# Approximation Properties of Durrmeyer Type of(p,q)-Bleimann, Butzer, and Hahn Operators

Authors: Qing-Bo Cai; İsmet Yüksel; Ülkü Dinlemez Kantar; Bayram Çekim
Journal: Journal of Function Spaces
 (2019)

---

## Abstract

In this study, we introduce a Durrmeyer type of Bleimann, Butzer, and Hahn operators (BBH) on(p,q)-integers. We derive the some approximation properties for these operators. We also give some graphs and numerical examples to illustrate the convergence properties of these operators to certain functions.


---

## Body

## 1. Introduction

In the literature, especiallyq-calculus in approximation theory is an important area and there are some applications of q-calculus in this theory (see [1–4]). Therefore, the researchers in the area of approximation theory put a particular importance to q-calculus. After that, the results in approximation theory related to q-calculus have been extended to the results in (p,q)-calculus. The researchers have presented the (p,q)-extensions of linear positive operators (see [5, 6]) and they have also begun to use these operators for applying in many different areas of science (see [7–9]).We first give some notations related to(p,q)-calculus as in [10–13]. For 0<q<p≤1 and each natural number n, the (p,q)-integer of n denoted by np,q,p,q-factorial denoted by np,q! was defined by(1)np,q=pn-1+pn-2q+pn-3q2+⋯+pqn-2+qn-1=pn-qnp-q,p≠q≠1;npn-1,p=q≠1;nq,p=1;n,p=q=1,and(2)np,q!=np,qn-1p,q⋯1p,q,n≥11,n=0,respectively. Also, the binomial representation was given by(3)∑k=0nnkp,qpn-k2qk2xk=1⊕xp,qn,where (4)nkp,q=np,q!n-kp,q!kp,q!and (5)1⊕xp,qn=1+xp+qx⋯pn-1+qn-1x. In [1], q-beta function was defined by(6)Bqr,s=KA,r∫0∞/Atr-11+tqr+sdqt,where(7)KA,r=Ar1+A1+1Aqr1+Aq1-r,A>0.We have K(A,r)=qr(r-1)/2 for the special case of r∈N and K(A,0)=1. Through formula (6), in [14], the authors define the (p,q)-beta function Bp,q(k,n) which is a generalization of Bq(k,n) as follows:(8)Bp,qk,n=pn2qk2∫0∞/Atk-11⊕tp,qn+kdp,qt,A>0,n,k∈N∖0.Now we note that the definitions of (p,q)-integral are as follows: (9)∫0∞ftdp,qt=q-p∑k=-∞∞pkqk+1fpkqk+1;pq<1,∫0∞ftdp,qt=p-q∑k=-∞∞qkpk+1fqkpk+1;qp<1for an arbitrary function f in [15]. The BBH operators have been studied in [16–24] in the approximation theory. For this reason, we define a new (p,q)-Durrmeyer generalization of the BBH operators and obtain the approximation theorems involving these operators.


