# Approximation of Durrmeyer Type Operators Depending on Certain Parameters

Authors: Neha Malik; Serkan Araci; Man Singh Beniwal
Journal: Abstract and Applied Analysis
 (2017)

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## Abstract

Motivated by a number of recent investigations, we consider a new analogue of Bernstein-Durrmeyer operators based on certain variants. We derive some approximation properties of these operators. We also compute local approximation and Voronovskaja type asymptotic formula. We illustrate the convergence of aforementioned operators by making use of the software MATLAB which we stated in the paper.


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## Body

## 1. Introduction

In approximation theory, the use of quantum calculus (q-calculus) has gained momentum in the last decade. In the year 1987, Lupaş [1] pioneered the work on q-versions of the Bernstein polynomials. After ten years, Phillips gave another q-variant of Bernstein polynomials. Since then, numerous operators have been generalized to their quantum variants and their approximation behaviours have been studied; we indicate the recent books [2, 3] on this topic. Also, see [4–6]. Lately, the further generalization of q-calculus, namely, the postquantum calculus, symbolized by p,q-calculus has become very contributing. In [7], Mursaleen et al. proposed the p,q-variant of Bernstein polynomials, which was further improved in [8]. Further, generalizations of p,q-Bernstein polynomials are due to Kantorovich and Durrmeyer, which were, respectively, studied in [9, 10]. Many papers pertaining to approximation theory and special functions have been presented recently (cf. [11–21]). The first q-variant and p,q-variant of Bernstein-Durrmeyer operators were given in [22] and [10], respectively. For postquantum calculus, some basic theorems and definitions are as follows (cf. [10, 23–25]).Thep,q-number is defined as (1)rp,q=pr-qrp-q,p,q∈0,1.It has been observed that rp,q=pr-1rq/p. By this identity, the results obtained in (p,q)-calculus cannot be obtained directly from q-calculus.Thep,q-binomial coefficient is known as (2)rkp,q=rp,q!r-kp,q!kp,q!,0⩽k⩽r,where (3)rp,q!=∏k=1rkp,q,r⩾1,0p,q!=1.The(p,q)-analogue of x-ωr is defined by (4)x⊖ωp,qr=x-ωpx-qωp2x-q2ω⋯pr-1x-qr-1ω.The followingΓp,qr+1(5)Γp,qr+1=p⊖qp,qrp-qr=rp,q!,0<q<p,r≥0is (p,q)-analogue of Gamma function.The(p,q)-analogue of derivative of a function f is defined by (6)Dp,qfx=fpx-fqxp-qx,x≠0.In the case when p=1, the (p,q)-derivative reduces to known q-derivative. Like q-derivative, (p,q)-derivative has the following properties which are product rules for two functions: (7)Dp,qfxgx=fpxDp,qgx+gqxDp,qfx,Dp,qfxgx=gpxDp,qfx+fqxDp,qgx.Letf be an arbitrary function and ω∈R. The (p,q)-integral of f(x) on 0,ω (see [24]) is defined as (8)∫0ωfxdp,qx=q-pω∑k=0∞pkqk+1fpkqk+1ω,ifp<q,∫0ωfxdp,qx=p-qω∑k=0∞qkpk+1fqkpk+1ω,ifq<p.Letm,n∈N, the p,q-Beta function of first kind be given by (9)Bp,qm,n=∫01pxm-1p⊖pqxp,qn-1dp,qx.The(p,q)-Gamma and (p,q)-Beta functions satisfy (see [10]) (10)Bp,qm,n=pn2m+n-2+n-2/2Γp,qmΓp,qnΓp,qm+n,where m,n∈N.In the present article, we extend the studies of [10] and discuss their ordinary approximation properties. These include direct estimates in terms of modulus of smoothness and Voronovskaja type asymptotic formula. Moreover, the convergence behaviour is shown graphically using MATLAB.


