In this paper, a kind of new analogue of Gamma type operators based on (p,q)-integers is introduced. The Voronovskaja type asymptotic formula of these operators is investigated. And some other results of these operators are studied by means of modulus of continuity and Peetre K-functional. Finally, some direct theorems concerned with the rate of convergence and the weighted approximation for these operators are also obtained.
National Natural Science Foundation of China116260311. Introduction
In recent years, with the rapid development of q-calculus, the study of approximation theory with q-integer has been discussed widely. Afterwards, with the generalization from q-calculus to (p,q)-calculus, it has been used efficiently in many areas of sciences such as algebras [1, 2] and CAGD [3]. And, recently, approximation by sequences of linear positive operators has been transferred to operators with (p,q)-integer. Some useful notations and definitions about q-calculus and (p,q)-calculus in this paper are reviewed in [4–6].
Let 0<q<p≤1. For each nonnegative integer n, the (p,q)-integer [n]p,q and (p,q)-factorial [n]p,q! are defined as(1)np,q=pn-qnp-qn=1,2,…and(2)np,q!=1p,q2p,q⋯np,q,n≥11,n=0
Further, the (p,q)-power basis is defined as(3)x⊕yp,qn=x+ypx+qyp2x+q2y⋯pn-1x+qn-1y.And(4)x⊖yp,qn=x-ypx-qyp2x-q2y⋯pn-1x-qn-1y.
Let n be a nonnegative integer; the (p,q)-Gamma function is defined as(5)Γp,qn+1=p⊖qp,qnp-qn=np,q!,0<q<p≤1.
Aral and Gupta [7] proposed (p,q)-Beta function of second kind for m,n∈N as(6)Bp,qm,n=∫0∞xm-11⊕pxp,qm+ndp,qx.And the relationship between (p,q)-analogues of Beta and Gamma functions is as follows:(7)Bp,qm,n=qΓp,qmΓp,qnpm+1qm-1m/2Γp,qm+n.Particularly, when p=q=1, B(m,n)=Γ(m)Γ(n)/Γ(m+n). It may be observed that, in (p,q)-setting, order is important, which is the reason why (p,q)-variant of Beta function does not satisfy commutativity property; that is, Bp,q(m,n)≠Bp,q(n,m).
In [8], Mazhar studied some approximation properties of the Gamma operators as follows:(8)F-nf;x=2n!xn+1n!n-1!∫0∞tn-1x+t2n+1ftdt,n>1,x>0.
Recently, Mursaleen first applied (p,q)-calculus in approximation theory and introduced the (p,q)-analogue of Bernstein operators [9], (p,q)-Bernstein-Stancu operators [10], and (p,q)-Bernstein-Schurer operators [11] and investigated their approximation properties. And many well-known approximation operators with (p,q)-integer have been introduced, such as (p,q)-Bernstein-Stancu-Schurer-Kantorovich operators [12], (p,q)-Szász-Baskakov operators [13], and (p,q)-Baskakov-Beta operators [14]. As we know, many researchers have studied approximation properties of the Gamma operators and their modifications (see [15–21], etc.). All this achievement motivates us to construct the (p,q)-analogue of the Gamma operators (8). First, we introduce (p,q)-analogue of Gamma operators as follows.
Definition 1.
For n∈N, n>1, x∈(0,∞), and 0<q<p≤1, the (p,q)-Gamma operators can be defined as(9)Fnp,qf;x=xn+1pn2qn2+nBp,qn,n+1∫0∞tn-1pnqnx⊕tp,q2n+1ftdp,qt
The paper is organized as follows. In the first section, we give the basic notations and the definition of (p,q)-Gamma operators. In the second section, we present the moments of the operators. In the third section, we obtain Voronovskaja type asymptotic formula. In the fourth section, we present a direct result of (p,q)-Gamma operators in terms of first- and second-order modulus of continuity. In the last section, we study the rate of convergence and the weighted approximation of the (p,q)-Gamma operators.
2. Auxiliary Results
In order to obtain the approximation properties of the operators Fnp,q(f;x), we need the following lemma and remarks.
Lemma 2.
The following equalities hold:
Fnp,q(1;x)=1.
Fnp,q(t;x)=x, for n>1.