## 2. Durrmeyer Type of(p,q)-BBH Operators

There are many papers mentioned about Durrmeyer type positive linear operators, such as [25–28] and so on. The Durrmeyer type of (p,q)-BBH operators can be introduced via equations (1), (2), (3), (4), (5), and (8) as follows:(10)Dn,p,qf,x=n+1p,q∑k=1nbn,kp,q,x∫0∞/Aγn,k-1p,q,qptftdp,qt+f0pn21⊕xp,qn-1,where(11)bn,kp,q,x=nkp,qxk1⊕xp,qnand(12)γn,k-1p,q,x=pn-k2+n-kqk2-k+1pn+qnxpn+1+qn+1xbn,k-1p,q,x.Let us give the following lemma to evaluate the values of the test functions.Lemma 1.
Ife0(t)=1, e1(t)=t/1+t, and em(t)=tm/1+t(pn+3+qn+3t)⋯(pn+1+m+qn+1+mt) for integer m with m≥2, then, we get the following result:(13)∫0∞/Aγn,k-1p,q,qptemtdp,qt=pn-k2qk2n+1p,q,m=0q/pk-1pn+1pn-k2qk2-2m+1k-m2+m+2/2k+m-1p,q!np,q!k-1p,q!n+m+1p,q!,m≥1.Proof.
Firstly, let us give the proof ofm=0. Writing (14)γn,k-1p,q,qpt=pn-k2+n-kqk2-k+1pn+qnq/ptpn+1+qn+1q/ptnk-1p,qq/ptk-11⊕q/ptp,qn instead of γn,k-1p,q,q/pt, we have (15)∫0∞/Aγn,k-1p,q,qptdp,qt=pn-k2+n-kqk2-k+1nk-1p,q∫0∞/A1pn+qnq/ptpn+1+qn+1q/ptq/ptk-11⊕q/ptp,qndp,qt. Substitute u=q/pt, and use dp,qu=q/pdp,qt and (16)pn+qnupn+1+qn+1u1⊕up,qn=1⊕up,qn+2 in the following integral; we get the result (17)∫0∞/Aγn,k-1p,q,qptdp,qt=pn-k2+n-k+1qk2-knk-1p,q∫0∞/p/qAuk-11⊕up,qn+2dp,qu. And then by (8) and the equation (18)Bp,qk,n+2-k=k-1p,q!n-k+1p,q!n+1p,q!, respectively, we get the proof of m=0(19)∫0∞/Aγn,k-1p,q,qptdp,qt=pn-k2+n-k+1qk2-knk-1p,qBp,qk,n+2-kpn+2-k2qk2=pn-k2qk2n+1p,q. Secondly, let us give the proof of m=1. Writing (20)γn,k-1p,q,qpt=pn-k2+n-kqk2-k+1pn+qnq/ptpn+1+qn+1q/ptnk-1p,qq/ptk-11⊕q/ptp,qn instead of γn,k-1p,q,q/pt, we have (21)∫0∞/Aγn,k-1p,q,qptt1+tdp,qt=pn-k2+n-kqk2-k+1nk-1p,q∫0∞/A1pn+qnq/ptpn+1+qn+1q/ptq/ptk-11⊕q/ptp,qnt1+tdp,qt. Now using (22)1+t1⊕qptp,qnpn+qnqptpn+1+qn+1qpt=1⊕tp,qn+3pn+2, we get (23)∫0∞/Aγn,k-1p,q,qptt1+tdp,qt=pn-k2+n-kqk2-k+1pn+2qpk-1nk-1p,q∫0∞/Atk1⊕tp,qn+3dp,qt. And then by (8) and the equation (24)Bp,qk+1,n+2-k=kp,q!n-k+1p,q!n+2p,q!, respectively, we get the proof of m=1: (25)∫0∞/Aγn,k-1p,q,qptt1+tdp,qt=pn-k2+n-kqk2-k+1pn+2qpk-1nk-1p,qBp,qk+1,n+2-kpn+2-k2qk+12=q/pk-1pn+1pn-k2qk2-3k+2/2kp,q!np,q!k-1p,q!n+2p,q!. Lastly, let us give the proof of m>1. Writing (26)γn,k-1p,q,qpt=pn-k2+n-kqk2-k+1pn+qnq/ptpn+1+qn+1q/ptnk-1p,qq/ptk-11⊕q/ptp,qn instead of γn,k-1p,q,q/pt, we have (27)∫0∞/Aγn,k-1p,q,qpttm1+tpn+3+qn+3t⋯pn+1+m+qn+1+mtdp,qt=pn-k2+n-kqk2-k+1nk-1p,q×∫0∞/A1pn+qnq/ptpn+1+qn+1q/ptq/ptk-11⊕q/ptp,qntm1+tpn+3+qn+3t⋯pn+1+m+qn+1+mtdp,qt. Now using (28)1+t1⊕qptp,qnpn+qnqptpn+1+qn+1qptpn+3+qn+3t⋯pn+1+m+qn+1+mt=1⊕tp,qn+m+2pn+2, in the following integral, we get the result (29)∫0∞/Aγn,k-1p,q,qpttm1+tpn+3+qn+3t⋯pn+1+m+qn+1+mtdp,qt=pn-k2+n-kqk2-k+1pn+2qpk-1nk-1p,q∫0∞/Atk+m-11⊕tp,qn+m+2dp,qt And then by (8) and the equation (30)Bp,qk+m,n+2-k=k+m-1p,q!n-k+1p,q!n+m+1p,q!, respectively, we get the proof of m>1: (31)∫0∞/Aγn,k-1p,q,qpttm1+tpn+3+qn+3t⋯pn+1+m+qn+1+mtdp,qt=pn-k2+n-kqk2-k+1pn+2qpk-1nk-1p,qBp,qk+m,n+2-kpn+2-k2qk+m2=q/pk-1pn+1pn-k2qk2-2m+1k-m2+m+2/2k+m-1p,q!np,q!k-1p,q!n+m+1p,q!.Lemma 2.