## 2.p,q-Durrmeyer-Stancu Operators and Moments

Theq-Durrmeyer operators were considered in [22] and also studied in [26]. Inspired by the work of [10, 27], we consider the postquantum Stancu variant of the eminent Durrmeyer operators. The p,q-analogue of Durrmeyer-Stancu operators for x∈0,1,  0⩽α⩽β and 0<q<p⩽1 is defined as(11)Dn,α,βp,qf,x=n+1p,q∑k=0np-n2+3n-k2-k/2bn,kp,q1,x∫01bn,kp,qp,pqtfnp,qt+αnp,q+βdp,qt,where (12)bn,kp,qp,pqt=nkp,qptkp⊖pqtp,qn-k,bn,kp,q1,x=nkp,qpkk-1-nn-1/2xk1⊖xp,qn-k.This type of generalization may be useful because of its flexibility and approximation behaviour may differ for different values ofα and β. For α=β=0, we observe that these operators reduce to the one given in [10].Lemma 1 (see [28]).
Form⩾1,  ei=ti(i=0,1,2,…), (13)Un,mp,qx≔Bn,p,qem,x=∑k=0nbn,kp,q1,xpn-kkp,qnp,qm,we have the following recurrence relation:(14)np,qUn,m+1p,qpx=pn1-pxDp,qUn,mp,qx+np,qpxUn,mp,qpx.
Further,(1)
(15)Bn,p,qe0,x=1,(2)
(16)Bn,p,qe1,x=x,(3)
(17)Bn,p,qe2,x=x2+pn-1x1-xnp,q,(4)
(18)Bn,p,qe3,x=x3+pn-1x21-xnp,q+2np,qpn-2x21-x+p2n-2x1-x1-2xp-1np,q2,(5)
(19)Bn,p,qe4,x=x4+x31-xpn-3p2+2p+3np,q+x21-xp2n-21-2xp-1+p-22-3xp-1p+2np,q2+x1-xp3n-4p-4xp-1-2x+6x2p-2np,q3·Lemma 2.
Forx∈0,1, 0<q<p⩽1, we have(1)
(20)Dn,α,βp,q1,x=1,(2)
(21)Dn,α,βp,qt,x=pnnp,q+αn+2p,q+qnp,q2xnp,q+βn+2p,q,(3)
(22)Dn,α,βp,qt2,x=p2n2p,qnp,q2+α2n+2p,qn+3p,q+2αpnnp,qn+3p,q+pnqp+2qnp,q3+2αqnp,q2n+3p,qx+q3np,q3x2np,q+pn-1x1-xnp,q+β2n+2p,qn+3p,q-1,(4)
(23)Dn,α,βp,qt3,x=p3nnp,q3p3+2p2q+2pq2+q3+3αp2n2p,qnp,q2n+4p,q+3α2pnnp,qn+3p,qn+4p,q+α3n+2p,qn+3p,qn+4p,q+p2n4pq3+p3q+3p2q2+3q4np,q4+3αpnpq+2q2np,q3n+4p,q+3α2qnp,q2n+3p,qn+4p,qx+pn2pq4+p2q3+3q5np,q4+3αq3np,q3n+4p,qx2np,q+pn-1x1-x+q6np,q4x3np,q2+pn-1x21-xnp,q+2np,qpn-2x21-x+p2n-2x1-x1-2xp-1np,q+β3n+2p,qn+3p,qn+4p,q-1,(5)
