# On Stancu-Type Generalization of Modifiedp,q-Szász-Mirakjan-Kantorovich Operators

Authors: Yong-Mo Hu; Wen-Tao Cheng; Chun-Yan Gui; Wen-Hui Zhang
Journal: Journal of Function Spaces
 (2021)

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## Abstract

In the present article, we constructp,q-Szász-Mirakjan-Kantorovich-Stancu operators with three parameters λ,α,β. First, the moments and central moments are estimated. Then, local approximation properties of these operators are established via K-functionals and Steklov mean in means of modulus of continuity. Also, a Voronovskaja-type theorem is presented. Finally, the pointwise estimates, rate of convergence, and weighted approximation of these operators are studied.


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## Body

## 1. Introduction

During this decades, the applications ofp,q-calculus transpired as a new area in the field of operator approximation theory. Many researchers constructed and discussed many positive linear operators based on p,q-integers, p,q-exponential functions, p,q-Gamma functions [1], p,q-Beta functions, and so on. Since Mursaleen et al. first constructed p,q-Bernstein operators [2] and p,q-Bernstein-Stancu operators [3], several generalizations of well-known positive linear operators based on p,q-calculus have been introduced and studied (see [4–11]). In [12], Acar first proposed p,q-Szász-Mirakjan operators defined on 0,∞. In [13], Kara et al. constructed a modified p,q-Szász-Mirakjan as follows:
(1)Snp,qf;t=∑k=0∞sn,kp,qtfpn−kkp,qnp,q,t∈0,∞,where 0<q<p≤1, f∈C0,∞ and sn,kp,qt=pkk−n/qkk−1/2np,qktk/kp,q!ep,q−np,qpk−n+1q−kt. Certain basic notations of p,q-calculus are mentioned below (for details see [14]): For each real number λ, p,q-analogue of λ named λp,q is defined by
(2)λp,q=pλ−qλp−q,p≠q.And for each nonnegative integern, the p,q-integer np,q and p,q-factorial np,q! are defined by
(3)np,q=pn−1+pn−2q+pn−3q2+⋯+pqn−2+qn−1=pn−qnn−n,p≠q;npn−1,p=q;nq,p=1;n,p=q=1,np,q!=1p,q2p,q⋯np,q,n≥1;1,n=0.Thep,q-analogue of the exponential function is defined by
(4)ep,qt=∑n=0∞pnn−1/2tnnp,q!.Letf be an arbitrary function and a∈ℝ. The p,q-Jackson integral [15] was defined by
(5)∫0afudp,qu=p−qa∑i=0∞qipi+1fqipi+1,0<q<p≤1.And thep,q-Jackson integral over an interval a,ba<b can be defined by
(6)∫abfudp,qu=∫0bfudp,qu−∫0afudp,qu.We easily know thatp,q-Jackson integral (6) is not positive unless it is assumed that f is a nondecreasing function. To solve this problem, Acar et al. [16] defined the p,q-integral of the arbitrary function f on interval a,ba<b as follows:
(7)∫abfudp,qu=p−qb−a∑n=0∞qnpn+1fa+b−aqnpn+1,0<q<p≤1.It is obvious that integral (6) and integral (7) of f on 0,1 are equivalence.The Kantorovich modification of positive linear operators on0,∞ is a method to approximate the Riemann integrable functions. The idea behind the Kantorovich modifications mainly depends on replacing the sample value fk/n by n∫k+1/nk/nfudu (see [17, 18]). By definite integral substitution, we have n∫k+1/nk/nfudu=∫01fk+u/ndu. However, two Kantorovich modifications may be not equivalence or cannot use definite integral substitution in q-calculus and p,q-calculus. For the researches about p,q-Szász-Mirakjan-Kantorovich-operators, we can see [19–21]. Meantime, the idea behind the Stancu modifications mainly depends on replacing the sample value fk/n by fk+α/n+β with two parameters 0≤α≤β (see [22]). For the researches about the Stancu modification of p,q-operators, we can see [23, 24]. All these achievements motivate us to construct the Stancu and Kantorovich generalizations of p,q-Szász-Mirakjan (1) with three parameters λ,α,β as follows:Definition 1.
Forn∈ℕ, 0<q<p≤1, λ>0, 0≤α≤β and f∈C0,∞, the p,q-Szász-Mirakjan-Kantorovich-Stancu operators can be defined by
(8)Sn,α,βp,q,λf;t=∑k=0∞sn,kp,qt∫01fpn−kkp,q+uλ+αnp,q+βdp,qu,t∈0,∞.


