# On Durrmeyer Typeλ-Bernstein Operators via (p, q)-Calculus

Authors: Qing-Bo Cai; Guorong Zhou
Journal: Journal of Function Spaces
 (2020)

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## Abstract

In the present paper, Durrmeyer typeλ-Bernstein operators via (p, q)-calculus are constructed, the first and second moments and central moments of these operators are estimated, a Korovkin type approximation theorem is established, and the estimates on the rate of convergence by using the modulus of continuity of second order and Steklov mean are studied, a convergence theorem for the Lipschitz continuous functions is also obtained. Finally, some numerical examples are given to show that these operators we defined converge faster in some λ cases than Durrmeyer type (p, q)-Bernstein operators.


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## Body

## 1. Introduction

In 2016, Mursaleen et al. [1] proposed the following (p, q)-analogue of Bernstein operators:
(1)Bn,p,qf;x=∑k=0nbn,kx;p,qfkp,qpk−nnp,q,x∈0,1,where bn,kx;p,qk=0,1,⋯,n are (p, q)-Bernstein basis functions and defined as
(2)bn,kx;p,q=1pnn−1/2nkp,qpkk−1/2xk1⊖xp,qn−k,x∈0,1.They also introduced and studied some important approximation properties of the Stancu type of operators (1) in [2]. After their construction, there are more and more papers on the study of (p, q)-analogue of Bernstein type operators, we mention some of them as [3–11], we also mention some other positive linear operators as [12–19].Very recently, Cai et al. [20] proposed the following λ-Bernstein operators based on (p, q)-integers as
(3)Bn,p,qλf;x=∑k=0nbn,kλx;p,qfkp,qpk−nnp,q,x∈0,1,where
(4)bn,0λx;p,q=bn,0x;p,q−λp1−nnp,q+1bn+1,1x;p,q,bn,kλx;p,q=bn,kx;p,q+λp1−nnp,q−2p1−kkp,q+1p2−2nnp,q2−1bn+1,kx;p,q−p1−nnp,q−2qp−kkp,q−1p2−2nnp,q2−1bn+1,k+1x;p,q,k=1,2,⋯,n−1bn,nλx;p,q=bn,nx;p,q−λp1−nnp,q+1bn+1,nx;p,q,bn,kx;p,qk=0,1,⋯,n are defined in (2), λ∈−1,1, n≥2, x∈0,1, and 0<q<p≤1. They also constructed the (p, q)-analogue of Kantorovich type λ-Bernstein operators and investigated their A-statistical convergence properties in [21].Inspired by above research, based on (3), we introduce Durrmeyer type λ-Bernstein operators via (p, q)-calculus as
(5)Dn,p,qλf;x=n+1p,qp−n∑k=0npkqkbn,kλx;p,q∫01bn,kqt;p,qftdp,qt,where x∈0,1, 0<q<p≤1, bn,kλx;p,qk=0,1,⋯,n are defined in (4) and bn,k·;p,qk=0,1,⋯,n are defined in (2). Apparently, when λ=0, operators Dn,p,q0f;x become to Dnp,qf;x which are defined as follows in [11],
(6)Dnp,qf;x=n+1p,qp−n2∑k=0nbn,kp,qxqp−k∫01bn,kp,qqtftdp,qt,where bn,kp,qx=pnn−1/2bn,kx;p,q. We will show that operators Dn,p,qλf;x converge to f faster than operators Dnp,qf;x in some cases of λ by some numerical examples in Section 4, say, we have more modeling flexibility when adding the parameter λ.We first mention some definitions based on (p, q)-integers; details can be found in [22–26]. For any fixed real number p>0 and q>0, the (p, q)-integers np,q are defined by
(7)np,q=pn−1+pn−2q+pn−3q2+⋯+pqn−2+qn−1=pn−qnp−q,p≠q≠1;1−qn1−q,p=1;n,p=q=1.(p, q)-factorial and (p, q)-binomial coefficients are defined as follows:
(8)np,q!=np,qn−1p,q⋯1p,q,n=1,2,⋯;1,n=0,nkp,q=np,q!kp,q!n−kp,q!.The (p, q)-power basis x⊕tp,qn and x⊝tp,qn are defined by
(9)x⊕tp,qn=x+tpx+qtp2x+q2t⋯pn−1x+qn−1t,x⊝tp,qn=x−tpx−qtp2x−q2t⋯pn−1x−qn−1t.Letf:0,a→ℝ, then (p, q)-integration of a funciton f is defined by
(10)∫0afxdp,qx=p−qa∑k=0∞qkpk+1fqkpk+1a,pq>1.This paper is mainly organized as follows: in Section2, we estimate some moments and central moments of Dn,p,qλf;x in order to obtain our main results; in Section 3, we study a Korovkin type approximation theorem and estimate the rate of convergence of Dn,p,qλf to f by using the second order modulus of smoothness, Peetre’s K-functional, Steklov mean function, and Lipschitz class function; in Section 4, we give some numerical experiments to verify our theoretical results; in the final section, a conclusion is given.


