# Generalizedp,q-Gamma-type operators

Authors: Wen-Tao Cheng; Qing-Bo Cai
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In the present paper, the generalizedp,q-gamma-type operators based on p,q-calculus are introduced. The moments and central moments are obtained, and some local approximation properties of these operators are investigated by means of modulus of continuity and Peetre K-functional. Also, the rate of convergence, weighted approximation, and pointwise estimates of these operators are studied. Finally, a Voronovskaja-type theorem is presented.


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## Body

## 1. Introduction

In [1], Mazhar introduced gamma operators preserving linear functions as follows:
(1)G¯nf;x=∫0∞gnx;udu∫0∞gn−1u;tftdt=2n!xn−1n!n−1!∫0∞tn−1x+t2n+1ftdt,where gnx;u=xn+1/n!e−xuun, n>1, x>0. In [2], Karsli considered new gamma operators preserving x2 as follows:
(2)G~nf;x=∫0∞gn+2x;udu∫0∞gnu;tftdt=2n+3!xn+3n!n+2!∫0∞tnx+t2n+4ftdt,where x>0. In [3], Mao defined generalized gamma operators as follows:
(3)Gn,kf;x=∫0∞gnx;udu∫0∞gn−ku;tftdt=2n−k+1!xn+1n!n−k!∫0∞tn−kx+t2n−k+2ftdt,where x>0,n≥k. Obviously, Gn,1f;x=G¯nf;x and Gn,2f;x=G~n−2f;x. In [4, 5], some approximation properties of operators ((1)–(3)) were discussed.In Bernstein polynomials, some of their modifications and corresponding operators have been studied in many papers (see [6–10]). The q-analogue of well-known positive probability operators were widely studied and discussed (see books [11–13]) since Bernstein polynomials were proposed by Lupas [14] and Phillips [15]. In [16], the q-analogue of the operators (1) was defined and discussed. In [17], Cai and Zeng constructed and studied a q-analogue of the operators (2). Meantime, modifications and generalizations of the operators (2) were introduced and researched in [18–20]. In [21], Karsli constructed a q-analogue of the operators (3) and extended the works of [16, 17, 20]. Recently, many operators are constructed with two-parameter p,q-integers based on postquantum calculus (p,q-calculus) which have been used widely in many areas of sciences such as Lie group, different equations, hypergeometric series, and physical sciences. First, we recall some useful concepts and notations from p,q-calculus, which can be found in [11–13]. The p,q-integers mp,q are defined by
(4)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.By some simple calculation, for anyi,j∈ℕ, we have the following relation:
(5)qij−ip,q=jp,q−pj−iip,q.Thep,q-factorial is defined by
(6)mp,q!=1p,q2p,q⋯mp,q,m≥1,1,m=0.Thep,q-power basis is defined by
(7)x⊕tp,qm=x+tpx+qtp2x+q2t⋯pm−1x+qm−1t,x!tp,qm=x−tpx−qtp2x−q2t⋯pm−1x−qm−1t.Letf be an arbitrary function. The improper p,q-integral of f on 0,∞ is defined as (see [22])
(8)∫0∞fxdp,qx=p−q∑m=−∞∞qmpm+1fqmpm+1,0<qp<1.Letm be a nonnegative integer. The p,q-gamma function is defined as
(9)Γp,qm+1=p!qp,qmp−qm=mp,q!,0<q<p≤1.Aral and Gupta [23] proposed a p,q-beta function of the second kind for i,j∈ℕ+≔1,2,⋯, as
(10)Bp,qi,j=∫0∞xi−11⊕pxp,qi+jdp,qx,and gave the relation of the p,q-analogues of beta and gamma functions:
(11)Bp,qi,j=qΓp,qiΓp,qjpi+1qi−1i/2Γp,qi+j.As a special case, ifp=q=1, Bi,j=ΓiΓj/Γi+j. It is obvious that the order is important for the p,q-setting, which is the reason why the p,q-variant of beta function does not satisfy commutativity property, i.e., Bp,qi,j≠Bp,qj,i.Since Mursaleen et al. firstly introducedp,q-calculus in approximation theory and constructed the p,q-analogue of Bernstein operators [24] and p,q-Bernstein-Stancu operators [25], generalizations of many well-known approximation operators based on p,q-calculus were widely introduced and discussed by several authors (see p,q-Szász-Mirakjan operators [26], p,q-Baskakov-Durrmeyer-Stancu operators [27], p,q-Bernstein-Stancu-Schurer-Kantorovich operators [28], p,q-Baskakov-beta operators [29], p,q-Lorentz polynomials [30], p,q-Szász-Mirakjan Kantorovich operators [31], p,q-Bleimann-Butzer-Hahn operators [32], p,q-Bernstein operators [33, 34], and so on). In [35], Cheng and Zhang constructed a p,q-analogue of the operators (1) using the p,q-beta function of the second kind and studied their approximation properties. Later, Cheng et al. defined the p,q-analogue of the operators (2) and researched their approximation properties in [36]. All these achievements motivate us to construct the p,q-analogue of the gamma operator (3) and generalize the works of [35, 36]. Now, we construct generalized p,q-gamma-type operators as follows:Definition 1.
Letk∈ℕ+, n=k,k+1,⋯, x∈0,∞, and 0<q<p≤1. Forf∈C0,∞, then the p,q-analogue of the gamma operator (3) can be defined by
(12)Gn,kp,qf;x=xn+1pn2−k−1n/2+k−1/2qn2+3−kn/2−k−1/2Bp,qn−k+1,n+1∫0∞tn−kpqn+1−k/2x⊕tp,q2n−k+2ftdp,qt.
In the casep=q=1,k=1, we obtain the operators (1); in the case p=q=1,k=2, we obtain the operators (2); in the case p=q=1, we obtain the operators (3); in the case p=1,k=1, we obtain the operators [16]; in the case p=1,k=2, we obtain the operators [17]; in the case p=1, we obtain the operators [21]; in the case k=1, we obtain the operators [35]; and in the case k=2, we obtain the operators [36].The paper is organized as follows: In Section1, we introduce the history of gamma-type operators and recall some basic notations about p,q-calculus; then, we construct the generalized p,q-gamma operators with the p,q-beta function. In Section 2, we obtain the auxiliary lemmas and corollaries about the moment computation formulas. And the second- and fourth-order central moment computation formula and limit equalities are also obtained. In Section 3, we discuss the local approximation about the operators by means of modulus of continuity and Peetre K-functional. In Sections 4 and 5, the rate of convergence and weighted approximation for these operators are researched. In Section 6, two pointwise estimates are given by using the Lipschitz-type maximal function. In Section 7, the Voronovskaja-type asymptotic formula is presented.


