# Phillips-Typeq-Bernstein Operators on Triangles

Authors: Asif Khan; M. S. Mansoori; Khalid Khan; M. Mursaleen
Journal: Journal of Function Spaces
 (2021)

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## Abstract

The purpose of the paper is to introduce a new analogue of Phillips-type Bernstein operatorsBm,qufu,v and Bn,qvfu,v, their products Pmn,qfu,v and Qnm,qfu,v, their Boolean sums Smn,qfu,v and Tnm,qfu,v on triangle Th, which interpolate a given function on the edges, respectively, at the vertices of triangle using quantum analogue. Based on Peano’s theorem and using modulus of continuity, the remainders of the approximation formula of corresponding operators are evaluated. Graphical representations are added to demonstrate consistency to theoretical findings. It has been shown that parameter q provides flexibility for approximation and reduces to its classical case for q=1.


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## Body

## 1. Introduction and Essential Preliminaries

In 1912, Bernstein constructed polynomials to provide a constructive proof of the Weierstrass approximation theorem [1, 2] using probabilistic interpolation, which is now known as Bernstein polynomials in approximation theory. In computer-aided geometric design (CAGD), the basis of Bernstein polynomials plays a significant role to preserve the shape of the curves and surfaces.Further, with the development ofq-calculus (quantum analogue), the first q-analogue of Bernstein operators (rational) was constructed by Lupas in [3]. In 1997, Phillips [4] initiated another generalization of Bernstein polynomials based on the q-integers (quantum analogue) called q-Bernstein polynomials. The q-Bernstein polynomials attracted a lot of attention and were studied broadly by several researchers. One can find a survey of the obtained results and references on the subject in [5].Computer-aided geometric design (CAGD) is a discipline which deals with computational aspects of geometric objects. It emphasizes on the mathematical development of curves and surfaces such that it becomes compatible with computers. Popular programs, like Adobe’s Illustrator and Flash, and font imaging systems, such as Postscript, utilize Bernstein polynomials to form what are known as Bézier curves [6–9].The approximating operators on triangles and their basis have important applications in finite element analysis and computer-aided geometric design [10] etc. Starting with the paper [11] of Barnhill et al., the blending interpolation operators were considered in the papers [12–14].In this paper, we construct new operators based on quantum analogue of Phillips. Bernstein-type operators also interpolate the value of a given function on the boundary of the triangle. Also, we will discuss some particular cases. Using modulus of continuity and Peano’s theorem, the remainders of the corresponding approximation formulas are evaluated. The accuracy of the approximation is also illustrated by graphics of given functions with suitable Bernstein-type approximation. For more information regarding such operators, their properties and their remainders one can refer to [15–28].In this paper, we would like to draw attention to the Phillipsq-analogue of the Bernstein operators and obtain new results using q-analogue on triangles. To present results by Phillips, we recall the following definitions. For other relevant works, one can see [29].Letq>0. For any m=0,1,2,⋯, the q-integer mq is defined by
(1)mq≔1+q+⋯+qm−1,m=1,2,⋯,0q≔0,and the q-factorial mq! by
(2)mq!≔1q2q⋯mq,m=1,2,⋯,0q!=1.For integers0≤i≤m, the q-binomial or the Gaussian coefficient is defined by
(3)miq≔mq!iq!m−iq!.Clearly, forq=1,
(4)m1=m,m1!=m!,mi1=mi.Theq-binomial coefficients are involved in Cauchy’s q-binomial theorem (cf. [30], Chapter 10, Section 10.2). The first one is a q-analogue as an extension to Newton’s binomial formula:
(5)au+bvqm≔∑i=0mqii−1/2miqam−ibium−ivi,(6)1+u1+qu··⋯1+qm−1u=∑i=0mmiqqii−1/2ui.Following Phillips, we denote(7)bm,iu,v=miq∏s=0m−i−11−qsu.It follows from (6) that
(8)∑i=0mbm,iq;u=1,u∈0,1,for integers k≥i≥0. These recurrence relations are satisfied by q-binomial coefficients
(9)k+1iq=qk−i+1ki−1q+kiq,k+1iq=ki−1q+qikiq,when q=1, both the relations reduce to the Pascal identity. In the next section, we construct quantum analogue of operators studied in [31] on triangles.