## 2. Some Lemmas

Before giving our main results, we need the following lemmas.Lemma 1.
Let0<q<p≤1, we have
(11)∫01bn,kqt;p,qdp,qt=qpkpnn+1p,q;(12)∫01tbn,kqt;p,qdp,qt=qpkp−kp2nk+1p,qn+1p,qn+2p,q;(13)∫01t2bn,kqt;p,qdp,qt=qpkp−2kp3nk+1p,qk+2p,qn+1p,qn+2p,qn+3p,q.Proof.
According to the following equations of Lemma1 in [11]
(14)∫01bn,kp,qqttsdp,qt=qpkp−kspnn+2s+1/2np,q!k+sp,q!kp,q!n+s+1p,q!,s=0,1,2,3⋯,and the fact that bn,kp,qx=pnn−1/2bn,kx;p,q, we get the proof of Lemma 1 easily.Lemma 2.
Letekt=tkk=0,1,2, λ∈−1,1, x∈0,1, and 0<q<p≤1, then for the operators Dn,p,qλf;x, we have
(15)Dn,p,qλe0;x=1;(16)Dn,p,qλe1;x=xq+pnq−2p,qxqn+2p,q+2λqnp,qn+1p,qx21−xn−11−q/ppnn+2p,qp2−2nnp,q2−1+λpn1−xn+1n+2p,qp1−nnp,q−1−λn+1p,qx1−xn1−q/pn+2p,qp1−nnp,q−1−2λqn+1p,qx1−xnpn+2p,qp2−2nnp,q2−1−λ1⊝xp,qn+1pnn−1/2n+2p,qp1−nnp,q−1;(17)Dn,p,qλe2;x=q3np,q2x2n+2p,qn+3p,q+pn−1qnp,qx2p,q2−q2x+2p,qp2nn+2p,qn+3p,q+λq21−q2/p2np,qn+1p,qx2p1−nnp,q+1n+2p,qn+3p,q2q2n−1p,qx1−xn−2pnp,q−pn−1−xn−1+2λqnp,qn+1p,qx21−xn−1p−3p,qq2/p3p2−2nnp,q2−1n+2p,qn+3p,q+2p,qλp2n1−xn+1p1−nnp,q−1n+2p,qn+3p,q+λpn−2qn+1p,qx1−xnn+2p,qn+3p,qp2+q2p1−nnp,q+1+p1−p/qp+2qp1−nnp,q−1−2p,qλpn1⊝xp,qn+1pnn−1/2p1−nnp,q−1n+2p,qn+3p,q−8λpn−1q2n+1p,qx1−xnp2−2nnp,q2−1n+2p,qn+3p,q+2λpn−1q2q−pn+1p,qx1!xp,qnpnn−1/2p2−2nnp,q2−1n+2p,qn+3p,q.Proof.
By (5), (11), and Lemma 2 of [20], we have
(18)Dn,p,qλ1;x=n+1p,qp−n∑k=0npkqkbn,kλx;p,q∫01bn,kqt;p,qdp,q=n+1p,qp−n∑k=0npkqkbn,kλx;p,qqkpkpnn+1p,q=∑k=0nbn,kλx;p,q=1.
Next, by the fact thatk+1p,q=pk+qkp,q, (5) and (12), we get
(19)Dn,p,qλt;x=n+1p,qp−n∑k=0npkqkbn,kλx;p,q∫01bn,kqt;p,qtdp,qt=n+1p,qp−n∑k=0npkqkbn,kλx;p,qqkpkp−kp2nk+1p,qn+1p,qn+2p,q=∑k=0nbn,kλx;p,qpnk+1p,qpkn+2p,q=qnp,qn+2p,qBn,p,qλt;x+pnn+2p,qBn,p,qλ1;x.
Then, the desired of (16) can be obtained by Lemma 2 and Lemma 3 of [20] and easy computations. Finally, by (5) and (13), we have
(20)Dn,p,qλt2;x=n+1p,qp−n∑k=0npkqkbn,kλx;p,q∫01bn,kqt;p,qt2dp,qt=n+1p,qp−n∑k=0npkqkbn,kλx;p,qqkpkp−2kp3nk+1p,qk+2p,qn+1p,qn+2p,qn+3p,q=∑k=0nbn,kλx;p,qp2nk+1p,qk+2p,qp2kn+2p,qn+3p,q.
Usingk+1p,qk+2p,q=q3kp,q2+pkq2q+pkp,q+2p,qp2k, we obtain
(21)Dn,p,qλt2;x=q3np,q2n+2p,qn+3p,qBn,p,qλt2;x+pnqp+2qnp,qn+2p,qn+3p,qBn,p,qλt;x+2p,qp2nn+2p,qn+3p,q.
We can get (17) by Lemma 2–4 of [20] and some computations. Lemma 2 is proved.Lemma 3.
LetΦxkt=t−xkk=1,2, λ∈−1,1, x∈0,1, and 0<q<p≤1, then we have