The following results for the Durrmeyer type of(p,q)-generalization of the BBH operators are verified:(i)
Dn,p,q(e0,x)=1,(ii)
Dn,p,qe1,x=p2np,q/n+2p,qx/1+x,(iii)
Dn,p,qe2,x=p2n-1p,qnp,q/n+2p,qn+3p,qx2/1+xpn-1+qn-1x+pn+1q-22p,qnp,q/n+2p,qn+3p,qx/pn-1+qn-1x.Proof.
(i) Firstly, let us give the proof of (i). Using (10), (3), and Lemma 1, we get the following result:(32)Dn,p,qe0,x=∑k=1npn-k2qk2bn,kp,q,x+pn21⊕xp,qn-1=∑k=0nnkp,qpn-k2qk2xk1⊕xp,qn=1.
(ii) Secondly, let us prove (ii). From (10) and Lemma 1, we have(33)Dn,p,qe1,x=∑k=1nkp,qn+2p,qqpk-1pn+1pn-k2qk2-3k+2/2bn,kp,q,x=∑k=1nn-1k-1p,qq/pk-1pn+1pn-k2qk2-3k+2/2np,qn+2p,qxk1⊕xp,qn=np,qn+2p,q∑k=0n-1n-1kp,qpn+1pn-1-k2qk+12-3k+1+2/2q/pkxk+11+xp+qx⋯pn-1+qn-1x=np,qn+2p,qx1+x∑k=0n-1n-1kp,qpn+1pn-1-k2qk+12-3k+1+2/2q/pxkp+qx⋯pn-1+qn-1x=p2np,qn+2p,qx1+x∑k=0n-1n-1kpn-1-k2qk2q/pxk1⊕q/pxp,qn-1.Thus from (3), we have the desired relation in (ii).
(iii) Lastly, let us prove (iii). We use (10) and Lemma 1 to get (34)Dn,p,qe2,x=∑k=1nbn,kp,q,xq/pk-1pn+1pn-k2qk2-5k/2kp,qk+1p,qn+2p,qn+3p,q.Using the relation k+1p,q=q2k-1p,q+pk-12p,q in the above equation, we reach(35)Dn,p,qe2,x=q2∑k=2nnp,q!k-2p,q!n-kp,q!q/pk-1pn+1pn-k2qk2-5k/2n+2p,qn+3p,qxk1⊕xp,qn+2p,q∑k=1nnp,q!k-1p,q!n-kp,q!pk-1q/pk-1pn+1pn-k2qk2-5k/2n+2p,qn+3p,qxk1⊕xp,qn=q2n-1p,qnp,qn+2p,qn+3p,q∑k=0n-2n-2kp,qq/pk+1pn+1pn-2-k2qk+22-5k+2/2xk+21+x⋯pn-1+qn-1x+2p,qnp,qn+2p,qn+3p,q∑k=0n-1n-1kp,qpkq/pkpn+1pn-1-k2qk+12-5k+1/2xk+11+x⋯pn-1+qn-1x.Upon necessary arrangments that have been done, we have(36)Dn,p,qe2,x=q2n-1p,qnp,qn+2p,qn+3p,qx21+xpn-1+qn-1x×∑k=0n-2n-2kp,qq/pk+1pn+1pn-2-k2qk+22-5k+2/2xkp+qx⋯pn-2+qn-2x+2p,qnp,qn+2p,qn+3p,qxpn-1+qn-1x×∑k=0n-1n-1kp,qpkq/pkpn+1pn-1-k2qk+12-5k+1/2xk1+xp+qx⋯pn-2+qn-2x.Finally, using (3), we derive the desired relation:(37)Dn,p,qe2,x=p2n-1p,qnp,qn+2p,qn+3p,qx21+xpn-1+qn-1x×∑k=0n-2n-2kp,qpn-2-k2qk2q/pxk1⊕q/pxp,qn-2+pn+1q-22p,qnp,qn+2p,qn+3p,qxpn-1+qn-1x×∑k=0n-1n-1kp,qpn-1-k2qk2xk1⊕xp,qn-1.which completes the proof.Remark 3.
The following result is valid when0<qn<pn≤1, pn→1, qn→1, pnn→1, and qnn→1 as n→∞: (38)Dn,pn,qnt1+t-x1+x2,x=Dn,pn,qnt1+t2,x-2x1+xDn,pn,qne1,x+x1+x2Dn,pn,qne0,x≤Dn,pn,qne2,x-2x1+xDn,pn,qne1,x+x1+x2Dn,pn,qne0,x=pn2n-1pn,qnnpn,qnn+2pn,qnn+3pn,qnx21+xpnn-1+qnn-1x-2pn2npn,qnn+2pn,qnx1+x2+x1+x2+pnn+1qn-22pn,qnnpn,qnn+2pn,qnn+3pn,qnxpnn-1+qnn-1x⟶0.