## 2. Moment Estimates

In order to obtain the approximation properties of the operatorsGn,kp,qf;x, we need the following lemmas and corollaries.Lemma 2.
Forx∈0,∞, 0<q<p≤1, k∈ℕ+, n=k,k+1,⋯, and m=0,1,2,⋯,n, we have
(13)Gn,kp,qtm,x=pqmk−m2n−k+mp,q!n−mp,q!n−kp,q!np,q!xm.Proof.
SetC=n2−k−1n/2+k−1/2, D=n2+3−kn/2−k−1/2, and A=B=n+1−k/2, we have
(14)Gn,kp,qtm;x=xn+1pCqDBp,qn−k+1,n+1∫0∞tn−kpAqBx⊕tp,q2n−k+2tmdp,qt=xn+1pCqDBp,qn−k+1,n+11pAqBx2n−k+2∫0∞tn−k+m1⊕pt/p1+AqBxp,q2n−k+2dp,qt=xmpCqDp1+AqBn−k+m+1Bp,qn−k+1,n+1pAqB2n−k+2∫0∞t/p1+AqBxn−k+m1⊕pt/p1+AqBxp,q2n−k+2dp,qtp1+AqBx=xmpCqDp1+AqBn−k+m+1pAqB2n−k+2Bp,qn−k+m+1,n−m+1Bp,qn−k+1,n+1=xmpCqDp1+AqBn−k+m+1pAqB2n−k+2pn−k+2qn−kn−k+1/2pn−k+m+2qn−k+mn−k+m+1/2n−k+mp,q!n−mp,q!n−kp,q!np,q!=pqmk−m/2n−k+mp,q!n−mp,q!n−kp,q!np,q!xm.Then, the following corollary can be obtained immediately.Corollary 3.
Forx∈0,∞, 0<q<p≤1, and n≥maxk,4,k∈ℕ, the following equalities hold:
(15)Gn,kp,q1;x=1,Gn,kp,qt;x=pqk−1/2n−k+1p,qnp,qx,Gn,kp,qt2;x=pqk−2n−k+1p,qn−k+2p,qnp,qn−1p,qx2,Gn,kp,qt3;x=pq3k−3/2n−k+1p,qn−k+2p,qn−k+3p,qnp,qn−1p,qn−2p,qx3,Gn,kp,qt4;x=pq2k−4n−k+1p,qn−k+2p,qn−k+3p,qn−k+4p,qnp,qn−1p,qn−2p,qn−3p,qx4.Corollary 4.
Forx∈0,∞, 0<q<p≤1, and n≥maxk,4, using Corollary 3, we can easily obtain the following explicit formulas for the first and second central moments:
(16)An,kp,qx≔Gn,kp,qt−x;x=pqk−1/2n−k+1p,qnp,q−1x,Bn,kp,qx≔Gn,kp,qt−x2;x=pqk−2n−k+1p,qn−k+2p,qnp,qn−1p,q−2pqk−1/2n−k+1p,qnp,q+1x2.Corollary 5.
The sequencespn,qn satisfy 0<qn<pn≤1 such that pn⟶1,qn⟶1 and pnn⟶α∈0,1,qnn⟶β∈0,1,npn,qn⟶∞ as n⟶∞; then, for any x∈0,∞, we have
(17)limn→∞npn,qnAn,kpn,qnx=−k−12α+βx,(18)limn→∞npn,qnBn,kpn,qnx=α+βx2,(19)limn→∞npn,qnGn,kpn,qnt−x4;x=0.Proof.
Using (5), we have n−k+1pn,qn=qn1−knpn,qn−pnnpnqn1−kk−1pn,qn. Hence,
(20)limn→∞npn,qnAn,kpn,qnx=limn→∞npn,qnpnqnk−1/2n−k+1pn,qnnpn,qn−1x=limn→∞npn,qnpnqnk−1/2−1−pnnpnqn1−k/2k−1pn,qnnpn,qnx=limn→∞npn,qnpnk−1/2−qnk−1/2x−k−1αx=limn→∞pnn−qnnpn−qnpnk−1−qnk−1pnk−1/2x+qnk−1/2−k−1αx=12limn→∞pnn−qnnk−1pn,qn−k−1αx=−k−12α+βx.