## 2. Construction of New Univariate Operators on Triangle

In [31], the authors considered only the standard triangle sufficient due to affine invariance as
(10)Th=u,v∈ℝ2∣u≥0,v≥0,u+v≤h,forh>0.LetΔmu=ih−v/m,i=0,m¯ and Δnv=jh−u/n,j=0,n¯ be uniform partitions of the intervals 0,h−v and 0,h−u, respectively.In 2009, they [31] constructed some univariant Bernstein-type operators on triangle Th as follows:
(11)Bmufu,v=∑i=0mpm,iu,vfimh−v,v,Bnvfu,v=∑j=0nqn,ju,vfu,jnh−u,where
(12)pm,iu,v=miuih−u−vm−ih−vm,0≤u+v≤h,(13)qn,ju,v=njvjh−u−vn−jh−un,0≤u+v≤h,respectively.Consider a real-valued functionf defined on Th as done in [31]. Through the point u,v∈Th, one considers the parallel lines to the coordinate axes which intersect the edges Γi,i=1,2,3, of the triangle at the points 0,v and h−v,v, respectively u,0 and u,h−u ([31], Figure 1).LetΔmu=ih−v/m,i=0,m¯ and Δnv=jh−u/n,j=0,n¯ be uniform partitions of the intervals 0,h−v and 0,h−u, respectively.We define the new Phillips-type Bernstein operatorsBm,qu and Bn,qv on triangle by using quantum calculus as follows:
(14)Bm,qufu,v=∑i=0mp~m,iu,vfiqmqh−v,v,u,v∈Th\0,h,f0,h,0,h∈Th,Bn,qvfu,v=∑j=0nq~n,ju,vfu,jqnqh−u,u,v∈Th\h,0,fh,0,h,0∈Th,where
(15)p~m,iu,v=miqui∏s=0m−i−1h−v−qsuh−vm,0≤u+v≤hexceptthepoint0,h,(16)q~n,ju,v=njqvj∏t=0n−j−1h−u−qtvh−un,0≤u+v≤hexceptthepoint0,h,respectively. These operators reduce to Phillips-type operator on 0,1. One can note that the bases (15) and (16) of the operators constructed using quantum calculus are different from the bases (12) and (13) of the operators constructed by Blaga and Coman [31]. In case q=1, corresponding operators reduce to its classical case on triangles. Now, we generalize various results of [31] in quantum calculus frame.For the sake of convenience, we use the following notation onwards:(17)h−vm≔∑i=0mmiqui∏s=0m−i−1h−v−qsu,h−un≔∑i=0nniqvi∏s=0n−i−1h−u−qsv.Theorem 1.
Iff is a real-valued function defined on Th, then
(i)
Bm,quf=fonΓ2∪Γ3(ii)
Bm,quei0u,v=ui,i=0,1dexBm,qu=1(iii)
Bm,que20u,v=u2+uh−u−v/mq(18)Bm,queiju,v=vjui,i=0,1,j∈N,vju2+uh−u−vmq,i=2,j∈ℕ,where eiju,v=uivj and dex Bm,qu is the degree of exactness of the operator Bm,qu.Proof.
By definition,Bm,quf0,h=f0,h. So we will calculate the moments only on Th\0,h. The interpolation property i follows from the relations
(19)p~m,i0,v=1,ifi=0,0,i≠0,p~m,ih−v,v=1,ifi=m,0,i≠m.
Regarding the propertyii, we have