## 3. Genuine Type of(p,q)-Durrmeyer BBH Operators

Now, we define the genuine type of Durrmeyer BBH operators denoted byDn,p,q∗ for 0<q<p≤1 as follows:(39)Dn,p,q∗f,x≔Dn,p,qf,σnp,q,x,where(40)σnp,q,x≔xp2np,q/n+2p,q+p2np,q/n+2p,q-1xfor 0≤x<αn/1-αn,αn=p2np,q/n+2p,q.We first give the following lemma for the test functions.Lemma 4.
Forx∈[0,αn/1-αn), 0<q<p≤1, and ej(t)=t/1+tj, where j=0,1,2, the operators Dn,p,q∗(ej,x) satisfy the following properties: (i)
Dn,p,q∗(1,x)=1,(ii)
Dn,p,q∗t/1+t,x=x/1+x,(iii)
Dn,p,q∗t/1+t2,x≤n-1p,q/qn+1np,qx/1+x2+2p,q/qn+3np,qx/1+x.Lemma 5.
The operatorsDn,p,q∗ satisfy the following inequality for x∈[0,αn/1-αn): (41)0≤Dn,p,q∗t1+t-x1+x2,x≤n-1p,qqn+1np,q-1x1+x2+2p,qqn+3np,qx1+x≤n-1p,qqn+1np,q+2p,qqn+3np,q-1x1+x2+x1+x≤2n-1p,qqn+1np,q+2p,qqn+3np,q-1.