Fnp,q(t2;x)=[n+1]p,qx2/pq[n-1]p,q, for n>2.
Fnp,q(t3;x)=[n+1]p,q[n+2]p,qx3/(pq)3[n-1]p,q[n-2]p,q, for n>3.
Fnp,q(t4;x)=[n+1]p,q[n+2]p,q[n+3]p,qx4/(pq)6[n-1]p,q[n-2]p,q[n-3]p,q, for n>4.
Proof.
According to the properties of (p,q)-Beta function and (p,q)-Gamma function, we have(10)Fntk;x=xn+1pn2qn2+nBp,qn,n+1∫0∞tn+k-1pnqnx⊕tp,q2n+1dp,qt=xn+1pn2qn2+nBp,qn,n+1∫0∞1x2n+1pn2n+1qn2n+1tn+k-11⊕pt/xpn+1qnp,q2n+1dp,qt=xn+1pn2qn2+nBp,qn,n+1∫0∞xn+kpn+1n+kqnn+kx2n+1pn2n+1qn2n+1t/xpn+1qnn+k-11⊕pt/xpn+1qnp,q2n+1dp,qtxpn+1qn=xkpnk+kqnkBp,qn+k,n-k+1Bp,qn,n+1=xkpnk+kqnkqn-1pn+1n/2qn+k-1pn+k+1n+k/2Γp,qn+kΓp,qn-k+1Γp,qnΓp,qn+1=xkpq-kk-1/2n+k-1p,q!n-kp,q!n-1p,q!np,q!This proves Lemma 2.
Remark 3.
Let n>2, and x∈(0,∞); then, for 0<q<p≤1, we have the central moments as follows:
The sequences (pn) and (qn) satisfy 0<qn<pn<1 such that pn→1, qn→1, and pnn→a, qnn→b, and [n]pn,qn→∞ as n→∞, where 0≤a,b<1, and x∈(0,∞); then
limn→∞[n-1]pn,qnFnpn,qn((t-x)2;x)=(a+b)x2.
limn→∞[n-3]pn,qnFnpn,qn((t-x)4;x)=0.
Proof.
(1) Using Remark 3,(11)limn→∞n-1pn,qnFnpn,qnt-x2;x=limn→∞pnn-22pn,qnqn-pn-qnn-1pn,qnpnx2=limn→∞pnn-22pn,qnqn-pnn-1-qnn-1pnx2=2a-a-bx2=a+bx2
(2) Let k=n-3; we have(12)n+1pn,qnn+2pn,qnn+3pn,qn=qn4kpn,qn+pnk4pn,qnqn5kpn,qn+pnk5pn,qnqn6kpn,qn+pnk6pn,qn~qn15kpn,qn3+pnkqn96pn,qn+qn105pn,qn+qn114pn,qnkpn,qn2.Similarly, we have(13)n+1pn,qnn+2pn,qnn-3pn,qn~qn9kpn,qn3+pnkqn54pn,qn+qn45pn,qnkpn,qn2.(14)n+1pn,qnn-2pn,qnn-3pn,qn~qn5kpn,qn3+pnkqn4pn,qn+qn4kpn,qn2(15)n-1pn,qnn-2pn,qnn-3pn,qn~qn3kpn,qn3+pnkqn2pn,qn+qn2kpn,qn2Using Lemma 2, we obtain(16)Fnpn,qnt-x4;x~An+1kpn,qnBnx4where An=qn15-4pn3qn12+6pn5qn10-3pn6qnq and(17)Bn=pnkqn96pn,qn+qn105pn,qn+qn114pn,qn-4pn3qn3qn54pn,qn+qn45pn,qn+6pn5qn5qn4pn,qn+qn4-3pn6qn6qn2pn,qn+qn2.Combining with(18)kpn,qnAn~kpn,qnqn6-4pn3qn3+6pn5qn-3pn6=kpn,qn4pn3pn3-qn3-6pn5pn-qn-pn6-qn6=kpn,qn4pn33pn,qnpn-qn-6pn5pn-qn-6pn,qnpn-qn=kpn,qnpnn-qnnnpn,qn4pn33pn,qn-6pn5-6pn,qn~pnn-qnn4pn33pn,qn-6pn5-6pn,qn~a-b12-6-6=0and(19)Bn~4+5+6-4×4+5+6×4+1-3×2+1=0,we can obtain limn→∞[n-3]pn,qnFnpn,qn((t-x)4;x)=0.
