Approximation Methods: Theory and ApplicationsView this Special Issue
Research Article | Open Access
Asif Khan, M. S. Mansoori, Khalid Khan, M. Mursaleen, "Phillips-Type -Bernstein Operators on Triangles", Journal of Function Spaces, vol. 2021, Article ID 6637893, 13 pages, 2021. https://doi.org/10.1155/2021/6637893
Phillips-Type -Bernstein Operators on Triangles
The purpose of the paper is to introduce a new analogue of Phillips-type Bernstein operators and , their products and , their Boolean sums and on triangle , 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 provides flexibility for approximation and reduces to its classical case for .
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 of -calculus (quantum analogue), the first -analogue of Bernstein operators (rational) was constructed by Lupa in . In 1997, Phillips  initiated another generalization of Bernstein polynomials based on the -integers (quantum analogue) called -Bernstein polynomials. The -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 .
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  etc. Starting with the paper  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 Phillips -analogue of the Bernstein operators and obtain new results using -analogue on triangles. To present results by Phillips, we recall the following definitions. For other relevant works, one can see .
Let . For any the -integer is defined by and the -factorial by
For integers , the -binomial or the Gaussian coefficient is defined by
Clearly, for ,
The -binomial coefficients are involved in Cauchy’s -binomial theorem (cf. , Chapter 10, Section 10.2). The first one is a -analogue as an extension to Newton’s binomial formula:
Following Phillips, we denote
It follows from (6) that for integers . These recurrence relations are satisfied by -binomial coefficients when , both the relations reduce to the Pascal identity. In the next section, we construct quantum analogue of operators studied in  on triangles.
2. Construction of New Univariate Operators on Triangle
In , the authors considered only the standard triangle sufficient due to affine invariance as
Let and be uniform partitions of the intervals and , respectively.
In 2009, they  constructed some univariant Bernstein-type operators on triangle as follows: where respectively.
Consider a real-valued function defined on as done in . Through the point , one considers the parallel lines to the coordinate axes which intersect the edges of the triangle at the points and , respectively and (, Figure 1).
Let and be uniform partitions of the intervals and , respectively.
We define the new Phillips-type Bernstein operators and on triangle by using quantum calculus as follows: where respectively. These operators reduce to Phillips-type operator on . 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 . In case , corresponding operators reduce to its classical case on triangles. Now, we generalize various results of  in quantum calculus frame.
For the sake of convenience, we use the following notation onwards:
Theorem 1. If is a real-valued function defined on , then (i)(ii)(iii)where and dex is the degree of exactness of the operator .
Proof. By definition, . So we will calculate the moments only on . The interpolation property follows from the relations Regarding the property , we have or equivalently,
Remark 2. In the same way, it can be proved that if is a real-valued function defined on , then (i)(ii)(iii)Based on the following approximation formula we present the following results.
Theorem 3. If , then
where modulus of continuity of the function with respect to the variable is denoted by
Further, if , then
Proof. Since by definition, and hence remainder will be zero at due to interpolation. We have Since one obtains As it follows that For , we obtain
Theorem 4. If , then where
Proof. As , by Peano’s theorem, one obtains
where the kernel
does not change the sign By the Mean Value Theorem, it follows that
After an easy calculation, we get
By using it in Equation (32), we get
Remark 5. From (32), it follows that (i)if is a concave function, then , i.e.,(ii)if is a convex function, then , i.e.,for and .
Remark 6. For the remainder of the approximation formula We also have the following: (A)If , thenAnd for , (B)If , thenwhere
3. Product Operators
Let and be the products of operators and .
Remark 7. The nodes of the operator are the -analogue of the nodes, which are given in , Figure 2, for , and .
Theorem 8. The product operator satisfies the following relations:
(i)(ii)(iii)The above proofs follow from some simple computation.
The property or implies that .
Remark 9. The product operator interpolates the function at the vertex and on the hypotenuse of the triangle .
The product operator , given by has the nodes, which are -analogue of nodes given in , Figure 3, for , , , and the properties: (i)(ii)(iii)Let us consider the approximation formula
Theorem 10. If and , then
Proof. We have After some transformations, one obtains while It follows Since We have
4. Boolean Sum Operators
be the Boolean sums of the Phillips-type Bernstein operators and .
Theorem 11. For the real-valued function defined on , we have
Proof. We have
The interpolation properties of , together with properties (i)–(iii) of the operator imply that
for all .
Let be the remainder of the Boolean sum approximation formula
Theorem 12. If , then for all .
Remark 13. Analogous relations can be obtained for the remainders of the product approximation formula and for the Boolean sum formula
5. Graphical Analysis
Let us consider a function for graphical analysis. In Figure 1(a), we have presented the graph of function