## 4. Approximation Properties of the OperatorsDn,p,q∗

In this section, we will consider the space of all bounded and continuous functions on the interval0,∞ denoted by CB0,∞ and use the supremum norm fCB=supft for f∈CB0,∞. Let Hω be the space of all-real valued functions defined on 0,∞ satisfying the following condition:(42)ft-fx<ωf,t1+t-x1+x,where any t,x∈0,∞ and ω is the modulus of continuity in [22] satisfying the following conditions:(i)
ωisanonnegativeincreasingfunctionon0,∞,(ii)
ωf,δ1+δ2≤ωf,δ1+ωf,δ2,(iii)
limδ→0⁡ω(f,δ)=0,(iv)
ω(f,λδ)≤(1+λ)ω(f,δ).Readers might see [29] for details. Now, we remember the Korovkin theorem for the linear positive operators acting from Hω into CB0,∞ in [22].Theorem 6 (see [22]).
If the linear positive operatorsLn acting from Hω into CB0,∞ satisfy the conditions, (43)Lnt1+tγ,x-x1+xγCB⟶0,n⟶∞,for γ=0,1,2, then we get (44)Lnf,x-fCB⟶0,n⟶∞for any function f∈Hω.Theorem 7.
Assume thatpn and qn are two sequences with 0<qn<pn≤1, pn→1, qn→1,pnn→1, and qnn→1 as n→∞. Then the assertion (45)limn→∞supx∈0,αn/1-αnDn,pn,qn∗f,x-fx=0 holds for any function f∈Hω.Proof.
Considering the operators,(46)D~n,pn,qnf,x=Dn,pn,qn∗f,x;if0≤x<αn1-αnfx;ifx≥αn1-αn, we have(47)D~n,pn,qnf-fCB=supx∈0,αn/1-αnDn,pn,qn∗f,x-fx.It is clear that D~n,pn,qn(1,x)-1CB=0 and D~n,pn,qnt/1+t,x-x/1+xCB=0. Using Lemma 5, we have the following inequality: (48)D~n,pn,qnt1+t2,x-x1+x2CB=supx∈0,αn/1-αnDn,pn,qn∗t1+t2,x-x1+x2=supx∈0,αn/1-αnDn,pn,qn∗t1+t-x1+x2,x≤2n-1pn,qnqnn+1npn,qn+2pn,qnqnn+3npn,qn-1. Thus by (47), we get (49)limn→∞D~n,pn,qnt1+tγ,x-x1+xγCB=0,γ=0,1,2. Applying Theorem 6 for the operators D~n,pn,qn, we have (50)limn→∞D~n,pn,qnf,x-fxCB=0 for any function f∈Hω. Hence using (47), we obtain (51)limn→∞supx∈0,αn/1-αnDn,pn,qn∗f,x-fx=0. Thus, we finish the proof of Theorem 7.Theorem 8.
Assume thatpn and qn are two sequences with 0<qn<pn≤1, pn→1, qn→1, pnn→1, and qnn→1 as n→∞. For the operators Dn,pn,qn∗, we get the following estimation property: (52)Dn,pn,qn∗f,x-fx≤2ωf,δn,pn,qn, where (53)δn,pn,qn≔2n-1pn,qnqnn+1npn,qn+2pn,qnqnn+3npn,qn-1, for each x∈[0,αn/1-αn) and f∈Hω.Proof.
UsingDn,pn,qn∗(1,x)=1, we can write(54)Dn,pn,qn∗f,x-fx≤Dn,pn,qn∗ft-fx,x.From the properties of the modulus of continuity, we have(55)ft-fx<ωf,t1+t-x1+x≤1+t/1+t-x/1+xδωf,δ.From (54) and (55), we get (56)Dn,pn,qn∗f,x-fx≤ωf,δ1+1δDn,pn,qn∗t1+t-x1+x,x. Applying Cauchy-Schwarz inequality, we have (57)Dn,pn,qn∗f,x-fx≤ωf,δ1+1δDn,pn,qn∗t1+t-x1+x2,x1/2.Choosing δ=δn,pn,qn as in (53) we obtain the result of Theorem 8.From [30] we give W~α,E as(58)W~α,E=f:sup1+xαfαx,y≤M11+yα,x≥0,y∈E,the definition of general Lipschitz-type maximal functions on E⊂0,∞, where f is a bounded and continuous function on 0,∞, M is a positive constant, 0<α≤1, (59)fαx,y=fx-fyx-yα.The distance of x and E is defined by(60)dx,E=infx-y:y∈E.Theorem 9.
Assume thatpn and qn are two sequences satisfying the conditions 0<qn<pn≤1, pn→1, qn→1, pnn→1, and qnn→1 as n→∞. Then for each f∈W~α,E,we get the following inequality: (61)Dn,pn,qn∗f,x-fx≤Mδ~n,pn,qnα/2+2dx,Eα, for each x∈[0,αn/1-αn) and δ~n,pn,qn=2n-1pn,qn/qnn+1npn,qn+2pn,qn/qnn+3npn,qn-1.Proof.
The closure of the setE is denoted by E¯. Then we have x0∈E¯ such that x-x0=d(x,x0) for x∈[0,αn/1-αn). Thus we have (62)ft-fx≤ft-fx0+fx0-fx.Using f∈W~α,E and linear and positive operator Dn,pn,qn∗, we get(63)Dn,pn,qn∗f,x-f≤Dn,pn,qn∗ft-fx0,x+fx0-fx≤MDn,pn,qn∗t1+t-x01+x0α,x+Mx-x0α1+xα1+x0α.Using the well-known inequality (a+b)α≤aα+bα for a≥0,b≥0, for 0<α≤1 and t∈0,∞, we have (64)t1+t-x01+x0α≤t1+t-x1+xα+x1+x-x01+x0α. Thus we can write (65)Dn,pn,qn∗t1+t-x01+x0α,x≤Dn,pn,qn∗t1+t-x1+xα,x+x-x0α1+xα1+x0α. Because Dn,pn,qn∗(1,x)=1,using Hölder inequality, we get (66)Dn,pn,qn∗t1+t-x01+x0α,x≤Dn,pn,qn∗t1+t-x1+x2,xα/2+x-x0α1+xα1+x0α. Using (63), the proof of Theorem 9 is completed.