3. Voronovskaja Type Theorem
We give a Voronovskaja type asymptotic formula for Fnp,q(f;x) by means of the second and fourth central moments.
Theorem 5.
Let f be bounded and integrable on the interval x∈(0,∞); second derivative of f exists at a fixed point x∈(0,∞); the sequences (pn) and (qn) satisfy 0<qn<pn<1 such that pn→1, qn→1, and pnn→a, qnn→b, and [n]pn,qn→∞ as n→∞, where 0≤a,b<1; then(20)limn→∞n-3pn,qnFnpn,qnf;x-fx=a+b2x2f′′x.
Proof.
Let x∈(0,∞) be fixed. In order to prove this identity, we use Taylor’s expansion:(21)ft-fx=t-xf′x+t-x2f′′x2+Φpn,qnt,x,where Φpn,qn(x,t) is bounded and limt→xΦpn,qn(t,x)=0. By applying the operator Fnpn,qn(f;x) to the equality above, we obtain(22)Fnpn,qnf;x-fx=f′xFnpn,qnt-x;x+12f′′xFnpn,qnt-x2;x+Fnpn,qnΦpn,qnt,xt-x2;x=12f′′xFnpn,qnt-x2;x+Fnpn,qnΦpn,qnt,xt-x2;x.Since limt→xΦpn,qn(t,x)=0, for all ϵ>0, there exists δ>0 such that |t-x|<δ and it will imply |Φpn,qn(t,x)|<ϵ for all fixed x∈(0,∞) as n is sufficiently large. Meanwhile, if |t-x|≥δ, then |Φpn,qn(t,x)|≤C/δ2(t-x)2, where C>0 is a constant. Using Remark 4, we have(23)limn→∞n-3Fnpn,qnt-x2;x=limn→∞n-1Fnpn,qnt-x2;x=a+bx2and(24)n-3pn,qnFnpn,qnΦpn,qnt,xt-x2;x≤ϵn-3pn,qnFnpn,qnt-x2;x+Cδ2n-3pn,qnFnpn,qnt-x4;x→0n→∞.The proof is completed.
4. Local Approximation
We denote the space of all real valued continuous bounded functions f defined on the interval [0,+∞) by CB[0,+∞). The norm · on the space CB[0,+∞) is given by(25)f=supfx:x∈0,+∞.
Let us consider the following K-functional:(26)Kf,δ=infg∈CB20,∞f-g+δg′′,where δ>0 and CB20,∞=g∈CB0,∞:g′,g′′∈CB0,∞. By [22] (p. 177, Theorem 2.4), there exists an absolute constant C>0 such that(27)Kf,δ≤Cω2f,δwhere(28)ω2f,δ=sup0<h≤δsupx∈0,∞fx+2h-2fx+h+fxis the second-order modulus of smoothness of f. By(29)ωf,δ=sup0<h≤δsupx∈0,∞fx+h-fx,we denote the usual modulus of continuity of f∈CB[0,∞).
Our first result is a direct local approximation theorem for the operators Fnp,q(f;x).
Theorem 6.
Let f∈CB[0,+∞); 0<q<p≤1; then, for every x∈(0,∞) and n>2, we have(30)Fnp,qf;x-fx≤Cω2f,Ax,where C is some positive constant.
Proof.
For all g∈CB2[0,∞), using Taylor’s expansion for x∈(0,∞), we have(31)gt=gx+g′xt-x+∫xtt-ug′′udu.Applying the operators Fnp,q to both sides of the equality above and using Remark 3, we get(32)Fnp,qg;x-gx=Fnp,q∫xtt-ug′′udu;x≤Fnp,q∫xtt-ug′′udu;x≤Fnp,qg′′t-x2;x≤Axg′′.Using Fnp,qf;x≤f, we have(33)Fnp,qf;x-fx≤Fnp,qf-g;x-f-gx+Fnp,qg;x-gx≤2f-g+Axg′′Lastly, taking infimum on both sides of the inequality above over all g∈CB2[0,∞),(34)Fnp,qf;x-fx≤2Kf;Axfor which we have the desired result by (27).