## 2. Auxiliary Results

In order to obtain the approximation properties of the operatorsSn,α,βp,q,λf;t, we need the following lemmas and corollaries.Lemma 2.
Fort∈0,∞, 0<q<p≤1, λ>0, we have ∫01tλdp,qt=1/λ+1p,q.Proof.
Using (7),
(9)∫01tλdp,qt=p−q∑n=0∞qnpn+1λ+1=p−qpλ+1∑n=0∞qλ+1pλ+1n=1λ+1p,q.Lemma 3.
([13], Lemma 4) For 0<q<p≤1, n∈ℕ, and t∈0,∞, we have
(10)Snp,q1;t=1,Snp,qu;t=t,Snp,qu2;t=t2+pn−1np,qt,Snp,qu3;t=t3+2p+qpn−2np,qt2+p2n−2np,q2t,Snp,qu4;t=t4+3p2+2qp+q2pn−3np,qt3+3p2+3qp+q2p2n−4np,q2t2+p3n−3np,q3t.The following lemma will tell us the relation between the moment of the operatorsSnp,q and the moment of the operators Sn,α,βp,q,λ:Lemma 4.
Fort∈0,∞, n,m∈ℕ, 0<q<p≤1, λ>0, 0≤α≤β, we have the following recursive relation:
(11)Sn,α,βp,q,λum;t=1np,q+βm∑i=0m∑j=0m−im!i!j!m−i−j!np,qiSnp,qui;tαm−i−jλj+1p,q.Proof.
By direct computation, we have(12)Sn,α,βp,q,λum;t=∑k=0∞sn,kp,qt∫01pn−kkp,q+uλ+αnp,q+βmdp,qu=1np,q+βm∑k=0∞∫01∑i=0m∑j=0m−im!i!j!m−i−j!pn−kkp,qiuλjαm−i−jdp,qu=1np,q+βm∑i=0m∑j=0m−inp,qi∑k=0∞sn,kp,qtpn−kkp,qnp,qiαm−i−jλj+1p,q=1np,q+βm∑i=0m∑j=0m−im!i!j!m−i−j!np,qiSnp,qui;tαm−i−jλj+1p,q.
Hence, the proof of Lemma4 is completed.Then, the following lemma can be obtain immediately:Lemma 5.
Fort∈0,∞, 0<q<p≤1, λ>0, 0≤α≤β, we have(13)Sn,α,βp,q,λ1;t=1,Sn,α,βp,q,λu;t=np,qnp,q+βt+1np,q+β1λ+1p,q+α,Sn,α,βp,q,λu2;t=np,q2np,q+β2t2+pn−1np,qt+2np,qnp,q+β21λ+1p,q+αt+1np,q+β212λ+1p,q+2αλ+1p,q+α2.Lemma 6.
Under the condition of Lemma5, we can easily obtain the following formulas for the first and second central moments:
(14)An,α,βp,q,λt≔Sn,α,βp,q,λu−t;t=1np,q+β1λ+1p,q+α−βt,Bn,α,βp,q,λt≔Sn,α,βp,q,λu−t2;t=pn−1np,qtnp,q+β2+1np,q+β2βt−α−1λ+1p,q2+12λ+1p,q−1λ+1p,q2.Lemma 7.
The sequencespn, qn satisfy 0<qn<pn≤1, such that qn⟶1, pnn⟶η∈0,1, npn,qn⟶∞ as n⟶∞; then for any t∈0,∞, 0<q<p≤1, λ>0, 0≤α≤β, we have
(15)limn→∞npn,qnAn,α,βpn,qn,λt=−βt+α+1λ+1,(16)limn→∞npn,qnBn,α,βpn,qn,λt=ηt,(17)limn→∞npn,qnSn,α,βpn,qn,λu−t4;t=0.Proof.
Byλ+1pn,qn=λ+1ξnλ,ξn∈qn,pn, we have limn→∞λ+1pn,qn=λ+1. Thus, we easily obtain (15) and (16). As n⟶∞, we can rewrite
(18)Snpn,qnu3;t=t3+2+qnpn−1pnn−1npn,qnt2+o1npn,qn,Snpn,qnu4;t=t4+3+2qnpn−1+qn2pn−2pnn−1npn,qnt3+o1npn,qn.
SetAn=1/λ+1pn,qn+α. Applying Lemma 4 and npn,qn/npn,qn+βi=1−iβ/npn,qn+β+o1/npn,qn, i=1,2,3,4, we can also rewrite
(19)Sn,α,βpn,qn,λu;t=npn,qnnpn,qn+βt+Annpn,qn+β=1−βnpn,qn+βt+Annpn,qn+β+o1npn,qn,Sn,α,βpn,qn,λu2;t=npn,qn2npn,qn+β2t2+pn−1npn,qnt+2npn,qnAnnpn,qn+β2t=1−2βnpn,qn+βt2+pn−1npn,qnt+2Annpn,qn+βt+o1npn,qn,Sn,α,βpn,qn,λu3;t=npn,qn3npn,qn+β3t3+2+qnpn−1pnn−1npn,qnt2+3npn,qn2Annpn,qn+β3t2+o1npn,qn=1−3βnpn,qn+βt3+2+qnpn−1pn−1npn,qnt2+3Annpn,qn+βt2+o1npn,qn,Sn,α,βpn,qn,λu4;t=npn,qn4npn,qn+β4t4+3+2qnpn−1+qn2pn−2pnn−1npn,qnt3+4npn,qn3Annpn,qn+β4t3+o1npn,qn=1−4βnpn,qn+βt4+3+2qnpn−1+qn2pn−2pn−1npn,qnt3+4Annpn,qn+βt3+o1npn,qn.
CombiningSn,α,βpn,qn,λu−t4;t=∑m=04m4−1mSn,α,βpn,qn,λu4−m;ttm, we can obtain
(20)npn,qnSn,α,βpn,qn,λu−t4;t=1−2qnpn−1+qn2pn−2pnn−1t3+o1⟶0,asn⟶∞,we obtain the required result.Lemma 8.
LetCB0,∞ be the set of real-valued continuous bounded functions defined on 0,∞ endowed with the norm ∥f∥=supx∈0,∞∣fx∣. Under the condition of Lemma 5, for any f∈CB0,∞, we have
(21)Sn,α,βp,q,λf;t≤f.Proof.
In view of (8) and Lemma 5, the proof of this lemma can be obtained easily.