Usingn−k+1pn,qn/npn,qn=qn1−k−pnnpnqn1−kk−1pn,qn/npn,qn and n−k+2pn,qn/n−1pn,qn=qn3−k−pnn−1pnqn3−kk−3pn,qn/n−1pn,qn, we have
(21)limn→∞npn,qnGn,kpn,qntt−x;x=limn→∞npn,qnpnqnk−2qn1−k−pnnpnqn1−kk−1pn,qnnpn,qnqn3−k−pnn−1pnqn3−kk−3pn,qnn−1pn,qn−pnqnk−1/2−pnnpnqn1−k/2k−1pn,qnnpn,qnx2=limn→∞npn,qnpnqnk−2−pnqnk−1/2x2−k−3αx2=limn→∞npn,qnpnk−3/2−qnk−3/2−k−3αx2=limn→∞k−32pn,qnpnn−qnnx2−k−3αx2=−k−32α+βx2.
Thus,(22)limn→∞npn,qnBn,kpn,qnx=limn→∞npn,qnGn,kpn,qnt−xt;x−xlimn→∞npn,qnGn,kpn,qnt−x;x=α+βx2.
Now, we prove the limit equality (19) while k≥4,k=1,2,3 is similar. Set rn=pn/qn and i≤k, by (5); we can obtain
(23)pnqnk−i/2n−k+ipn,qnnpn,qn=rnk−i/2qnk−in−k−ipn,qnnpn,qn=rnk−i/2npn,qn−pnn−k−ik−ipn,qnnpn,qn=rnk−i/21−pnn−k−ik−ipn,qnnpn,qn=rnk−i/2−pnqn−k−i/2pnnk−ipn,qnnpn,qn.
Hence, we can rewrite(24)Gn,kpn,qnt;x=rnk−i/2−pnqn−k−i/2pnnk−1pn,qnnpn,qnx=rnk−i/2−k−1αnpn,qn+o1npn,qnx,Gn,kpn,qnt2;x=∏l=01rnk−2/2−pnqn−k−2/2pnn−lk−2pn,qnn−lpn,qnx2=rnk−2−2k−2αnpn,qn+o1npn,qnx2,Gn,kpn,qnt3;x=∏l=02rnk−3/2−pnqn−k−3/2pnn−lk−3pn,qnn−lpn,qnx3=rn3k−3/2−3k−3αnpn,qn+o1npn,qnx3,Gn,kpn,qnt4;x=∏l=03rnk−4/2−pnqn−k−4/2pnn−lk−4pn,qnn−lpn,qnx4=rn2k−4−4k−4αnpn,qn+o1npn,qnx4.
Further, we can easily get(25)Gn,kpn,qnt−x4;x=rn2k−4−4rn3k−3/2+6rnk−2−4rnk−1/2+1+o1npn,qnx4.
For anyl∈ℕ+, npn,qnrnl−1~npn,qnpnl−qnl~α−βl. We can obtain
(26)limn→∞npn,qnGn,kpn,qnt−x4;x=limn→∞npn,qnrn2k−4−4rn3k−3/2+6rnk−2−4rnk−1/2+1+o1npn,qnx4=limn→∞npn,qnrn2k−4−1−4rn3k−3/2−1+6rnk−2−1−4rnk−1/2−1x4=α−β2k−4−6k−3+6k−2−2k−1x4=0,we obtain the required result.Corollary 6.
Let us denote the normf=supx∈0,∞fx on CB0,∞ (the class of real valued continuous bounded functions on 0,∞). For any f∈CB0,∞, we have
(27)Gn,kp,qf;x≤f.Proof.
In view of (12) and Corollary 3, the proof of this corollary can be obtained easily.