(22)Dn,p,qλΦx1;x=1−qqx+pnq−2p,qxqn+2p,q+2λqnp,qn+1p,qx21−xn−11−q/ppnn+2p,qp2−2nnp,q2−1+λpn1−xn+1n+2p,qp1−nnp,q−1−λn+1p,qx1−xn1−q/pn+2p,qp1−nnp,q−1−2λqn+1p,qx1−xnpn+2p,qp2−2nnp,q2−1−λ1⊝xp,qn+1pnn−1/2n+2p,qp1−nnp,q−1≔Ωn,p,qλx;(23)≤1−qq+1n+2p,q+2p1−nnp,q−1+3pnn−1/2n+2p,qp1−nnp,q−1≔Θn;p,q;(24)Dn,p,qλΦx2;x≤1−q2q2+8qn+2p,q+4p1−nnp,q−1+1n+3p,q+8q2n+2p,qn+3p,q+11pnn−1/2n+2p,qp1−nnp,q−1+12pnn−1/2n+2p,qn+3p,qp1−nnp,q−1≔Ψn;p,q.Proof.
We can obtain (22) easily by (15) and (16). For λ∈0,1, we have
(25)Dn,p,qλΦx1;x≤1−qq+pnn+2p,q+2qnp,qn+1p,q1−q/ppnn+2p,qp2−2nnp,q2−1+pnn+2p,qp1−nnp,q−1.
Forλ∈−1,0, we have
(26)Dn,p,qλΦx1;x≤1−qq+pnn+2p,q+n+1p,q1−q/pn+2p,qp1−nnp,q−1+2n+1p,qn+2p,qp2−2nnp,q2−1+1pnn−1/2n+2p,qp1−nnp,q−1.
On one hand, since(27)np,qpnp1−nnp,q+1=np,qpnp,q+pn≤np,qpnp,q+qn≤np,qn+1p,q≤1,we get
(28)2qnp,qn+1p,q1−q/ppnn+2p,qp2−2nnp,q2−1≤2qn+1p,q1−q/pn+2p,qp1−nnp,q−1.
On the other hand, we have(29)n+1p,qn+2p,qp2−2nnp,q2−1≤pnn+2p,qp1−nnp,q−1≤1pnn−1/2n+2p,qp1−nnp,q−1with the fact that n+1p,q/p1−nnp,q+1=pnn+1p,q/pnp,q+pn≤pn. Combing (25)–(29), we have
(30)Dn,p,qλΦx1;x≤1−qq+pnn+2p,q+2n+1p,q1−q/pn+2p,qp1−nnp,q−1+3pnn−1/2n+2p,qp1−nnp,q−1≤1−qq+1n+2p,q+2p1−nnp,q−1+3pnn−1/2n+2p,qp1−nnp,q−1.
Thus, the desired result of (23) is proved. Finally, by Lemma 2 and the linear property of Dn,p,qλf, we have
(31)Dn,p,qλΦx2;x=Dn,p,qλe2;x−2xDn,p,qλe1;x+x2≤q3np,q2n+2p,qn+3p,q−2q+1x2+pn−1q2p,q2np,qxn+2p,qn+3p,q+2p,qp2nn+2p,qn+3p,q+22p,qpnx2qn+2p,q+Λn,p,qλx≤1−q2q2+q2p,q2+22p,qqn+2p,q+2p,qq2+3p,qq2n+2p,qn+3p,q+Λn,p,qλx,where Λn,p,qλx is some function related to λ, np,q and x, and we will estimate it in two cases. For λ∈0,1, we have
(32)Λn,p,qλx≤4p1−nnp,q−1+2p,qn+2p,qn+3p,q+4pn+2p,qp1−nnp,q−1+2pnn−1/2n+2p,qp1−nnp,q−1+2p,qp1−nnp,q−1n+2p,qn+3p,q+2pnn−1/2p1−nnp,q−1n+2p,qn+3p,q.
Forλ∈−1,0, we have
(33)Λn,p,qλx≤4p1−nnp,q−1+1n+3p,q+23p,q+2n+2p,qp1−nnp,q−1+3qp1−nnp,q−1n+2p,qn+3p,q+8p1−nnp,q−1n+2p,qn+3p,q+2+2p,qpnn−1/2p1−nnp,q−1n+2p,qn+3p,q.
From the above two equations (32) and (33), we obtain
(34)Λn,p,qλx≤4p1−nnp,q−1+1n+3p,q+23p,q+2pn+2p,qp1−nnp,q−1+3pnn−1/2n+2p,qp1−nnp,q−1+8p1−nnp,q−1n+2p,qn+3p,q+2+2p,qpnn−1/2p1−nnp,q−1n+2p,qn+3p,q.
Combing (31), (33), and (34), we get
(35)Dn,p,qλΦx2;x≤1−q2q2+8qn+2p,q+4p1−nnp,q−1+1n+3p,q+8q2n+2p,qn+3p,q+11pnn−1/2n+2p,qp1−nnp,q−1+12pnn−1/2n+2p,qn+3p,qp1−nnp,q−1.
Thus, we arrive at (24). Lemma 3 is proved.Lemma 4.
(See [6]).
Let sequencesq≔qn=1−αn, p≔pn=1−βn such that 0≤βn<αn<1, αn→0, βn→0asn→∞. The following statements are true
(A)
Iflimn→∞enβn−αn=1 and enβn/n→0, then npn,qn→∞.(B)
Iflim¯n→∞enβn−αn<1 and enβnαn−βn→0, then npn,qn→∞.(C)
Iflim¯n→∞enβn−αn<1,lim¯n→∞enβn−αn=1 and maxenβn/n,enβnαn−βn→0, then npn,qn→∞.


