0m⟶∞.
The proof is completed.


## 4. Point-Wise Estimate

In this section, we establish two point-wise estimate of the operatorsDmp,qg;t. First, a function g∈C0,∞ is said to satisfy the Lipschitz condition Lipγ on D (named g∈LipCg,γγ,E), γ∈0,1, D⊂0,∞ if
(27)gu−gt≤Cg,γu−tγ,u∈0,∞andt∈D,whereCg,γ is an absolute positive constant depending only on g and γ.Theorem 8.
Let0<q<p≤1, γ∈0,1 and E be any bounded subset on 0,∞. Ifg∈CB0,∞∩LipCg,γγ,E, then, for all t∈0,∞, we have
(28)Dmp,qg;t−gt≤Cg,γBmp,qtγ/2+2dγt;E,
wheredt;E denotes the distance between t and E defined by
(29)dt;E=infu−t:u∈E.Proof.
LetE¯ be the closure of E. Using the properties of infimum, there is at least a point u0∈E¯ such that dt;E=∣t−u0∣. By the triangle inequality
(30)gu−gt≤gu−gu0+gt−gu0,we can obtain
(31)Dmp,qg;t−gt≤Dmp,qgu−gu0;t+Dmp,qgt−gu0;t≤Cg,γDmp,qu−u0γ;x+∣t−u0γ≤Cg,γDmp,qu−tγ+t−u0γ;t+∣t−u0γ=Cg,γDmp,qu−tγ;t+2∣t−u0γ.
Choosingp1=2/γ and p2=2/2−γ and using the well-known Hölder inequality, we have
(32)Dmp,qf;x−fx≤Cg,γDmp,qt−xp1γ;x1/p1Dmp,q1p2;x1/p2+2dγx;E≤Cg,γDmp,qt−x2;xγ/2+2dγx;E≤Cg,γBmp,qtγ/2+2dγx;E.
Next, we obtain the local direct estimate of the operatorsDmp,q, using the Lipcshitz-type maximal function of the order γ introduced by Lenze [13] as
(33)ω~γg;t=supt,u∈0,∞,t≠u∣gu−gt∣u−tγ,γ∈0,1.Theorem 9.
Letg∈CB0,∞ and γ∈0,1. Then, for all t∈0,∞, we have
(34)Dmp,qg;t−gt≤ω~γg;tBmp,qtγ/2.Proof.
From equation (33), we have
(35)Dmp,qg;t−gt≤ω~γg;tDmp,qu−tγ;t.
Applying the well-known Hölder inequality, we can get(36)Dmp,qg;t−gt≤ω~γg;tDmp,qu−t2;tγ/2≤ω~γg;tBmp,qtγ/2.


## 5. Local Approximation

In this section, we establish local approximation theorem forp,q-Szász-Mirakjan-Durrmeyer operators. Let us consider the following K-functional:
(37)Kg;δ=infh∈S2g−h+δh′′,where δ∈0,∞ and S2=h∈CB0,∞:h′,h″∈CB0,∞. The usual modulus of continuity and the second-order modulus of smoothness of g can be defined by
(38)ωg;δ=sup0<u≤δsupt∈0,∞gt+u−gt,ω2g;δ=sup0<u≤δsupt∈0,∞gt+2u−2gt+u+gt.By ([14], p.177, Theorem 2.4), there exists an absolute constant C>0 such
(39)Kg;δ2≤Cω2g;δ,δ>0.Theorem 10.
Letpm, qm be the sequences defined in Lemma 6 and g∈CB0,∞. Then, for all m∈ℕ, there exists an absolute positive C1=4C such that
(40)Dmpm,qmg;t−gt≤C1ω2g;Ampm,qmt2+Bmpm,qmt+ωg;Ampm,qmt.Proof.
For a given functiong∈CB0,∞, let us define the following new operators:
(41)Dmpm,qmg;t=Dmpm,qmg;t−gAmpm,qmt+t+gt,g∈0,∞.
By Lemma4 and Lemma 5, we obtain
(42)Dmpm,qm1;t=1andDmpm,qmu;t=t.
Lett,u∈0,∞ and h∈S2. By Taylor’s expansion formula, we get
(43)hu=ht+h′tu−t+∫tuh′′vu−vdv.
ApplyingDmpm,qm to the above equality, we can write
(44)Dmpm,qmh;t−ht=Dmpm,qm∫tuh′′vu−vdv;t≤Dmpm,qm∫tuu−vh′′vdv;t+∫tAmpm,qmt+th′′uAmpm,qmt+t−vdv≤Dmpm,qmu−t2;th′′+Ampm,qmt2h′′=Ampm,qmt+Bmpm,qmt2h′′.
On the other hand, since∣Dmpm,qmg;t∣≤3∥g∥. Hence,
(45)Dmpm,qmg;t−gt=Dmpm,qmg;t+gAmpm,qmt+t−2gt≤Dmpm,qmg−h;t−g−ht+Dmpm,qmh;x−hx+gAmpm,qmx+x−gx≤4g−h+Ampm,qmt+Bmpm,qmt2+h′′+ωg;Ampm,qmt.
Taking infinum on the right hand side over allh∈S2 and using (39), we obtain the desired assertion.Corollary 11.
Letpm, qm be the sequences defined in Lemma 6 and g∈CB0,∞. Then, for any finite interval I⊂0,∞, the sequence Dmpm,qmg;t converges to g uniformly on I.