## 3. Local Approximation

For anyf∈CB0,∞, let us consider the following K-functional:
(28)Kf;δ=infg∈S2f−g+δg′′,where δ∈0,∞ and S2=g∈CB0,∞:g′,g′′∈CB0,∞. The usual modulus of continuity and the second-order modulus of smoothness of f can be defined by
(29)ωf;δ=sup0<t≤δsupx∈0,∞fx+t−fx,ω2f;δ=sup0<t≤δsupx∈0,∞fx+2t−2fx+t+fx.By [37] (p.177, Theorem 2.4), there exists an absolute constant C>0 such that
(30)Kf;δ2≤Cω2f;δ,δ>0.Theorem 7.
Letpn,qn be the sequences defined in Corollary 5 and f∈CB0,∞. Then, for all k=1,2,3,⋯,n≥maxk,2, there exists an absolute positive C1=4C such that
(31)Gn,kpn,qnf;x−fx≤C1ω2f;An,kpn,qnx2+Bn,kpn,qnx+ωf;An,kpn,qnx.Proof.
Define the following new operators:(32)Gn,kpn,qnf;x=Gn,kpn,qnf;x−fAn,kpn,qnx+x+fx,x∈0,∞.
Letx,t∈0,∞ and g∈S2. By Taylor’s expansion formula, we get
(33)gt=gx+g′xt−x+∫xtg′′ut−udu.
ApplyingGn,kpn,qn to the above equality and Gn,kpn,qnt−x;x=0, we can obtain
(34)Gn,kpn,qng;x−gx=Gn,kpn,qn∫xtg′′ut−udu;x≤Gn,kpn,qn∫xt∣t−u∣g′′udu;x+∫xAn,kpn,qnx+xg′′uAn,kpn,qnx+x−udu≤Gn,kpn,qnt−x2;xg′′+An,kpn,qnx2g′′=An,kpn,qnx+Bn,kpn,qnx2g′′.
By Corollary6 and (32), we easily know ∣Gn,kpn,qnf;x∣≤3∥f∥. Hence,
(35)Gn,kpn,qnf;x−fx=Gn,kpn,qnf;x+fAn,kpn,qnx+x−2fx≤Gn,kpn,qnf−g;x−f−gx+Gn,kpn,qng;x−gx+fAn,kpn,qnx+x−fx≤4f−g+An,kpn,qnx+Bn,kpn,qnx2+g′′+ωf;An,kpn,qnx.
Taking the infimum on the right-hand side over allg∈S2 and using (30), we obtain the desired assertion.Corollary 8.
Letpn,qn be the sequences defined in Corollary 5 and f∈CB0,∞. Then, for any finite interval I⊂0,∞, the sequence Gn,kpn,qnf;x converges to f uniformly on I.