## 3. Rate of Convergence

In the sequel, let sequencesq≔qn and p≔pn satisfy the conditions of Lemma 4. We first give a Korovkin type approximation theorem for Dn,p,qλf.Theorem 5.
Letf be a continuous function on 0,1, λ∈−1,1 and n>1, then Dn,p,qλf;x converge uniformly to fon0,1.Proof.
Since the hypothesis of sequencesp and q, we know that n+ip,q→∞i=1,2,3 as n→∞. It is easy to get Dn,p,qλek;x→xkk=0,1,2 combining the relation n+ip,q=ip,qpn+qinp,qi=0,1,2. Therefore, we obtain the desired result due to the well-known Korovkin theorem (see [27], pp. 8-9).
Letf be a continuous function on 0,1 and endowed with the norm f=supx∈0,1fx. Peetre’s K-functional is defined by
(36)K2f;δ=infg∈C2f−g+δg′′,where δ>0 and C2=g∈C0,1:g′,g′′∈C0,1. The second order modulus of smoothness is defined as
(37)ω2f;δ=sup0<h≤δsupx,x+h,x+2h∈0,1fx+2h−2fx+h+fx.
We know that there is a relationship betweenK2f;δ and ω2f;δ, that is
(38)K2f;δ≤Cω2f;δ,where C is a positive constant. The modulus of continuity is denoted by
(39)ωf;δ=sup0<h≤δsupx,x+h∈0,1fx+h−fx.
Then, the rate of convergence ofDn,p,qλf to f is given as follows.Theorem 6.
Letf be a continuous function on 0,1, λ∈−1,1, and n>1, we have
(40)Dn,p,qλf;x−fx≤Cω2f;12Θn;p,q2+Φn;p,q+ωf;Θn;p,q,where C is a positive constant, Θn;p,q and Ψn;p,q are defined in (23) and (24).Proof.
Let us define auxiliary operatorsD^n,p,qλf;x which preserve linear functions as
(41)D^n,p,qλf;x=Dn,p,qλf;x−fx+Ωn,p,qλx+fx,where Ωn,p,qλx is defined in (22). Obviously,
(42)D^n,p,qλt−x;x=0.
Setg∈C2, by Taylor’s expansion, we have
(43)gt=gx+g′t−x+∫xtt−ug′′udu,x,t∈0,1.
ApplyingD^n,p,qλg;x to (43) and by (42), we obtain
(44)D^n,p,qλg;x−gx=D^n,p,qλ∫xtt−ug′′udu;x.
Thus, by (41), we have
(45)D^n,p,qλg;x−gx≤Dn,p,qλ∫xtt−ug′′udu;x+∫xx+Ωn,p,qλxx+Ωn,p,qλx−ug′′udu≤Dn,p,qλΦx2;x+Ωn,p,qλx2g′′≤Θn;p,q2+Φn;p,qg′′,where Θn;p,q and Ψn;p,q are defined in (23) and (24). According to (41), (5), and (15), we have
(46)D^n,p,qλf;x≤Dn,p,qλf;x+2f≤3f.Using (41), (45), and (46), we get
(47)Dn,p,qλf;x−fx≤D^n,p,qλf−g;x−f−gx+D^n,p,qλg;x−gx+fx+Ωn,p,qλx−fx≤4f−g+Θn;p,q2+Φn;p,qg′′+ωf;Θn;p,q.Taking the infimum on the right hand side over allg∈C2, we have