(24)Dn,α,βp,qt4,x=p4nnp,q4p6+3p5q+5p4q2+6p3q3+5p2q4+3pq5+q6+4αnp,q3n+5p,qp3np3+2p2q+2pq2+q3+6α2p2n2p,qnp,q2n+4p,qn+5p,q+4α3pnnp,qn+3p,qn+4p,qn+5p,q+α4n+2p,qn+3p,qn+4p,qn+5p,q+p3n12p3q4+12p2q5+9pq6+4q7+8p4q3+p6q+4p5q2np,q5+4αp2n4pq3+p3q+3p2q2+3q4np,q4n+5p,q+6α2pnpq+2q2np,q3n+4p,qn+5p,q+4α3qnp,q2n+3p,qn+4p,qn+5p,qx+p2n9pq7+7p3q5+9p2q6+6q8+3p4q4+p5q3np,q5+4αpn2pq4+p2q3+3q5np,q4n+5p,qx2np,q+pn-1x1-x+pn3pq8+2p2q7+p3q6+4q9np,q5+4αq6np,q4n+5p,qx3np,q2+pn-1x1-xnp,q+2pn-2x21-xnp,q+p2n-2x1-x1-2xp-1+q10np,q5x4np,q3+x31-xpn-3p2+2p+3np,q2+x21-xp2n-21-2xp-1+p-22-3xp-1p+2np,q+x1-xp3n-4p-4xp-1-2x+6x2p-2np,q+β4n+2p,qn+3p,qn+4p,qn+5p,q-1·
Proceeding along the lines of [[10], Lemma  3.1] and using the linearity property of the operators (11), the result follows immediately. We omit the details.Lemma 3.
Denote(25)μn,α,β,mp,qx=n+1p,q∑k=0np-n2+3n-k2-k/2bn,kp,q1,x∫01bn,kp,qp,pqtnp,qt+αnp,q+β-xmdp,qt.
Then, the central moments are given as follows:(1)
(26)μn,α,β,1p,qx=qnp,q2-np,q+βn+2p,qx+pnnp,q+αn+2p,qnp,q+βn+2p,q,(2)
(27)μn,α,β,2p,qx=p2n2p,qnp,q2+α2n+2p,qn+3p,q+2αpnnp,qn+3p,q+xpnqp+2qnp,q3+2αqnp,q2n+3p,q+pn-1q3np,q3-2np,qpn+αn+2p,qnp,q+βn+3p,q+x2q3np,q4-pn-1q3np,q3+np,q+β2n+2p,qn+3p,q-2qnp,q2np,q+βn+3p,qnp,q+β2n+2p,qn+3p,q-1,(3)
(28)μn,α,β,3p,qx=Dn,α,βp,qt3,x-3xDn,α,βp,qt2,x+3x2Dn,α,βp,qt,x-x3Dn,α,βp,q1,x=p3nnp,q3p3+2p2q+2pq2+q3+3αp2n2p,qnp,q2n+4p,q+3α2pnnp,qn+3p,qn+4p,q+α3n+2p,qn+3p,qn+4p,q+p2n4pq3+p3q+3p2q2+3q4np,q4+3αpnpq+2q2np,q3n+4p,q+3α2qnp,q2n+3p,qn+4p,qx+pn2pq4+p2q3+3q5np,q4+3αq3np,q3n+4p,qx2np,q+pn-1x1-x+q6np,q4x3np,q2+pn-1x21-xnp,q+2np,qpn-2x21-x+p2n-2x1-x1-2xp-1np,q+β3n+2p,qn+3p,qn+4p,q-1-3xp2n2p,qnp,q2+α2n+2p,qn+3p,q+2αpnnp,qn+3p,q+pnqp+2qnp,q3+2αqnp,q2n+3p,qx+q3np,q3x2np,q+pn-1x1-xnp,q+β2n+2p,qn+3p,q-1+3x2pnnp,q+αn+2p,q+qnp,q2xnp,q+βn+2p,q-x3≔f1n,α,β,p,q,x,(4)