## 4. Rate of Convergence

Let(36)Bw0,∞=f:fx≤Mfwx,Cw0,∞=f:f∈Bw0,∞∩C0,∞,Cw00,∞=f:f∈Cw0,∞,limx→∞fxwx<∞,where wx is the weighted function given by wx=1+x2 and Mf is an absolute constant depending only on f. Cw00,∞ is equipped with the norm fw=supx∈0,∞fx/wx. As is known, if f∈C0,∞ is not uniform, we cannot obtain limδ→0+ωf;δ=0. In [38], Ispir defined the following weighted modulus of continuity:
(37)Ωf;δ=sup0<t≤δ,x∈0,∞fx+t−fxwxwt,and proved the properties of monotone increasing about Ωf;δ as δ>0, limδ→0+Ωf;δ=0, and the inequality
(38)Ωf;λδ≤21+λ1+δ2Ωf;δ,while λ>0 and f∈Cw00,∞. Meantime, we recall the modulus of continuity of f on the interval 0,κ⊂0,∞ by
(39)ωκf;δ=supx,t∈0,κ,x−t≤δft−fx,δ>0.Theorem 9.
Letf∈Cw0,∞, 0<q<p≤1, and κ>0, we have
(40)Gn,kp,qf;x−fxC0,κ≤4Mf3+2κ2Bn,kp,qκ+2ωκ+1f;Bn,kp,qκ.Proof.
For anyx∈0,κ and t>κ+1, we easily have 1≤t−κ2≤t−x2; thus,
(41)ft−fx≤ft+fx≤Mf2+t2+x2=Mf2+x2+t−x+x2≤Mf2+2x2+t−x2≤Mf3+2x2t−x2≤Mf3+2κ2t−x2,and for any x∈0,κ, t∈0,κ+1, and δ>0, we have
(42)ft−fx≤ωκ+1t−x;x≤1+∣t−x∣δωκ+1f;δ.
For (41) and (42), we can get
(43)ft−fx≤Mf3+2κ2t−x2+1+∣t−x∣δωκ+1f;δ.
By Schwarz’s inequality, for anyx∈0,κ, we can get
(44)Gn,kp,qf;x−fx≤Gn,kp,qft−fx;x≤Mf3+2κ2Gn,kp,qt−x2;x+Gn,kp,q1+∣t−x∣δ;xωκ+1f;δ≤Mf3+2κ2Gn,kp,qt−x2;x+ωκ+1f;δ1+1δGn,kp,qt−x2;x≤Mf3+2κ2Bn,kp,q+ωκ+1f;δ1+1δBn,kp,qx≤Mf3+2κ2Bn,kp,qκ+ωκ+1f;δ1+1δBn,kp,qκ.
By takingδ=Bn,kp,qκ and the supremum over all x∈0,κ, we accomplish the proof of Theorem 9.