## 3. Local Approximation

In this section, we will establish local approximation theorem for the operators. For anyf∈CB0,∞, we consider the following K-functional:
(22)Kf;δ=infh∈W2∥f−h∥+δ∥h′′∥,where δ∈0,∞ and w2=h∈CB0,∞:h′,h″∈CB0,∞. The usual modulus of continuity and the second-order modulus of smoothness of f can be defined as
(23)ωf;δ=sup0<u<δsupx∈0,∞ft+u−ft,ω2f;δ=sup0<u<δsupx∈0,∞ft+2u−ft+u+ft.By ([25], p.177, Theorem 2.4), there exists an absolute positive constant C such that
(24)Kf;δ≤Cω2f;δ,δ>0.In the meantime, forf∈CB0,∞ and h>0, the Steklov mean is defined as
(25)fht=4h2∫0h/2∫0h/22ft+u+v−ft+2u+vdudv.Thus,fh∈CB0,∞, and we can write
(26)fht−ft=4h2∫0h/2∫0h/22ft+u+v−ft+2u+v−ftdudv.It is obvious that∣fht−ft∣≤ω2f;h and ∥fh−f∥≤ω2f;h. If f is continuous, then fh′,fh′′∈CB0,∞ and
(27)fh′t=4h22∫0h/2ft+u+h2−ft+udu−12∫0h/2ft+h+2u−ft+2udu.Thus, we have∥fh′∥≤5/hωf;h. Similarly, ∥fh′′∥≤9/h2ω2f;h.Theorem 9.
Under the condition of Lemma7, then for all f∈CB0,∞ and t∈0,∞, we have
(28)Sn,α,βpn,qn,λf;t−ft≤2ωf;Bn,α,βpn,qn,λt.Proof.
For anyδ>0, we have ∣fu−ft∣≤ωf;∣u−t∣≤1+∣u−t∣/δωf;δ. Applying Sn,α,βpn,qn,λ to both ends and using Lemma 5, we can obtain
(29)Sn,α,βpn,qn,λf;t−ft≤Sn,α,βpn,qn,λfu−ft;t≤1+1δSn,α,βpn,qn,λ∣u−t∣;tωf;δ.
By using the Chauchy-Schwarz inequality and takingδ=Bn,α,βpn,qn,λt, we have
(30)Sn,α,βpn,qn,λf;t−ft≤1+1δSn,α,βpn,qn,λu−t2;tωf;δ≤2ωf;Bn,α,βpn,qn,λt.
Theorem9 is proved.Theorem 10.
Under the condition of Lemma7, then for all f∈CB0,∞ and t∈0,∞, there exists an absolute positive constant C1=4C such that
(31)Sn,α,βpn,qn,λf;t−ft≤C1ω2f;An,α,βpn,qn,λt2+Bn,α,βpn,qn,λt+ωf;An,α,βpn,qn,λt.Proof.
First, we define the following new positive linear operators as follows:(32)Tn,α,βpn,qn,λf;t=Sn,α,βpn,qn,λf;t−fAn,α,βpn,qn,λt+t+ft,t∈0,∞.
It is apparent from Lemma5, Lemma 6, and Lemma 8 that
(33)Tn,α,βpn,qn,λ1;t=1;Tn,α,βpn,qn,λu−t;t=0,(34)Tn,α,βpn,qn,λf;t≤3f.
Now for any given functionh∈W2 and u,t∈0,∞, we write Taylor’s expansion formula as follows:
(35)hu=ht+h′tu−t+∫tuh′′vu−vdv.
By applyingTn,α,βpn,qn,λ operators to both sides of the above equality, we can obtain
(36)Tn,α,βpn,qn,λh;t=Tn,α,βpn,qn,λht+h′tu−t+∫tuh′′vu−vdv;t=ht+Tn,α,βpn,qn,λh′tu−t;t+Tn,α,βpn,qn,λ∫tuh′′vu−vdv;t.