(29)μn,α,β,4p,qx=Dn,α,βp,qt4,x-4xDn,α,βp,qt3,x+6x2Dn,α,βp,qt2,x-4x3Dn,α,βp,qt,x+x4Dn,α,βp,q1,x=p4nnp,q4p6+3p5q+5p4q2+6p3q3+5p2q4+3pq5+q6+4αnp,q3n+5p,qp3np3+2p2q+2pq2+q3+6α2p2n2p,qnp,q2n+4p,qn+5p,q+4α3pnnp,qn+3p,qn+4p,qn+5p,q+α4n+2p,qn+3p,qn+4p,qn+5p,q+p3n12p3q4+12p2q5+9pq6+4q7+8p4q3+p6q+4p5q2np,q5+4αp2n4pq3+p3q+3p2q2+3q4np,q4n+5p,q+6α2pnpq+2q2np,q3n+4p,qn+5p,q+4α3qnp,q2n+3p,qn+4p,qn+5p,qx+p2n9pq7+7p3q5+9p2q6+6q8+3p4q4+p5q3np,q5+4αpn2pq4+p2q3+3q5np,q4n+5p,qx2np,q+pn-1x1-x+pn3pq8+2p2q7+p3q6+4q9np,q5+4αq6np,q4n+5p,qx3np,q2+pn-1x1-xnp,q+2pn-2x21-xnp,q+p2n-2x1-x1-2xp-1+q10np,q5x4np,q3+x31-xpn-3p2+2p+3np,q2+x21-xp2n-21-2xp-1+p-22-3xp-1p+2np,q+x1-xp3n-4p-4xp-1-2x+6x2p-2np,q+β4n+2p,qn+3p,qn+4p,qn+5p,q-1-4xp3nnp,q3p3+2p2q+2pq2+q3+3αp2n2p,qnp,q2n+4p,q+3α2pnnp,qn+3p,qn+4p,q+α3n+2p,qn+3p,qn+4p,q+p2n4pq3+p3q+3p2q2+3q4np,q4+3αpnpq+2q2np,q3n+4p,q+3α2qnp,q2n+3p,qn+4p,qx+pn2pq4+p2q3+3q5np,q4+3αq3np,q3n+4p,qx2np,q+pn-1x1-x+q6np,q4x3np,q2+pn-1x21-xnp,q+2np,qpn-2x21-x+p2n-2x1-x1-2xp-1np,q+β3n+2p,qn+3p,qn+4p,q-1+6x2p2n2p,qnp,q2+α2n+2p,qn+3p,q+2αpnnp,qn+3p,q+pnqp+2qnp,q3+2αqnp,q2n+3p,qx+q3np,q3x2np,q+pn-1x1-xnp,q+β2n+2p,qn+3p,q-1-4x3pnnp,q+αn+2p,q+qnp,q2xnp,q+βn+2p,q-1+x4≔f2n,α,β,p,q,x.
The proof follows immediately by applying Lemma2.Remark 4.
In order to understand the convergence behaviour of the sequence of(p,q)-Durrmeyer operators, let q=(qn) and p=(pn), such that 0<qn<pn⩽1 and pn→1,  qn→1,  pnn→a,  qnn→b,  0<b<a⩽1 for n sufficiently large. With this restriction, we have limn→∞[n]pn,qn=limn→∞pnn-qnn/pn-qn=a-b/1-1=∞.Remark 5.
From Lemma3 and Remark 4, we have(30)limn→∞μn,α,β,3pn,qnx=limn→∞f1n,α,β,pn,qn,x=0,(31)limn→∞μn,α,β,4pn,qnx=limn→∞f1n,α,β,pn,qn,x=0.