Theorem 7.
Let 0<γ≤1 and let E be any bounded subset of the interval [0,∞). If f∈CB[0,∞) is locally in Lip(γ), i.e., the condition(35)fx-ft≤Lx-tγ,t∈Eandx∈0,∞holds, then, for each x∈(0,∞), we have(36)Fnp,qf;x-fx≤LAxγ/2+2dx;Eγ,where L is a constant depending on γ and f; and d(x;E) is the distance between x and E defined by(37)dx;E=inft-x:t∈E.
Proof.
From the properties of infimum, there is at least one point t0 in the closure of E; that is, t0∈E¯, such that(38)dx;E=t0-x.Using the triangle inequality, we have(39)Fnp,qf;x-fx≤Fnp,qft-fx;x≤Fnp,qft-ft0;x+Fnp,qft0-fx;x≤LFnp,qt-t0γ;x+Fnp,qt0-xγ;x≤LFnp,qt-xγ;x+2t0-xγChoosing a1=2/γ and a2=2/2-γ and using the well-known Hölder inequality,(40)Fnp,qf;x-fx≤LFnp,qt-xγa1;x1/a1Fnp,q1a2;x1/a2+2t0-xγ≤LFnp,qt-x2;xγ/2+2t0-xγ≤LAγ/2x+2dx;EγThis completes the proof.
5. Rate of Convergence and Weighted Approximation
Let Cρ[0,∞) be the set of all functions defined on [0,∞) satisfying the condition |f(x)|≤Cfρ(x), where Cf>0 is a constant depending only on f and ρ(x) is a weight function. Let Cρ[0,∞) be the space of all continuous functions in Cρ[0,∞) with the norm fρ=supx∈[0,∞)|f(x)|/ρ(x) and Cρ0[0,∞)=f∈Cρ[0,∞):limx→∞|f(x)|/ρ(x)<∞. We consider ρ(x)=1+x2 in the following two theorems. Meanwhile, we denote the modulus of continuity of f on the closed interval [0,a], a>0, by(41)ωaf,δ=supt-x≤δsupx,t∈0,aft-fx.Obviously, for the function f∈Cρ[0,∞), the modulus of continuity ωa(f,δ) tends to zero. Then we establish the following theorem on the rate of convergence for the operators Fnp,q(f;x).
Theorem 8.
Let f∈Cρ[0,∞), 0<q<p≤1, and ωa+1(f,δ) be its modulus of continuity on the finite interval [0,a+1]⊂[0,∞), where a>0. Then, for every n>2, x>0,(42)Fnp,qf;x-fxC0,a≤4Cf1+a2Ax+2ωa+1f,Ax
Proof.
For all x∈[0,a] and t>a+1, we easily have (t-x)2≥(t-a)2≥1; therefore,(43)ft-fx≤ft+fx≤Cf2+x2+t2=Cf2+x2+x-t-x2≤Cf2+3x2+2x-t2≤Cf4+3x2t-x2≤4Cf1+a2t-x2.And, for all x∈[0,a], t∈[0,a+1], and δ>0, we have(44)ft-fx≤ωa+1f,t-x≤1+t-xδωa+1f,δFrom (43) and (44), we get(45)ft-fx≤4Cf1+a2t-x2+1+t-xδωa+1f,δ.By Schwarz’s inequality and Remark 3, we have(46)Fnp,qf;x-fx≤Fnp,qft-fx;x≤4Cf1+a2Fnp,qt-x2;x+Fnp,q1+t-xδ;xωa+1f,δ≤4Cf1+a2Fnp,qt-x2;x+ωa+1f,δ1+1δFnp,qt-x2;x≤4Cf1+a2Ax+ωa+1f,δ1+1δAxBy taking δ=A(x), we get the proof of Theorem 8.
The following is a direct estimate in weighted approximation.
Theorem 9.
Let (pn) and (qn) satisfy 0<qn<pn≤1 such that pn→1, qn→1, pnn→a, qnn→b, and [n]pn,qn→∞. Then, for f∈Cρ0[0,∞), we have(47)limn→∞Fnpn,qnf;x-fxρ=0.