Using (32), (33), and the following inequality,
(37)∫tuh′′vu−vdv≤∫tuh′′vu−vdv≤∥h′′∥∫tuu−vdv≤u−t2∥h′′∥,we can get
(38)Tn,α,βpn,qn,λh;t−ht=Tn,α,βpn,qn,λ∫tuh′′vu−vdv;t≤Sn,α,βpn,qn,λ∫tuh′′v∣u−v∣dv;t+∫tAn,α,βpn,qn,λt+th′′vAn,α,βpn,qn,λt+t−vdv≤Sn,α,βpn,qn,λu−t2;t∥h′′∥+An,α,βpn,qn,λt2∥h′′∥=An,α,βpn,qn,λt2+Bn,α,βpn,qn,λt∥h′′∥.
By using (32) and (34), we have
(39)Sn,α,βpn,qn,λh;t−ht=Tn,α,βpn,qn,λf;t+fAn,α,βpn,qn,λt+t−2ft≤Tn,α,βpn,qn,λf−h;t−f−gt+Tn,α,βpn,qn,λh;t−ht+fAn,α,βpn,qn,λt+t−ft≤4∥f−h∥+An,α,βpn,qn,λt2+Bn,α,βpn,qn,λt∥h′′∥+ωf;An,α,βpn,qn,λt.
Taking the infimum on the right-hand side over allh∈W2 and using (24), we complete the proof of Theorem 10.Theorem 11.
Under the condition of Lemma7, then for all f′∈CB0,∞ and t∈0,∞, we have
(40)Sn,α,βpn,qn,λf;t−ft≤An,α,βpn,qn,λtf′t+2Bn,α,βpn,qn,λtωf′;Bn,α,βpn,qn,λt.Proof.
ApplyingSn,α,βpn,qn,λ to both sides of the equality fu=ft+f′tu−t+fu−ft−f′tu−t, using mean value theorem and the Chauchy-Schwarz inequality and taking δ=Bn,α,βpn,qn,λt, we can obtain
(41)Sn,α,βpn,qn,λf;t−ft≤f′tSn,α,βpn,qn,λu−t;t+Sn,α,βpn,qn,λfu−ft−f′tu−t;t≤∣f′t∣Sn,α,βpn,qn,λu−t;t+Sn,α,βpn,qn,λ∣u−t∣1+∣u−t∣δωf′;δ;t≤∣f′t∣An,α,βpn,qn,λt+ωf′;δSn,α,βpn,qn,λ∣u−t∣;t+Sn,α,βpn,qn,λu−t2;tδ≤∣f′t∣An,α,βpn,qn,λt+ωf′;δSn,α,βpn,qn,λu−t2;t1+Sn,α,βpn,qn,λu−t2;tδ≤An,α,βpn,qn,λtf′t+2Bn,α,βpn,qn,λtωf′;Bn,α,βpn,qn,λt.Theorem 12.
Under the condition of Lemma7, if f∈CB0,∞, then
(42)Sn,α,βpn,qn,λf;t−ft≤5npn,qnAn,α,βpn,qn,λtωf;1npn,qn+92npn,qnBn,α,βpn,qn,λt+2ω2f;1npn,qn.Proof.
Fort∈0,∞, using the Steklov mean function fh, we can write
(43)Sn,α,βpn,qn,λf;t−ft≤Sn,α,βpn,qn,λ∣f−fh∣;t+Sn,α,βpn,qn,λfh−fht;t+fht−ft.
By Lemma8 and properties of the Steklov mean, we can obtain
(44)Sn,α,βpn,qn,λ∣f−fh∣;t≤∥Sn,α,βpn,qn,λ∣f−fh∣∥≤∥f−fh∥≤ω2f;h.
By Taylor’s expansion formula, we have(45)Sn,α,βpn,qn,λfh−fht;t≤fh′tAn,α,βpn,qn,λt+12∥fh′′∥Bn,α,βpn,qn,λt≤5hωf;hAn,α,βpn,qn,λt+92h2ω2f;hBn,α,βpn,qn,λt.
Hence,(46)Sn,α,βpn,qn,λf;t−ft≤5hAn,α,βpn,qn,λtωf;h+92h2Bn,α,βpn,qn,λt+2ω2f;h.
Settingh=1/npn,qn, we can get the desired result.By the classic Korovkin theorem, we easily get the following corollary:Corollary 13.
Under the condition of Lemma7, then for all f∈CB0,∞ and any A>0, the the sequence Sn,α,βpn,qn,λf;t converges to f uniformly on 0,A.