## 6. Rate of Convergence

Let(46)B20,∞=g:gt≤Mg1+t2,C20,∞=g:g∈B20,∞∩C0,∞,C200,∞=g:g∈C20,∞andlimt→∞gt1+t2<∞,where Mg is an absolute constant depending only on g. C200,∞ is equipped with the norm g2=supt∈0,∞gt/1+t2. As is known, if f∈C0,∞ is not uniform, we cannot obtain limδ⟶0+ωg;δ=0. In [15], Ispir defined the following weighted modulus of continuity
(47)Ωg;δ=sup0<u≤δ,t∈0,∞gu+t−gt1+u21+t2,and proved the properties of monotone increasing aboutΩg;δ as δ>0, limδ⟶0+Ωg;δ=0 and the inequality
(48)Ωg;λδ≤21+λ1+δ2Ωg;δ,whileλ>0 and g∈C200,∞. Meantime, we recall the modulus of continuity of g on the interval 0,κ⊂0,∞ by
(49)ωκg;δ=supu,t∈0,κ,u−t≤δgu−gt,δ>0.Theorem 12.
Letg∈C20,∞, 0<q<p≤1, and κ>0, we have
(50)Dmp,qg;t−gC0,κ≤4Mg3+2κ2Bmp,qκ+2ωκ+1f;Bmp,qκ.Proof.
For anyt∈0,κ and u>κ+1, we easily have 1≤u−κ2≤u−t2, thus
(51)gu−gt≤gu+gt≤Mg2+u2+t2=Mg2+t2+u−t+t2≤Mg2+2t2+u−t2≤Mg3+2t2u−t2≤Mg3+2κ2u−t2,
and for anyt∈0,κ, u∈0,κ+1 and δ>0, we have
(52)gu−gt≤ωκ+1∣u−t∣;t≤1+∣u−t∣δωκ+1g;δ.
For (51) and (52), we can get
(53)gu−gt≤Mg3+2κ2u−t2+1+∣u−t∣δωκ+1g;δ.
By Schwarz’s inequality, for anyt∈0,κ, we can get
(54)Dmp,qg;t−gt≤Dmp,qgu−gt;t≤Mg3+2κ2Dmp,qu−t2;t+Dmp,q1+u−tδ;tωκ+1g;δ≤Mg3+2κ2Dmp,qu−t2;t+ωκ+1g;δ1+1δDmp,qu−t2;t≤Mg3+2κ2Bmp,qt+ωκ+1f;δ1+1δBmp,qt≤Mg3+2κ2Bmp,qκ+ωκ+1g;δ1+1δBmp,qκ.
By takingδ=Bmp,qκ and supremum over all t∈0,κ, we accomplish the proof of Theorem 12.