(20)Bm,que00u,v=∑i=0mmiqui∏s=0m−i−1h−v−qsuh−vm=h−vmh−vm=1,Bm,que10u,v=∑i=0mmiqui∏s=0m−i−1h−v−qsuh−vmiqmqh−v=∑i=0miq/mqmiqui∏s=0m−i−1h−v−qsuh−vm−1=∑i=0m−1m−1iqui+1∏s=0m−i−2h−v−qsuh−vm−1=u∑i=0m−1m−1iqui∏s=0m−1−i−1h−v−qsuh−vm−1=u,Bm,que20u,v=∑i=0mmiqui∏s=0m−i−1h−v−qsuh−vmiq2mq2h−v2=h−v2∑i=0m−1i+1q/mqmiqui+1∏s=0m−i−2h−v−qsuh−vm=h−v2u∑i=0m−11+qiq/mqmiqui∏s=0m−i−2h−v−qsuh−vm=h−vumq∑i=0m−1miqui∏s=0m−1−i−1h−v−qsuh−vm−1+h−v2u∑i=0m−1qm−1q/mqiq/m−1qm−1iqui∏s=0m−i−2h−v−qsuh−vm=h−vumq+qm−1qu2mq∑i=0m−2m−2iqui∏s=0m−2−i−1h−v−qsuh−vm−2,Bm,que20u,v=h−vumq+qm−1qu2mq,or equivalently,
(21)Bm,que20u,v=h−vumq+u21−1mq=u2+uh−u−vmq.Remark 2.
In the same way, it can be proved that iff is a real-valued function defined on Th, then
(i)
Bn,qvf=fonΓ1∪Γ3(ii)
Bn,qve0ju,v=vj,j=0,1dexBn,qv=1(iii)
Bn,qve02u,v=v2+vh−u−v/nq(22)Bn,qveiju,v=uivj,j=0,1,i∈N,uiv2+vh−u−vnq,j=2,i∈ℕ.
Based on the following approximation formula(23)f=Bm,quf+Rm,quf,we present the following results.Theorem 3.
Iff.,v∈C0,h−v, then
(24)Rm,qufu,v≤1+h2δmqwf.,v;δ,v∈0,h,
where modulus of continuity of the functionf with respect to the variable u is denoted by wf.,v;δ.
Further, ifδ=1/mq, then
(25)∣Rm,qufu,v∣≤1+h2wf.,v;1mq,v∈0,h.Proof.
Since by definition,Bm,quf0,h=f0,h and hence remainder will be zero at 0,h due to interpolation. We have
(26)Rm,qufu,v≤∑i=0mp~m,iu,vfu,v−fiqh−vmq,v.
Since(27)∣fu,v−fiqh−vmq,v∣≤1δ∣u−iqh−vmq∣+1wf.,v;δ,
one obtains(28)∣Rm,qufu,v∣≤∑i=0mp~m,iu,v1δ∣u−iqh−vmq∣+1wf.,v;δ≤1+1δ∑i=0mp~m,iu,vu−iqh−vmq21/2wf.,v;δ=1+1δuh−u−vmqwf.,v;δ.
As(29)maxThuh−u−v=h24,it follows that
(30)∣Rm,qufu,v∣≤1+h2δmqwf.,v;δ.
Forδ=1/mq, we obtain
(31)∣Rm,qufu,v∣≤1+h2wf.,v;1mq.Theorem 4.
Iff.,v∈C20,h, then
(32)Rm,qufu,v=−uh−u−v2mqf2,0ξ,y,ξ∈0,h−v,(33)Rm,qufu,v≤h28mqM20f,u,v∈Th,where
(34)Mijf=maxThfi,ju,v.Proof.
AsdexBm,qu=1, by Peano’s theorem, one obtains
(35)Rm,qufu,v=∫0h−vK20u,v;tf2,0t,vdt,where the kernel
(36)K20u,v;t≔Rm,quu−t+=u−t+−∑i=0mp~m,iu,viqh−vmq−t+does not change the signK20u,v;t≤0,u∈0,h−v. By the Mean Value Theorem, it follows that
(37)Rm,qufu,v=f2,0ξ,v∫0h−vK20u,v;tdt,ξ∈0,h−v.
After an easy calculation, we get(38)Rm,qufu,v=−uh−u−v2mqf2,0ξ,v,where ξ∈0,h−v.
By using it in Equation (32), we get
(39)∣Rm,qufu,v∣≤h28mqM20f,u,v∈Th.Remark 5.
From (32), it follows that
(i)
iff.,v is a concave function, then Rm,qufu,v≥0, i.e.,(40)Bm,qufu,v≤fu,v,(ii)
iff.,v is a convex function, then Rm,qufu,v≤0, i.e.,(41)Bm,qufu,v≥fu,v,
foru∈0,h−v and v∈0,h.Remark 6.
For the remainderRn,qvf of the approximation formula
(42)f=Bn,qvf+Rn,qvf.
We also have the following:(A)
Iffu,.∈C0,h−u, then(43)∣Rn,qvfu,v∣≤1+h2δnqwfu,.;δ,u∈0,h.
And forδ=1/nq,
(44)∣Rn,qvfu,v∣≤1+h2wfu,.;1nq,u∈0,h.(B)
Iffu,.∈C20,h, then(45)Rn,qvfu,v=−vh−u−v2nqf0,2u,η,η∈0,h−u,∣Rn,qvfu,v∣≤h28nqM02f,u,v∈Th,where
(46)Mijf=maxThfi,ju,v.