## 4. Voronovskaja-Type Theorem forSn,α,βpn,qn,λ

In this section, we show a Voronovskaja-type asymptotic formula for the operatorsSn,α,βpn,qn,λ by means of the first, second and fourth central moments.Theorem 14.
Under the condition of Lemma7, then for all f∈CB0,∞ satisfying f′′t that exists at a point t∈0,∞, we can obtain
(47)limn→∞npn,qnSn,α,βpn,qn,λf;t−ft=α−βt+1λ+1f′t+η2f′′tt.Proof.
By Taylor’s expansion formula forf, we have
(48)fu=ft+f′tu−t+12f′′tu−t2+ϕu;tu−t2,where
(49)ϕu;t=fu−ft−f′tu−t−1/2f′′tu−t2u−t2,u≠t;0,u=t.
Applying L’Hospital’s Rule,(50)limu→tϕu;t=12limu→tf′u−f′tu−t−12f′′t=0.
Thus,ϕ.;t∈CB0,∞. Consequently, we can write
(51)npn,qnSn,α,βpn,qn,λf;t−fx=npn,qnAn,α,βpn,qn,λt+12npn,qnBn,α,βpn,qn,λt+npn,qnSn,α,βpn,qn,λϕu;tu−t2;t.
By Schwarz’s inequality, we have(52)Sn,α,βpn,qn,λϕu;tu−t2;t≤Sn,α,βpn,qn,λϕ2u;t;tSn,α,βpn,qn,λu−t4;t.
We observe thatϕ2t;t=0 and ϕ2.;t∈CB0,∞. Then, it follows in Corollary 13 that
(53)limn→∞npn,qnSn,α,βpn,qn,λϕ2u;t;t=ϕ2t;t=0.
Hence, from (17), we can obtain
(54)limn→∞npn,qnSn,α,βpn,qn,λϕu;tu−t2;t=0.
Combining, we complete the proof of Theorem14.Corollary 15.
Under the condition of Lemma7, then for all f′,f′′∈CB0,∞, we have
(55)limn→∞npn,qnSn,α,βpn,qn,λf;t−ft=α−βt+1λ+1f′t+η2f′′tt,uniformly with respect to any finite interval I⊂0,∞.