## 5. Numerical Results

In this section, we will analyze the theoretical results presented in the previous sections by numerical examples.Example 1.
Letg2(x)=x2/(1+x)pn+3+qn+3x, the graphs of Dn,p,q(g2;x) with p=0.99999999, q=0.999999, and n=20,30,50,100 are shown in Figure 1. In Figure 2, fix n=100 and let p1=0.9999, q1=0.999; the curves of g2(x) (the black curve, denoted by g2(x,p1,q1) in Figure 2) and Dn,p1,q1(g2;x) (the red curve) are shown; let p2=0.9999, q2=0.9999; the curves of g2(x) (the purple curve, denoted by g2(x,p2,q2) in Figure 2) and Dn,p2,q2(g2;x) (the brown curve) are shown; let p3=0.999999, q3=0.99999; the curves of g2(x) (the green curve, denoted by g2(x,p3,q3) in Figure 2) and Dn,p3,q3(g2;x) (the brown curve) are shown. In Figure 3, the graphs of Dn,p,q(g2;x) and Dn,p,q∗(g2;x) with p=0.9999, q=0.999, and n=100 are shown. Tables 1 and 2 are the absolute error bound of Dn,p,q(g2;x) to g2(x) and Dn,p,q∗(g2;x) to g2(x) with different values of p and q. One can see from Tables 1 and 2, the latter operators Dn,p,q∗(g2;x) are better than the former ones Dn,p,q(g2;x).Table 1
The absolute error bound ofDn,p,q(g2;x) to g2 with p=1-10-m and q=1-10-m+1.
m g 2 ( x ) - D n , p , q ( g 2 ; x ) ∞ n = 100 n = 200 n = 500 n = 1000 n = 2000 5 0.0306017 0.01567566 0.00647738 0.00337185 0.00181546 6 0.03038806 0.01546181 0.00626191 0.00315343 0.00159127 7 0.03036675 0.01544054 0.00624065 0.00313215 0.00156992 8 0.03036463 0.01543841 0.00623853 0.00313002 0.0015678 9 0.03036441 0.0154382 0.00623832 0.00312981 0.00156759 10 0.03036439 0.01543818 0.0062383 0.00312979 0.00156757Table 2
The absolute error bound ofDn,p,q∗(g2;x) to g2 with p=1-10-m and q=1-10-m+1.
m g 2 ( x ) - D n , p , q ∗ ( g 2 ; x ) ∞ n = 100 n = 200 n = 500 n = 1000 n = 2000 5 0.00493872 0.00258649 0.00113532 0.00065426 0.00043724 6 0.00481934 0.00246397 0.00100536 0.00051145 0.00026317 7 0.00480761 0.00245211 0.00099337 0.00049933 0.00025083 8 0.00480644 0.00245093 0.00099281 0.00049814 0.00024963 9 0.00480632 0.00245081 0.00099206 0.00049802 0.00024951 10 0.00480631 0.0024508 0.00099205 0.00049801 0.0002495Figure 1
The figures ofg2(x) and Dn,p,q(g2;x) with p=0.99999999, q=0.999999 and different values of n.Figure 2
The figures ofg2(x) and Dn,pi,qi(g2;x) with different values of pi and qi, i=1,2,3.Figure 3
The figures ofg2(x), Dn,p,q(g2;x), and Dn,p,q∗(g2;x) with p=0.9999, q=0.999 and n=100.



---

*Source: 7047656-2019-03-03.xml*