## 3. Product Operators

LetPmn,q=Bm,quBn,qv and Qmn,q=Bn,qvBm,qu be the products of operators Bm,qu and Bn,qv.We have(47)Pmn,qfu,v=∑i=0m∑j=0np~m,iu,vq~n,jiqh−vmq,vfiqh−vmq,jqmq−iqh+iqvmqnq,u,v∈Th\0,h,h,0,f0,h,0,h∈Th,fh,0,h,0∈Th.Remark 7.
The nodes of the operatorPmn,q are the q-analogue of the nodes, which are given in [31], Figure 2, for i=0,m¯;j=0,n¯, and v∈0,h.Theorem 8.
The product operatorPmn,q satisfies the following relations:
(i)
Pmn,qfu,0=Bm,qufu,0(ii)
Pmn,qf0,v=Bn,qvf0,v(iii)
Pmn,qfu,h−u=fu,h−u,u,v∈0,h
The above proofs follow from some simple computation.
The propertyi or ii implies that Pmn,qf0,0=f0,0.Remark 9.
The product operatorPmn,q interpolates the function f at the vertex 0,0 and on the hypotenuse u+v=h of the triangle Th.
The product operatorQmn,q, given by
(48)Qnm,qfu,v=∑i=0m∑j=0np~m,iu,jqh−unqq~n,ju,vfiqnq−jqh+jqumqnq,jqh−unq,u,v∈Th\0,h,h,0,f0,h,0,h∈Th,fh,0,h,0∈Th,has the nodes, which are q-analogue of nodes given in [31], Figure 3, for i=0,m¯, j=0,n¯, u∈0,h, and the properties:
(i)
Qnm,qfu,0=Bm,qufu,0(ii)
Qnm,qf0,v=Bn,qvf0,v(iii)
Qnm,qfh−v,v=fh−v,v,u,v∈0,h
Let us consider the approximation formula(49)f=Pmn,qf+Rmn,qPf.Theorem 10.
Iff∈CTh and 0<q≤1, then
(50)∣Rmn,qPfu,v∣≤1+hwf;1mq,1nq,u,v∈Th.Proof.
We have(51)∣Rmn,qPfu,v∣≤1δ1∑i=0m∑j=0np~m,iu,vq~n,jiqh−vmq,v∣u−iqh−vmq∣+1δ2∑i=0m∑j=0np~m,iu,vq~n,jiqh−vmq,v∣v−jqmq−iqh+iqvmqnq∣+∑i=0m∑j=0np~m,iu,vq~n,jiqh−vmq,vwf;δ1,δ2.
After some transformations, one obtains(52)∑i=0m∑j=0np~m,iu,vq~n,jiqh−vmq,vu−iqh−vmq≤uh−u−vmq,∑i=0m∑j=0np~m,iu,vq~n,jiqh−vmq,vv−jqmq−iqh+iqvmqnq≤vh−u−vnq,while
(53)∑i=0m∑j=0np~m,iu,vq~n,jiqh−vmq,v=1.
It follows(54)∣Rmn,qPfu,v∣≤1δ1uh−u−vmq+1δ2vh−u−vnq+1wf;δ1,δ2.
Since(55)uh−u−vmq≤h24mq,vh−u−vnq≤h24nq,forallu,v∈Th.We have
(56)∣Rmn,qPfu,v∣≤h2δ1mq+h2δ2nq+1wf;δ1,δ2∣Rmn,qPfu,v∣≤1+hwf;1mq,1nq.