## 5. Pointwise Estimates

In this section, we establish two pointwise estimates of the operatorsSn,α,βp,q,λ. First, we compute the rate of convergence locally by using functions belonging to the Lipschitz class. We denote that f∈CB0,∞ is in LipMγ,D, γ∈0,1, D⊂0,∞ if it satisfies the following condition:
(56)fu−ft≤Mu−tγ,u∈D,t∈0,∞,where M is a positive constant depending only on γ and f.Theorem 16.
The sequencespn, qn satisfy 0<qn<pn≤1, γ∈0,1 and D be any bounded subset on 0,∞. If f∈CB0,∞∩LipMγ,D, then for any t∈0,∞, we have
(57)Sn,α,βpn,qn,λf;t−fx≤MBn,α,βpn,qn,λtγ/2+2dγt;D,where dt;D=inf∣u−t∣:u∈D denotes the distance between t and D.Proof.
LetD¯ be the closure of D. Using the properties of infimum, and there is at least a point t0∈D¯ such that dt;E=∣t−t0∣. By the triangle inequality
(58)fu−ft≤fu−ft0+ft−ft0.
By the monotonicity ofSn,α,βpn,qn,λ, we get
(59)Sn,α,βpn,qn,λf;t−fx≤Sn,α,βpn,qn,λ∣fu−ft0∣;t+Sn,α,βpn,qn,λ∣ft−ft0∣;t≤MSn,α,βpn,qn,λu−t0γ;t+Sn,α,βpn,qn,λt−t0γ;t≤MSn,α,βpn,qn,λu−tγ+t−t0γ;t+t−t0γ=MSn,α,βpn,qn,λu−tγ;t+2t−t0γ.
Applying the well-known Hölder inequality witha1=2/γ, a2=2/2−γ, we obtain
(60)Sn,α,βpn,qn,λf;t−fx≤MSn,α,βpn,qn,λu−ta1γ;t1/a1+2dγt;D≤MSn,α,βpn,qn,λu−t2;t1/a1+2dγt;D=MBn,α,βpn,qn,λtγ/2+2dγt;D.
Second, we will give a local direct estimation of the operatorsSn,α,βpn,qn,λ by using the Lipschitz-type maximal function of the order γ introduced by Lenze [26] as
(61)ω~γf;t=supu≠t,u∈0,∞∣fu−ft∣u−tγ,t∈0,∞andγ∈0,1.Theorem 17.
The sequencespn, qnsatisfy0<qn<pn≤1andγ∈0,1. Iff∈CB0,∞, then for anyt∈0,∞, we have(62)Sn,α,βpn,qn,λf;t−ft≤ω~γf;tBn,α,βpn,qn,λtγ/2.Proof.
Using the equality (61), we obtain
(63)Sn,α,βpn,qn,λf;t−ft≤ω~γf;tSn,α,βpn,qn,λu−tγ;t.
By the well-known Hölder inequality, we have(64)Sn,α,βpn,qn,λf;t−ft≤ω~γf;tSn,α,βpn,qn,λu−t2;tγ/2≤ω~γf;tBn,α,βpn,qn,λtγ/2.
Thus, the proof of Theorem17 is completed.