(48)Dn,p,qλf;x−fx≤4K2f;14Θn;p,q2+Φn;p,q+ωf;Θn;p,q.Therefore, we obtain(49)Dn,p,qλf;x−fx≤Cω2f;12Θn;p,q2+Φn;p,q+ωf;Θn;p,q.where C is a positive constant, Θn;p,q and Ψn;p,q are defined in (23) and (24). Theorem 6 is proved.Letf be a continuous function on 0,1, the Steklov mean function is defined asOne can write(50)fhx−fx=4h2∫0h/2∫0h/22fx+u+v−fx+2u+v−fxdudvby the fact that fh is continuous on 0,1. It is obvious that
(51)fh−f≤ω~2f,h,where ω~2f,δ=supx,u,v≥0sup∣u−v∣≤δ∣fx+2u−2fx+u+v+fx+2v∣. If f is continuous on 0,1, so are fh′,fh′′ and
(52)fh′≤5hω~f,h,fh′′≤9h2ω~2f,h,where ω~f,h=supx,u,v≥0,∣u−v∣<h∣fx+u−fx+v∣. Details can be found in [28].Now, we apply Steklov mean to prove the following theorem.Theorem 7.
Letf be a continuous function on 0,1, λ∈−1,1, and n>1, we have
(53)Dn,p,qλf;x−fx≤5ω~f,Θn;p,q+132ω~2f,Φn;p,q,where Θn;p,q and Ψn;p,q are defined in (23) and (24).Proof.
Since(54)Dn,p,qλf;x−fx≤Dn,p,qλf−fh;x+Dn,p,qλfh−fhx;x+fhx−fx.
By (5), (15), and (51), we have
(55)Dn,p,qλf−fh;x≤Dn,p,qλf−fh≤f−fh≤ω~2f,h.
By Taylor’s expansion, Lemma3 and (52), we obtain
(56)Dn,p,qλfh−fhx;x≤fh′xDn,p,qλΦx1;x+12fh′′Dn,p,qλΦx2;x≤5Θn;p,qh1ω~f,h1+9Ψn;p,q2h22ω~2f,h2.
Therefore, by (54)–(56), we have
(57)Dn,p,qλf;x−fx≤5Θn;p,qh1ω~f,h1+9Ψn;p,q2h22+2ω~2f,h2=5ω~f,Θn;p,q+132ω~2f,Φn;p,q,by choosing h1=Θn;p,q, h2=Ψn;p,q. Theorem 7 is proved.
Finally, we study the rate of convergence ofDn,p,qλf with the help of functions of Lipschitz class LipMξ, where M is a positive constant, 0<ξ≤1. A function f belongs to LipMξ if
(58)∣ft−fx∣≤Mt−xξt,x∈0,1.
We have the following theorem.Theorem 8.
Letf∈LipMξ, λ∈−1,1, and n>1, we have
(59)Dn,p,qλf;x−fx≤MΨn;p,qξ/2,where Ψn;p,q is defined in (24).Proof.
Sincef∈LipMξ and Dn,p,qλf are linear positive operators, using Hölder’s inequality, we have
(60)Dn,p,qλf;x−fx≤Dn,p,qλ∣ft−fx∣;x≤MDn,p,qλt−xξ;x=Mn+1p,qp−n∑k=0npkqkbn,kλx;p,q2−ξ/2pkqkbn,kλx;p,qξ/2∫01bn,kqt;p,qt−xξdp,qt≤Mn+1p,qp−n∑k=0npkqkbn,kλx;p,q∫01bn,kqt;p,qdp,qt2−ξ/2n+1p,qp−n∑k=0npkqkbn,kλx;p,q∫01bn,kqt;p,qt−x2dp,qtξ/2=MDn,p,qλt−x2;xξ/2.
Thus, Theorem8 can be obtained by (24).