Proof.
Using the Korovkin theorem in [23], we see that it is sufficient to verify the following three conditions:(48)limn→∞Fnpn,qntk;x-xkρ=0,k=0,1,2.Since Fnpn,qn(1;x)=1 and Fnpn,qn(t;x)=x, (48) holds true for k=0,1. By Remark 3, for n>2, we have(49)limn→∞Fnpn,qnf;x-fxρ=supx∈0,∞qnpn-1+pnn-22pn,qnqnn-1pn,qnx21+x2≤qnpn-1+pnn-22pn,qnqnn-1pn,qn→0n→∞.Thus the proof is completed.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research is supported by the National Natural Science Foundation of China (Grant no. 11626031).
BurbanI.Two-parameter deformation of the oscillator algebra and (p, q)-analog of two-dimensional conformal field theory199523-438439110.2991/jnmp.1995.2.3-4.18MR1370049BurbanI. M.KlimykA. U.P, Q differentiation, P, Q integration and P, Q hypergeometric functions related to quantum groups199421153610.1080/10652469408819035MR1421878KhanK.LobiyalD. K.Bèzier curves based on Lupas (p, q)-analogue of Bernstein functions in CAGD201731745847710.1016/j.cam.2016.12.016MR3606090AralA.GuptaV.AgarwalR. P.2013Berlin, GermanySpringerMR3075547GuptaV.AgarwalR. P.2014New York, NY, USASpringer10.1007/978-3-319-02765-4MR3135432Zbl1295.41002GuptaV.RassiasT. M.AgrawalP. N.AcuA. M.2018New York, NY, USASpringerAralA.GuptaV.(p, q)-type beta functions of second kind20161113414610.22034/aot.1609.1011MR3721330MazharS. M.Approximation by positive operators on infinite intervals19915299104MR1145883MursaleenM.AnsariK. J.KhanA.On (p, q)-analogue of Bernstein operators2015266874882(Erratum: Applied Mathematics and Computation, vol. 278, pp. 70-71, 2016)10.1016/j.amc.2015.04.0902-s2.0-84957827939MursaleenM.AnsariK. J.KhanA.Some approximation results by (p, q)-analogue of Bernstein-Stancu operators2015269744746(Erratum: Applied Mathematics and Computation, vol. 264, pp. 744-746, 2016)10.1016/j.amc.2015.07.118MR3396817MursaleenM.NasiruzzamanM.NurgaliA.Some approximation results on Bernstein-Schurer operators defined by (p, q)-integers20152015, article 24910.1186/s13660-015-0767-4Zbl1334.41036CaiQ. B.ZhouG.On (p, q)-analogue of Kantorovich type Bernstein-Stancu-Schurer operators2016276122010.1016/j.amc.2015.12.006MR3451993GuptaV.(p, q)-Szász-Mirakyan-Baskakov operators2018121172510.1007/s11785-015-0521-4MR3741665MalikN.GuptaV.Approximation by (p, q)-Baskakov-Beta operators2017293495610.1016/j.amc.2016.08.005MR3549651ChenS. N.ChengW. T.ZengX. M.Stancu type generalization of modified Gamma operators based on q-integers201754235937310.4134/BKMS.b140625MR3632441ChenW. Z.GuoS. S.On the rate of convergence of the Gamma operator for functions of bounded variation1985158596MR858739KarsliH.Rate of convergence of new Gamma type operators for functions with derivatives of bounded variation2007455-661762410.1016/j.mcm.2006.08.001MR22873092-s2.0-33751408326Zbl1165.41316TotikV.The Gamma operators in Lp spaces1985321-24355MR810590ZengX. M.Approximation properties of Gamma operators2005311238940110.1016/j.jmaa.2005.02.051MR2168404XuX. W.WangJ. Y.Approximation properties of modified Gamma operators2007332279881310.1016/j.jmaa.2006.10.065MR2324302ZhaoC.ChengW. T.ZengX. M.Some approximation properties of a kind of q-Gamma-Stancu operators20142014, article 94MR3346807DeVoreR. A.LorentzG. G.1993Berlin, GermanySpringerGadjievA. D.Theorems of the type of P. P. Korovkin's theorems1976205781786MR0493081