## 6. Rate of Convergence

LetB20,∞ be the set of all functions f defined on 0,∞ satisfying the condition ∣ft∣≤Cf1+t2 with an absolute constant Cf>0 which may depend only on f. C20,∞ denotes the subspace of all continuous functions f∈B20,∞ with the norm ∥f∥2=supx∈0,∞ft/1+t2. By C200,∞, and we denote the subspace of all functions f∈C20,∞ for which limx→+∞ft/1+t2 is finite. Meantime, we denote the modulus of continuity of f on the interval 0,a, a>0 by
(65)ωaf;δ=sup∣u−t∣≤δsupu,t∈0,afu−ft.Theorem 18.
Letf∈C20,∞, 0<q<p≤1, and a>0. Then, for all t∈0,a, we have
(66)Sn,α,βp,q,λf;t−ft≤Cf4+3a2Bn,α,βp,q,λt+2ωa+1f;Bn,α,βp,q,λt.Proof.
For anyt∈0,a and u>a+1, we easily have 1≤u−a2≤u−t2; thus
(67)fu−ft≤fu+ft≤Cf2+u2+t2=Cf2+t2+u−t+t2≤Cf2+3t2+2u−t2≤Cf4+3t2u−t2≤Mf4+3a2u−t2,and for any t∈0,a, u∈0,a+1 and δ>0, we have
(68)fu−ft≤ωa+1∣u−t∣;t≤1+∣u−t∣δωa+1f;δ.
For (67) and (68), we can get
(69)fu−ft≤Cf4+3a2u−t2+1+∣u−t∣δωa+1f;δ.
Applying the Cauchy-Schwarz inequality and choosingδ=Bn,α,βp,q,λt, we have
(70)Sn,α,βp,q,λf;t−ft≤Sn,α,βp,q,λ∣fu−ft∣;t≤Cf4+3a2Sn,α,βp,q,λu−t2;t+Sn,α,βp,q,λ1+∣u−t∣δ;tωa+1f;δ≤Cf4+3a2Bn,α,βp,q,λt+ωa+1f;δ1+Bn,α,βp,q,λtδ=Cf4+3a2Bn,α,βp,q,λt+2ωa+1f;Bn,α,βp,q,λt.
This completes the proof of Theorem18.