## 4. Boolean Sum Operators

Let(57)Smn,q≔Bm,qu⨁Bn,qv=Bm,qu+Bn,qv−Bm,quBn,qv,Tnm,q≔Bn,qv⨁Bm,qu=Bn,qv+Bm,qu−Bn,qvBm,qu,be the Boolean sums of the Phillips-type Bernstein operatorsBm,qu and Bn,qv.Theorem 11.
For the real-valued functionf defined on Th, we have
(58)Smn,qf∂Th=f∂Th.Proof.
We have(59)Smn,qf=Bm,qu+Bn,qv−Bm,quBn,qvf.
The interpolation properties ofBm,qu,Bn,qv together with properties (i)–(iii) of the operator Pmn,q imply that
(60)Smn,qfu,0=Bm,qufu,0+fu,0−Bm,qufu,0=fu,0,Smn,qf0,v=f0,v−Bn,qvf0,v+Bn,qvf0,v=f0,v,Smn,qfu,h−u=fu,h−u+fu,h−u−fu,h−u=fu,h−u,
for allu,v∈0,h.
LetRmn,qSf be the remainder of the Boolean sum approximation formula
(61)f=Smn,qf+Rmn,qSf.Theorem 12.
Iff∈CTh, then
(62)Rmn,qSfu,v≤1+h2wf.,v;1mq+1+h2wfu,.;1nq+1+hwf;1mq,1nq,
for allu,v∈Th.Proof.
From the equality(63)f−Smn,qf=f−Bm,quf+f−Bn,qvf−f−Pmn,qf,we get
(64)Rmn,qSfu,v≤Rm,qufu,v+Rn,qvfu,v+Rmn,qPfu,v.
Now, from (25), (44), and (50), we follow the proof (62).Remark 13.
Analogous relations can be obtained for the remainders of the product approximation formula(65)f=Qnm,qf+Rnm,qQf=Bn,qvBm,quf+Rnm,qQf,and for the Boolean sum formula
(66)f=Tnm,qf+Rnm,qTf=Bn,qv⊕Bm,quf+Rnm,qTf.


## 5. Graphical Analysis

Let us consider a function for graphical analysis. In Figure1(a), we have presented the graph of function fu,v=sin10u+cos5v on triangular domain. The graph of Phillips Bernstein operator Bm,quf based on quantum analogue on triangular domain is shown in Figure 1(b). Similarly, other operators Bn,qvf, Pmn,qf, and Smn,qf approximating function are shown in Figures 1(c)–1(e) for various values of q, m, n, and h. One can observe from Figures 1–5 that operators are approximating function better as q approaches to 1 for fixed value of m and n.Figure 1
OperatorsBm,quf, Bn,qvf, Pmn,qf, and Smn,qf approximating function on triangular domain for h=1, m=6, n=6, and q=0.70.
(a)
fu,v=sin10u+cos5v(b)
OperatorBm,quf(c)
Bn,qvf(d)
Pmn,qf(e)
Smn,qfFigure 2
OperatorsBm,quf, Bn,qvf, Pmn,qf, and Smn,qf approximating function on triangular domain for h=1, m=6, n=6, and q=0.99.
(a)
fu,v=sin10u+cos5v(b)
OperatorBm,quf(c)
Bn,qvf(d)
Pmn,qf(e)
Smn,qfFigure 3
OperatorsBm,quf, Bn,qvf, Pmn,qf, and Smn,qf approximating function on triangular domain for h=1, m=15, n=15, and q=0.70.
(a)
fu,v=sin10u+cos5v(b)
OperatorBm,quf(c)
Bn,qvf(d)
Pmn,qf(e)
Smn,qfFigure 4
OperatorsBm,quf, Bn,qvf, Pmn,qf, and Smn,qf approximating function on triangular domain for h=1, m=15, n=10, and q=0.99.
(a)
fu,v=sin10u+cos5v(b)
OperatorBm,quf(c)
Bn,qvf(d)
Pmn,qf(e)
Smn,qfFigure 5
OperatorsBm,quf, Bn,qvf, Pmn,qf, and Smn,qf approximating function on triangular domain for h=2, m=10, n=10, and q=0.99.
(a)
fu,v=sin10u+cos5v(b)
OperatorBm,quf(c)
Bn,qvf(d)
Pmn,qf(e)
Smn,qfAlso from these figures, one can observe that operator is approximating function better with increasing values ofm and n and by fixing q on triangular domain.Thus, we have constructed Phillips-typeq-Bernstein operators over triangular domain which hold the end point interpolation property on some edges and vertices of triangle.Hence, it can be concluded that after introducing one extra parameterq in Lupas Bernstein operators, we have more modeling flexibility for approximation on triangular domain.



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