## 7. Weighted Approximation

In this section, we will discuss the following three theorems about weighted approximation for the operatorsDmpm,qmg;t.Theorem 13.
Letf∈C200,∞ and the sequences pm, qm be the sequences defined in Lemma 6; then, for any t∈0,∞, there exists N∈ℕ+ such that for all m>N, the inequality
(55)Dmpm,qmg;t−gt1+t22≤10Ωf;1mpm,qmholds.Proof.
Using (47) and (48), we can write
(56)gu−gt≤1+u−t21+t2Ωf;u−t≤21+u−tδ1+δ2Ωf;δ1+u−t21+t2≤41+δ221+t2Ωf;δ,u−t≤δ,41+δ21+t2Ωf;δ∣u−t∣+u−t3δ,u−t>δ.
For anyδ∈0,1/2 and t,u∈0,∞, (56) can be rewritten
(57)gt−gx≤51+t2Ωf;δ54+∣u−t∣+u−t3δ.
Using (20) and (21), there exists N∈ℕ+ such that for any m>N,
(58)Dmpm,qmu−t2;t≤2mpm,qmt,Dmpm,qmu−t4;t≤t32.
By Schwarz’s inequality, we can obtain(59)Dmpm,qmu−t;t≤Dmpm,qmu−t2;t≤2tmpm,qm,(60)Dmpm,qmu−t3;t≤Dmpm,qmu−t2;tDmpm,qmu−t4;t≤t2mpm,qm.
SinceDmpm,qm is linear and positive, using (58)–(60), we can obtain
(61)Dmpm,qmg;t−gt≤51+t2Ωg;δ54+Dmpm,qm∣u−t∣+u−t3;tδ≤51+t254+2t+t2δmpm,qmΩf;δ.
Choosingδ=1/mpm,qm, we have
(62)Dmpm,qmg;t−gt≤51+t254+2t+t2Ωf;1mpm,qm≤101+t22Ωf;1mpm,qm.the conclusion holds.Theorem 14.
Letpm, qm be the sequences defined in Lemma 6. Then, for any g∈C200,∞, we have
(63)limm⟶∞Dmpm,qmg;t−g2=0.Proof.
By weighted Korovkin theorem in [16], we see that it is sufficient to verify the following three conditions:
(64)limm⟶∞Dmpm,qmuk;t−tk2=0,k=0,1,2.
SinceDmpm,qm1;x=1, then (64) holds true for k=0. By Lemma 4, we can obtain
(65)Dmpm,qmu;t−t2=supt∈0,∞Dmpm,qmu;t−t1+t2≤pmmqm−1mpm,qmsupt∈0,∞t1+x2=pmmqm−1mpm,qm⟶0,m⟶∞,Dmpm,qmu2;t−t22=supt∈0,∞Dmpm,qmu2;t−t21+t2≤supt∈0,∞t1+t2pmm2pm,qm2pm−1qm−2mpm,qm+supt∈0,∞t1+t2pm2mqm−32pm,qmmpm,qm2=pmm2pm,qm2pm−1qm−2mpm,qm+pm2mqm−32pm,qmmpm,qm2⟶0,n⟶∞.
Thus, the proof of Theorem14 is completed.Theorem 15.
Letpm, qm be the sequences defined in Lemma 6. Then, for any g∈C200,∞ and λ>0, we have
(66)limm⟶∞supt∈0,∞Dmpm,qmg;t−gt1+t21+λ=0.Proof.
Lett0∈0,∞ be arbitrary but fixed. Then,
(67)supt∈0,∞Dmpm,qmg;t−gt1+t21+λ≤supt∈0,t0Dmpm,qmg;t−gt1+t21+λ+supt∈t0,∞Dmpm,qmg;t−gt1+t21+λ≤Dmpm,qmg;t−gC0,t0+g2supt∈t0,∞Dmpm,qm1+u2;t1+t21+λ+supt∈t0,∞gt1+t21+λ.
Sincegt≤Mg1+t2, we have supt∈t0,∞gt/1+t21+λ≤Mgg2/1+t02λ. Let ε>0 be arbitrary, we can choose t0 to be so large that
(68)Mgg21+t02λ<ε3.
In view of Lemma4, while t∈t0,∞, we can obtain
(69)f2limt⟶∞limm⟶∞Dmpm,qm1+u2;x1+t21+λ=f2limt⟶∞11+t2λ=0.
Hence, we can chooseN and t0 to be so large such that for any m>N the inequality
(70)supt∈t0,∞g2Dmpm,qm1+u2;t1+t21+λ<ε3.
holds. Also, the first term of the above inequality tends to zero by Theorem12, that is
(71)Dmpm,qmg;t−gC0,t0<ε3.
Thus, combining (68)–(71), we obtain the desired result.



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*Source: 6649570-2021-03-11.xml*