## 5. Weighted Approximation

In this section, we will discuss the following three theorems about weighted approximation for the operatorsGn,kpn,qnf;x:Theorem 10.
Letf∈Cw00,∞ and the sequences pn,qn satisfy 0<qn<pn≤1 such that pn⟶1,qn⟶1,pnn⟶α∈0,1,qnn⟶β∈0,1,npn,qn⟶∞ as n⟶∞; then, there exists N∈ℕ+ such that for all n>N and ρ>0, the inequality
(45)supx∈0,∞Gn,kpn,qnf;x−fx1+x25/2+ρ≤82+2Ωf;1npn,qnholds.Proof.
Using (37) and (38), we can write
(46)ft−fx≤1+t−x21+x2Ωf;t−x≤21+∣t−x∣δ1+δ2Ωf;δ1+t−x21+x2≤41+δ221+x2Ωf;δ,t−x≤δ,41+δ21+x2Ωf;δt−x+t−x3δ,t−x>δ.
For anyδ∈0,1 and x,t∈0,∞, (46) can be rewritten:
(47)ft−fx≤81+x2Ωf;δ2+t−x+t−x3δ.
Using (18) and (19), there exists N∈ℕ+ such that for any n>N,
(48)Gn,kpn,qnt−x2;x≤2npn,qnx2,Gn,kpn,qnt−x4;x≤x4.
By Schwarz’s inequality, we can obtain(49)Gn,kpn,qnt−x;x≤Gn,kpn,qnt−x2;x≤2xnpn,qn,(50)Gn,kpn,qnt−x3;x≤Gn,kpn,qnt−x2;xGn,kpn,qnt−x4;x≤2x3npn,qn.
SinceGn,kpn,qn is linear and positive, using (47), (49), and (50), we can obtain
(51)Gn,kpn,qnf;x−fx≤81+x2Ωf;δ2+Gn,kpn,qnt−x+t−x3;xδ≤81+x22+2x+x3δnpn,qnΩf;δ.
By choosingδ=1/npn,qn, the conclusion holds.Theorem 11.
Letpn,qn be the sequences defined in Theorem 10. Then, for any f∈Cw00,∞, we have
(52)limn→∞Gn,kpn,qnf;x−fw=0.Proof.
By the weighted Korovkin theorem in [39], we see that it is sufficient to verify the following three conditions:
(53)limn→∞Gn,kpn,qntk;x−xkw=0,k=0,1,2.
SinceGn,kpn,qn1;x=1, then (53) holds true for k=0. By Corollary 3, we can obtain
(54)Gn,kpn,qnt;x−xw=supx∈0,∞Gn,kpn,qnt;x−x1+x2≤pnqnk−1/2n−k+1pn,qnnpn,qn−1supx∈0,∞x1+x2=12pnqnk−1/2n−k+1pn,qnnpn,qn−1→0,n→∞,Gn,kpn,qnt2;x−x2w=supx∈0,∞Gn,kpn,qnt2;x−x21+x2≤supx∈0,∞x21+x2pnqnk−2n−k+1pn,qnn−k+2pn,qnnpn,qnn−1pn,qn−1=pnqnk−2n−k+1pn,qnn−k+2pn,qnnpn,qnn−1pn,qn−1→0,n→∞.
Thus, the proof of Theorem11 is completed.Theorem 12.
Letpn,qn be the sequences defined in Theorem 10. Then, for any f∈Cw00,∞ and λ>0, we have
(55)limn→∞supx∈ℝ+Gn,kpn,qnf;x−fx1+x21+λ=0.Proof.
Letx0∈0,∞ be arbitrary but fixed. Then,
(56)supx∈0,∞Gn,kpn,qnf;x−fx1+x21+λ≤supx∈0,x0Gn,kpn,qnf;x−fx1+x21+λ+supx∈x0,∞Gn,kpn,qnf;x−fx1+x21+λ≤Gn,kpn,qnf;x−fC0,x0+fwsupx∈x0,∞Gn,kpn,qn1+t2;x1+x21+λ+supx∈x0,∞fx1+x21+λ.
Since∣fx∣≤Mf1+x2, we have supx∈x0,∞∣fx∣/1+x21+λ≤Mffw/1+x02λ. Let ε>0 be arbitrary; we can choose x0 to be so large such that
(57)Mffw1+x02λ<ε3.
In view of Corollary3, while x∈x0,∞, we can obtain
(58)fwlimx→∞limn→∞Gn,kpn,qn1+t2;x1+x21+λ=fwlimx→∞11+x2λ=0.
Hence, we can chooseN and x0 to be so large such that for any n>N, the inequality
(59)supx∈x0,∞fwGn,kpn,qn1+t2;x1+x21+λ<ε3holds. Also, the first term of the above inequality tends to zero by Theorem 9, that is,
(60)Gn,kpn,qnf;x−fC0,x0<ε3.
Thus, combining (57), (59), and (60), we obtain the desired result.


