Phillips-Type 
 q
 -Bernstein Operators on Triangles

<jats:p>The purpose of the paper is to introduce a new analogue of Phillips-type Bernstein operators <jats:inline-formula>
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                  </jats:inline-formula>, 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 <jats:inline-formula>
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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 computeraided 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 of q-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][7][8][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][13][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][16][17][18][19][20][21][22][23][24][25][26][27][28].
In this paper, we would like to draw attention to the Phillips q-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].
Let q > 0. For any m = 0, 1, 2, ⋯, the q-integer ½m q is defined by and the q-factorial ½m q ! by For integers 0 ≤ i ≤ m, the q-binomial or the Gaussian coefficient is defined by Clearly, for q = 1, The q-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: Following Phillips, we denote It follows from (6) that for integers k ≥ i ≥ 0. These recurrence relations are satisfied by q-binomial coefficients 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.

Construction of New Univariate Operators on Triangle
In [31], the authors considered only the standard triangle sufficient due to affine invariance as Let Δ u m = fiððh − vÞ/mÞ, i = 0, mg and Δ v n = fjððh − uÞ/nÞ, j = 0, ng be uniform partitions of the intervals ½0, h − v and ½0, h − u, respectively.
In 2009, they [31] constructed some univariant Bernstein-type operators on triangle T h as follows: where respectively. Consider a real-valued function f defined on T h as done in [31]. Through the point ðu, vÞ ∈ T h , one considers the parallel lines to the coordinate axes which intersect the edges Γ i , i = 1, 2, 3, of the triangle at the points ð0, vÞ and ðh − v, vÞ, respectively ðu, 0Þ and ðu, h − uÞ ([31], Figure 1).
We define the new Phillips-type Bernstein operators B u m,q and B v n,q on triangle by using quantum calculus as follows: Journal of Function Spaces respectively. These operators reduce to Phillips-type operator on ½0, 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: Theorem 1. If f is a real-valued function defined on T h , then where e ij ðu, vÞ = u i v j and dex ðB u m,q Þ is the degree of exactness of the operator B u m,q .
Proof. By definition, ðB u m,q f Þð0, hÞ = f ð0, hÞ. So we will calculate the moments only on T h \ ð0, hÞ. The interpolation property ðiÞ follows from the relations Regarding the property ðiiÞ, we have or equivalently,

Journal of Function Spaces
Remark 2. In the same way, it can be proved that if f is a realvalued function defined on T h , then Based on the following approximation formula we present the following results.
where modulus of continuity of the function f with respect to the variable u is denoted by wð f ð:,vÞ ; δÞ: Proof. Since by definition, ðB u m,q f Þð0, hÞ = f ð0, hÞ and hence remainder will be zero at ð0, hÞ due to interpolation. We have Since one obtains As it follows that Theorem 4. If f ð:,vÞ ∈ C 2 ½0, h, then where Proof. As dexðB u m,q Þ = 1, by Peano's theorem, one obtains Journal of Function Spaces where the kernel does not change the signðK 20 ðu, v ; tÞ ≤ 0, u ∈ ½0, h − vÞ: By the Mean Value Theorem, it follows that After an easy calculation, we get where ξ ∈ ½0, h − v. By using it in Equation (32), we get (ii) if f ð:,vÞ is a convex function, then ðR u m,q f Þðu, vÞ ≤ 0, i.e., for u ∈ ½0, h − v and v ∈ ½0, h.

Remark 6. For the remainder R v n,q f of the approximation formula
We also have the following: ð43Þ where

Product Operators
Let P mn,q = B u m,q B v n,q and Q mn,q = B v n,q B u m,q be the products of operators B u m,q and B v n,q . We have

Journal of Function Spaces
Remark 7. The nodes of the operator P mn,q are the q-analogue of the nodes, which are given in [31], Figure 2, for i = 0, m ; j = 0, n, and v ∈ ½0, h.
Theorem 8. The product operator P mn,q satisfies the following relations: The above proofs follow from some simple computation. The property ðiÞ or ðiiÞ implies that ðP mn,q f Þð0, 0Þ = f ð0, 0Þ.
Remark 9. The product operator P mn,q interpolates the function f at the vertex ð0, 0Þ and on the hypotenuse u + v = h of the triangle T h .
The product operator Q mn,q , given by has the nodes, which are q-analogue of nodes given in [31], Figure 3, for i = 0, m, j = 0, n, u ∈ ½0, h, and the properties: Let us consider the approximation formula Theorem 10. If f ∈ CðT h Þ and 0 < q ≤ 1, then Proof. We have After some transformations, one obtains It follows Journal of Function Spaces     Proof. We have

Boolean Sum Operators
The interpolation properties of B u m,q ,B v n,q together with properties (i)-(iii) of the operator P mn,q imply that for all u, v ∈ ½0, h. Let R S mn,q f be the remainder of the Boolean sum approximation formula Theorem 12. If f ∈ CðT h Þ, then for all ðu, vÞ ∈ T h .
Remark 13. Analogous relations can be obtained for the remainders of the product approximation formula and for the Boolean sum formula

Graphical Analysis
Let us consider a function for graphical analysis. In Figure 1(a), we have presented the graph of function f ðu, vÞ = sin ð10uÞ + cos ð5vÞ on triangular domain. The graph of Phillips Bernstein operator B u m,q f based on quantum analogue on triangular domain is shown in Figure 1(b). Similarly, other operators B v n,q f , P mn,q f , and S mn,q f 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.
Also from these figures, one can observe that operator is approximating function better with increasing values of m and n and by fixing q on triangular domain. 12 Journal of Function Spaces Thus, we have constructed Phillips-type q-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 parameter q in Lupas Bernstein operators, we have more modeling flexibility for approximation on triangular domain.

Data Availability
No data are available.

Conflicts of Interest
The authors declare that they have no competing interests.