## 4. Numerical Examples

In this section, we give several numerical examples to show the convergence ofDn,p,qλf;x and Dnp,qf;x to fx with different values of parameters.Example 9.
Letfx=x−1/4sin2πx and λ=−0.5. The graphs of fx and Dn,p,qλf;x with different values of parameters n=10,p=0.9,q=0.85;n=20,p=0.99,q=0.9;n=40,p=0.9999,q=0.99 are shown in Figure 1.Example 10.
Letfx=x−1/4cos2πx and n=10,p=0.9,q=0.8. The graphs of fx, Dnp,qf;x and Dn,p,qλf;x for λ=−1 and λ=1 are given in Figure 2. The error graphs of Dnp,qf;x−fx and Dn,p,qλf;x−fx for λ=−1 and λ=1 are given in Figure 3. Moreover, in Table 1, there are given the maximum errors of Dnp,qf;x−fx, Dn,p,q−1f;x−fx, and Dn,p,q1f;x−fx with different values of parameters, where p=1−1/n,q=1−2/n.Figure 1
The convergence ofDn,p,q−0.5f;x to fx.Figure 2
The approximation graphs ofDnp,qf;x, Dn,p,q−1f;x, and Dn,p,q1f;x.Figure 3
The error graphs ofDnp,qf;x−fx, Dn,p,q−1f;x−fx, and Dn,p,q1f;x−fx.Table 1
The maximum error ofDnp,qf;x−fx, Dn,p,q−1f;x−fx and Dn,p,q1f;x−fx with p=1−1/n,q=1−2/n.
n102030405070100maxDn,p,q−1f;x−fx0.43900.28680.21100.16720.13850.10330.0749maxDnp,qf;x−fx0.43450.28480.21010.16670.13820.10310.0748maxDn,p,q1f;x−fx0.43010.28270.20910.16610.13790.10300.0747


## 5. Conclusion

In the present paper, we proposed a class of Durrmeyer typeλ-Bernstein operators based on (p, q)-calculus. Due to the parameter λ, we have more flexibility in modeling. We studied the Korovkin type theorem, the estimated rate of convergence by using Peetres K-functional, the modulus of continuity of second order and Steklov mean; we also obtained a convergence theorem for the Lipschitz continuous functions. To make things more intuitive, we also give some numerical examples.



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*Source: 8832627-2020-11-27.xml*