## 3. Direct Estimates

LetW2=g∈C0,1:g′′∈C0,1, then K-functional is defined as(32)K2f,δ=inf⁡f-g+δg′′:g∈W2,where δ>0 and norm-· denotes the uniform norm on C0,1. Following the acclaimed inequality owing to DeVore and Lorentz [29], there exists a constant C>0, such that(33)K2f,δ⩽Cω2f,δ,where the second-order modulus of smoothness for f∈C0,1 is defined as (34)ω2f,δ=sup0<h⩽δsupx,x+h∈0,1fx+2h-2fx+h+fx.The usual modulus of continuity for f∈C0,1 is defined as (35)ωf,δ=sup0<h⩽δsupx,x+h∈0,1fx+h-fx.Our prime result is the subsequent local theorem.Theorem 6.
Letn>3 be a natural number and 0<q<p⩽1. Then, there exists a constant C>0, such that (36)Dn,α,βp,qf,x-fx⩽Cω2f,μn,α,β,2p,qx+μn,α,β,1p,qx2+ωf,μn,α,β,1p,qx,where f∈C0,1 and x∈0,1.Proof.
Forf∈C0,1, let (37)D~n,α,βp,qf,x=Dn,α,βp,qf,x+fx-fpnnp,q+αn+2p,qnp,q+βn+2p,q+qnp,q2xnp,q+βn+2p,q.Then, using Lemma 2, we immediately get (38)D~n,α,βp,q1,x=Dn,α,βp,q1,x=1,D~n,α,βp,qt,x=Dn,α,βp,qt,x+x-pnnp,q+αn+2p,qnp,q+βn+2p,q+qnp,q2xnp,q+βn+2p,q=x.Using Taylor’s formula (39)gt=gx+t-xg′x+∫xtt-ug′′udu,we obtain(40)D~n,α,βp,qg,x=gx+D~n,α,βp,q∫xtt-ug′′udu,x=gx+Dn,α,βp,q∫xtt-ug′′udu,x-∫xpnnp,q+αn+2p,q/np,q+βn+2p,q+qnp,q2x/np,q+βn+2p,qpnnp,q+αn+2p,qnp,q+βn+2p,q+qnp,q2xnp,q+βn+2p,q-ug′′udu.
Thus,(41)D~n,α,βp,qg,x-gx⩽Dn,α,βp,q∫xtt-ug′′udu,x+∫xpnnp,q+αn+2p,q/np,q+βn+2p,q+qnp,q2x/np,q+βn+2p,qpnnp,q+αn+2p,qnp,q+βn+2p,q+qnp,q2xnp,q+βn+2p,q-ug′′udu⩽μn,α,β,2p,qx+μn,α,β,1p,qx2g′′.Furthermore, for f∈C0,1, we have Dn,α,βp,qf⩽f and therefore(42)D~n,α,βp,qf,x⩽Dn,α,βp,qf,x+fx+fpnnp,q+αn+2p,qnp,q+βn+2p,q+qnp,q2xnp,q+βn+2p,q⩽3f∀f∈C0,1.
Now, forf∈C0,1 and g∈W2, we get(43)Dn,α,βp,qf,x-fx=D~n,α,βp,qf,x-fx+fpnnp,q+αn+2p,qnp,q+βn+2p,q+qnp,q2xnp,q+βn+2p,q-fx⩽D~n,α,βp,qf-g,x+D~np,qg,x-gx+gx-fx+fpnnp,q+αn+2p,qnp,q+βn+2p,q+qnp,q2xnp,q+βn+2p,q-fx⩽4f-g+μn,α,β,2p,qx+μn,α,β,1p,qx2g′′+ωf,μn,α,β,1p,qx, where we have used (41) and (42).
On taking infimum on the right hand side over allg∈W2, we attain (44)Dn,α,βp,qf,x-fx⩽4K2f,μn,α,β,2p,qx+μn,α,β,1p,qx2+ωf,μn,α,β,1p,qx. Lastly, in view of (33), we have (45)Dn,α,βp,qf,x-fx⩽Cω2f,μn,α,β,2p,qx+μn,α,β,1p,qx2+ωf,μn,α,β,1p,qx.The theorem is hence proved.Example 7.
We present comparisons and graphs for the convergence of operatorsDn,α,βp,q(f,x) for various values of the parameters α, β, p, and q, such that 0<q<p⩽1 and 0⩽α⩽β. For x∈0,1,  α=5,  β=20,  p=0.8, and q=0.3, the convergence behaviour of the difference of the operators Dn,α,βp,qf,x to the function f, where fx=9-6x-3x2, for various values of n, is presented in Figure 1 using MATLAB.Figure 1
The differenceDn,α,β0.8,0.3(f,x)-f(x) for α=5,  β=20, and f(x)=9-6x-3x2.Example 8.
The convergence behaviour of the operatorsDn,α,βp,q(f,x) to the function 9-6x-3x2 for α=0=β is illustrated in Figure 2. It may be observed that this shows the convergence for the operators defined in [10].Figure 2
The differenceDn,α,β0.8,0.3(f,x)-f(x) for α=0,  β=0, and f(x)=9-6x-3x2.Example 9.
We compute the error of approximation for the operatorsDn,α,βp,q(f,x) depending upon the parameters α and β at certain points from 0,1 as shown in Table 1. We consider n=100,  p=0.5,  q=0.4, and f(x)=9-6x-3x2.Table 1
Error of approximation.
x
D
n
,
α
,
β
p
,
q
f
,
x
-
f
x for α=2,  β=20
D
n
,
α
,
β
p
,
q
f
,
x
-
f
x for α=0,  β=0
0
0.6300
3.2640
0.01
0.5697