## 7. Weighted Approximation

As is known, iff∈C0,∞ is not uniform, the limit limδ→0+ωf;δ=0 may be not true. In [27], Ispir defined the following weighted modulus of continuity:
(71)Ωf;δ=supt∈0,∞,0<h≤δft+h−ft1+t21+h2forf∈C200,∞,and proved the properties of monotone increasing about Ωf;δ as δ>0, limδ→0+Ωf;δ=0 and the inequality
(72)Ωf;τδ≤21+τ1+δ2Ωf;δ,τ>0.Theorem 19.
Under the condition of Lemma7, f∈C200,∞, then for sufficiently large n, the inequality
(73)Sn,α,βpn,qn,λf;t−ft≤K1+t22+θΩf;1npn,qnholds, where θ≥1/2 and K is a positive constant depending only on f and n.Proof.
Applying (71) and (72), we can obtain
(74)fu−ft≤1+u−t21+t2Ωf;∣u−t∣≤21+∣u−t∣δ1+δ2Ωf;δ1+u−t21+t2≤41+δ221+t2Ωf;δ,∣u−t∣≤δ,41+δ21+t2Ωf;δ∣u−t∣+u−t3δ,∣u−t∣>δ.
Thus, for anyδ∈0,1/2 and u,t∈0,∞, the above inequality can be rewritten
(75)fu−ft≤51+t2Ωf;δ54+∣u−t∣+u−t3δ.
Applying (16) and (17), there exists sufficiently large n such that
(76)npn,qnSn,α,βpn,qn,λu−t2;t≤K121+t2,npn,qnSn,α,βpn,qn,λu−t4;t≤K221+t22.
By Schwarz’s inequality, we can obtain(77)Sn,α,βpn,qn,λ∣u−t∣;t≤Sn,α,βpn,qn,λu−t2;t≤K1npn,qn1+t2,Sn,α,βpn,qn,λu−t3;t≤Sn,α,βpn,qn,λu−t2;tSn,α,βpn,qn,λu−t4;t≤K2npn,qn1+t23.
UsingSn,α,βpn,qn,λ as linear and positive and choosing δ=1/npn,qn, we can obtain
(78)Sn,α,βpn,qn,λf;t−ft≤51+t2Ωf;1npn,qn54+K11+t2+K21+t23≤K1+t25/2Ωf;1npn,qn,for sufficiently large n and t∈0,∞, where K≔5max5/4,K1,K2.Theorem 20.
Under the condition of Lemma7, then for any f∈C200,∞, we have
(79)limn→∞∥Sn,α,βpn,qn,λf;t−f∥2=0.Proof.
Applying the Korovkin theorem [28], we only see that it is sufficient to prove the following three conditions:
(80)limn→∞∥Sn,α,βpn,qn,λum;t−tm∥2=0,m=0,1,2.
SinceSn,α,βpn,qn,λ1;t=1, the condition holds for m=0. By Lemma 6, we can obtain
(81)limn→∞Sn,α,βpn,qn,λu;t−t2=limn→∞An,α,βpn,qn,λt2≤1npn,qn+β1λ+1pn,qn+αsupt∈0,∞11+t2+βsupt∈0,∞t1+t2⟶0,asn⟶∞.
Hence, (80) holds for m=1. Similarly, by Lemma 5, we can write for m=2,
(82)limn→∞∥Sn,α,βpn,qn,λu2;t−t2∥2≤npn,qn2npn,qn+β2−1supt∈0,∞t21+t2+npn,qnnpn,qn+β2pnn−1+2λ+1pn,qn+2αsupt∈0,∞t1+t2+1npn,qn+β212λ+1pn,qn+2αλ+1pn,qn+α2supt∈0,∞11+t2⟶0,asn⟶∞.
Thus, (80) holds for m=2. Hence, the proof is completed.Theorem 21.
Under the condition of Lemma7, then for any f∈C200,∞ and κ>0, we have
(83)limn→∞supt∈0,∞Sn,α,βpn,qn,λf;t−ft1+t21+κ=0.Proof.
Lett0∈0,∞ be arbitrary but fixed.
(84)supt∈0,∞Sn,α,βpn,qn,λf;t−ft1+t21+κ≤supt∈0,t0Sn,α,βpn,qn,λf;t−ft1+t21+κ+supt∈t0,∞Sn,α,βpn,qn,λf;t−ft1+t21+κ≤∥Sn,α,βpn,qn,λf;t−f∥0,t0+∥f∥2supt∈t0,∞Sn,α,βpn,qn,λ1+u2;t1+t21+κ+supt∈t0,∞∣ft∣1+t21+κ≔I1+I2+I3.
Applying∣ft∣≤∥f∥21+t2, we have
(85)I3=supt∈t0,∞∣ft∣1+t21+κ≤supt∈t0,∞∥f∥21+t21+t21+κ≤∥f∥21+t02κ.
Letε>0. By Lemma 5, there exists N1∈ℕ, such that for all n>N1:
(86)f2Sn,α,βpn,qn,λ1+u2;t1+t21+κ≤∥f∥21+t21+κ1+t2+ε3∥f∥2≤∥f∥21+t2κ+ε3.
Hence(87)f2supt∈t0,∞Sn,α,βpn,qn,λ1+u2;t1+t21+κ≤∥f∥21+t02κ+ε3,∀n≥N1.
Thus(88)I2+I3<2∥f∥21+t02κ+ε3,∀n≥N1.
Next, for sufficiently larget0 such that ∥f∥2/1+t02κ<ε/6. Then, I2+I3<2ε/3,∀n≥N1. Applying Corollary 13, there exists N2∈ℕ, such that for all n>N2,
(89)Sn,α,βpn,qn,λf;t−f0,t0<ε3.
LetN=maxN1,N2. Combining (86), (88), and (89), we have
(90)supt∈0,∞Sn,α,βpn,qn,λf;t−ft1+t21+κ<ε,∀n≥N.
Hence, the proof of Theorem21 is completed.



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*Source: 6683004-2021-01-09.xml*