## 6. Pointwise Estimates

In this section, we establish two pointwise estimates of the operators (12). First, we give the relation between the local smoothness of f and local approximation. We denote that f∈C0,∞ is in LipMγ,E,γ∈0,1,E⊂0,∞ if it satisfies the following condition:
(61)ft−fx≤Mt−xγ,t∈0,∞,x∈E,where M is a constant depending only on γ and f.Theorem 13.
Let0<q<p≤1, γ∈0,1, and E be any bounded subset on 0,∞. If f∈CB0,∞⋂LipMγ,E, then, for all x∈0,∞, we have
(62)Gn,kp,qf;x−fx≤MBn,kp,qxγ/2+2dγx;E,where dx;E denotes the distance between x and E defined by
(63)dx;E=inft−x:t∈E.Proof.
LetE¯ be the closure of E. Using the properties of the infimum, there is at least a point t0∈E¯ such that dx;E=∣x−t0∣. By the triangle inequality
(64)ft−fx≤ft−ft0+fx−ft0,we can obtain
(65)Gn,kp,qf;x−fx≤Gn,kp,qft−ft0;x+Gn,kp,qfx−ft0;x≤MGn,kp,qt−t0γ;x+x−t0γ≤MGn,kp,qt−xγ+x−t0γ;x+x−t0γ=MGn,kp,qt−xγ;x+2x−t0γ.
Choosingp1=2/γ and p2=2/2−γ and using the well-known Hölder inequality, we have
(66)Gn,kp,qf;x−fx≤MGn,kp,qt−xp1γ;x1/p1Gn,kp,q1p2;x1/p2+2dγx;E≤MGn,kp,qt−x2;xγ/2+2dγx;E≤MBn,kp,qxγ/2+2dγx;E.
Next, we obtain the local direct estimate of the operatorsGn,kp,q, using the Lipschitz-type maximal function of the order γ introduced by Lenze [40] as
(67)ω~γf;x=supx,t∈0,∞,x≠tft−fxt−xγ,γ∈0,1.Theorem 14.
Letf∈CB0,∞ and γ∈0,1. Then, for all x∈0,∞, we have
(68)Gn,kp,qf;x−fx≤ω~γf;xBn,kp,qxγ/2.Proof.
From equation (67), we have
(69)Gn,kp,qf;x−fx≤ω~γf;xGn,kp,qt−xγ;x.
Applying the well-known Hölder inequality, we can get(70)Gn,kp,qf;x−fx≤ω~γf;xGn,kp,qt−x2;xγ/2≤ω~γf;xBn,kp,qxγ/2.


## 7. Voronovskaja-Type Theorem

In this section, we give a Voronovskaja-type asymptotic formula for the operators (12) by means of the second and fourth central moments.Theorem 15.
Letpn,qn be the sequences defined in Corollary 5 and f∈CB0,∞. Suppose that f′′x exists at a point x∈0,∞, then we can obtain
(71)limn→∞npn,qnGn,kpn,qnf;x−fx=α+β2x2f′′x−k−1xf′x.Proof.
Using Taylor’s expansion formula, we can obtain(72)ft=fx+f′xt−x+12f′′xt−x2+Rt,xt−x2,where Rt,x is the Peano form of the remainder and limt→xRt,x=0. Applying Gn,kpn,qn to the both sides of (72), we have
(73)npn,qnGn,kpn,qnζ;x−ζx=npn,qnf′xGn,kpn,qnt−x;x+npn,qnf′′x2Gn,kpn,qnt−x2;x+npn,qnGn,kpn,qnRt,xt−x2;x.
By Schwarz’s inequality, we have(74)Gn,kpn,qnRt,xt−x2;x≤Gn,kpn,qnR2t,x;xGn,kpn,qnt−x4;x.We observe that R2x,x=0 and R2·,x∈CB0,∞. Then, it follows from Corollary 8 that
(75)limn→∞Gn,kpn,qnR2t,x;x=R2x,x=0.
Hence, from (19), (74), and (75), we can obtain
(76)limn→∞npn,qnGn,kpn,qnRt,xt−x2;x=0.
Combining (17), (18), and (76), we obtain the required result.



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*Source: 8978121-2020-07-11.xml*