3.3624
0.02
0.5088
3.4612
0.03
0.4473
3.5603
0.04
0.3852
3.6599
0.05
0.3225
3.7598
0.06
0.2592
3.8601
0.07
0.1953
3.9608
0.08
0.1308
4.0619
0.09
0.0657
4.1634
0.1
0
4.2652Remark 10.
It may be observed from the above example that we may get better convergence depending upon the flexible values ofα and β. This is one of the reasons for discussing operators (11).We now present the Voronovskaja type asymptotic formula.Theorem 11.
Assumef∈C0,1. If f′′ exists at a point x∈α/npn,qn+β,npn,qn+α/npn,qn+β, then under the assumptions of Remark 4, we get(46)limn→∞⁡npn,qnDn,α,βpn,qnf,x-fx=a+α-2b+βxf′x+2ax-a+bx2f′′x2·Proof.
Employing Taylor's expansion off, we get(47)ft=fx+f′xt-x+f′′x2t-x2+ξt,xt-x2,where ξ(t,x)→0 as t→x. Applying Dn,α,βpn,qn to (47), we obtain (48)Dn,α,βpn,qnf,x-fx=Dn,α,βpn,qnt-x,xf′x+Dn,α,βpn,qnt-x2,xf′′x2+Dn,α,βpn,qnξt,xt-x2,x.Using Cauchy-Schwarz inequality and (31), we get(49)Dn,α,βpn,qnξt,xt-x2,x⩽Dn,α,βpn,qnξ2t,x,xDn,α,βpn,qnt-x4,x.As ξ2x,x=0 and ξ2·,x∈C0,1, we have(50)limn→∞⁡npn,qnDn,α,βpn,qnξ2t,x,x=0,uniformly with respect to x∈α/npn,qn+β,npn,qn+α/npn,qn+β. Therefore, from (49) and (50), we get (51)limn→∞npn,qnDn,α,βpn,qnξt,xt-x2,x=0. Thus, (52)limn→∞npn,qnDn,α,βpn,qnf,x-fx=limn→∞npn,qnDn,α,βpn,qnt-x,xf′x+12f′′xDn,α,βpn,qnt-x2,x+Dn,α,βpn,qnξt,xt-x2,x=a+α-2b+βxf′x+2ax-a+bx2f′′x2·Remark 12.
For theq-Durrmeyer operators discussed in [22], the recurrence relation was established for q-Bernstein-Durrmeyer operators (see Theorem  4.3 of [2] and references therein). However, for (p,q)-Durrmeyer operators Dn,α,βp,q, it is not analogous and maybe discussed somewhere else.Remark 13.
LetI=0,1. Then, for I2=I×I, let C(I2) be the space of all real valued continuous functions on I2 equipped with the norm fI=supx,y∈I2fx,y.
Forf∈C(I2) and 0<pn1<1, 0<pn2<1, 0<qn1<1, and 0<qn2<1; we construct the bivariate extension of the (p,q)-analogue of Durrmeyer-Stancu operators (11) as (53)Dn1,n2,α,βpn1,pn2,qn1,qn2f,x,y=n1+1pn1,qn1n2+1pn2,qn2∑k1=0n1∑k2=0n2pn1-n12+3n1-k12-k1/2pn2-n22+3n2-k22-k2/2bn1,n2,k1,k2pn1,pn2,qn1,qn21,x,y∫01bn1,n2,k1,k2pn1,pn2,qn1,qn2pn1,pn2,pn1qn1t,pn2qn2sfn1pn1,qn1t+αn1pn1,qn1+β,n2pn2,qn2s+αn2pn2,qn2+βdpn1,qn1tdpn2,qn2s,
where(x,y)∈I2 and (54)bn1,n2,k1,k2pn1,pn2,qn1,qn2pn1,pn2,pn1qn1t,pn2qn2s=n1k1pn1,qn1n2k2pn2,qn2pn1tk1pn2sk2×pn1⊖pn1qn1tpn1,qn1n1-k1pn2⊖pn2qn2spn2,qn2n2-k2,bn1,n2,k1,k2pn1,pn2,qn1,qn21,x,y=n1k1pn1,qn1n2k2pn2,qn2pn1k1k1-1-n1n1-1/2pn2k2k2-1-n2n2-1/2xk1yk21⊖xpn1,qn1n1-k11⊖ypn2,qn2n2-k2.The purpose of this study is to obtain approximation properties of the bivariate generalization of (p,q)-Bernstein-Durrmeyer operators defined by (11). We may discuss the properties elsewhere.


## 4. Conclusion

In the paper, we have proposed Bernstein-Durrmeyer type operators based on some certain variants. In the case whenα=β=0, our operators reduce to the acclaimed one as defined in [21]. We have derived some approximation properties of Bernstein-Durrmeyer type operators. From those properties, we have estimated its local approximation and Voronoskaja type asymptotic formula. Finally, we have shown some comparisons and illustrative graphs for the convergence of these operators by using the software MATLAB.



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*Source: 5316150-2017-05-30.xml*