On a New Construction of Generalized q -Bernstein Polynomials Based on Shape Parameter λ

: This paper deals with several approximation properties for a new class of q -Bernstein polynomials based on new Bernstein basis functions with shape parameter λ on the symmetric interval [ − 1,1 ] . Firstly, we computed some moments and central moments. Then, we constructed a Korovkin-type convergence theorem, bounding the error in terms of the ordinary modulus of smoothness, providing estimates for Lipschitz-type functions. Finally, with the aid of Maple software, we present the comparison of the convergence of these newly constructed polynomials to the certain functions with some graphical illustrations and error estimation tables.


Introduction
Quantum calculus, briefly q-calculus, has many applications in various disciplines such as mechanics, physics, mathematics, and so on. In approximation theory, a generalization of Bernstein polynomials based on q-calculus was firstly introduced by Lupaş [1]. In 1997, Phillips [2] proposed some convergence theorems and Voronovskaya-type asymptotic formulae for the most popular generalizations of the q-Bernstein polynomials. Now, before proceeding further, let us recall the certain notations on q-calculus (for more details, see [3]). Let 0 < q ≤ 1, for all integers j > 0; the q-integer [j] q is given by: [j] q := 1−q j 1−q , q = 1 j, q = 1 .
Basis functions with desirable properties have an important role in Computer-Aided Geometric Design (CAGD) and computer graphics in order to construct surfaces and curves. The Bernstein polynomial basis was studied and used in many papers. An extensive study about this topic was given by Farouki in the survey paper [18]. These basis functions are widely addressed in many applications areas such as the numerical solution of partial differential equations, CAGD, font design, and 3D modeling. For some applications in CAGD, we refer to [19][20][21][22].
Very recently, the Bernstein basis with shape parameter λ ∈ [−1, 1], which was introduced by Ye et al. [23], has attracted the interest of and in as short a time was studied by a number of researchers.
Inspired by all the above-mentioned works, we now construct the following polynomials with shape parameter λ ∈ [−1, 1] : where: [m+a] q [m+b] q ], 0 < q ≤ 1, and u m,j,a,b (y; q), defined as in (2). Many researchers have investigated iterated Boolean sums of positive operators since these operators have the possibility of accelerating the convergence with respect to the originating positive operators (see [41][42][43]). There are certain important papers in this context in which the authors established global direct, inverse, and saturation results for the convergence of iterated Boolean sums of Bernstein operators to the identity operator. One may obtain further results about the saturation order of approximating polynomials defined in this paper as a future study.
The present work is organized as follows: In Section 2, we calculate some preliminary results such as moments and central moments. In Section 3, we give a Korovkin-type convergence theorem, bind the error in terms of the ordinary modulus of smoothness, and provide estimates for Lipschitz-type functions. In the final section, with the help of the Maple software, we present the comparison of the convergence polynomials (4) with the different values of the a, b, m, q, and λ parameters with some graphs and error estimation tables.

Preliminaries
This section is devoted to calculating some necessary results such as the moments and central moments of polynomials (4).
In case a = b, the polynomials (4) reduce to the λ-Bernstein polynomials based on the q-integers studied by Cai et al. [24].
In case λ = 0 and a = b, the polynomials (4) reduce to the q-Bernstein polynomials introduced by Lupaş [1].
In case λ = 0 and q = 1, the polynomials (4) reduce to the new class of Bernstein polynomials proposed by Izgi [44].

Convergence Results of F m,a,b (µ; λ, q, y)
In what follows, let the sequence q := q m satisfy the conditions given by Remark 2. As is known, the space C[0, ] stands for the real-valued continuous function on [0, |µ(y)|.
In the following theorem, we show the uniform convergence of the polynomials (4).
The proof of (19) follows easily, from Lemmas 2-4. Hence, we obtain the required sequel.
Let us denote the usual moduli of continuity of µ ∈ C[0, ] as follows: Since η > 0, ω(µ; η) has some useful properties (see [46]). Furthermore, we present an element of the Lipschitz continuous function with Lip L (ζ), where L > 0 and 0 < ζ ≤ 1. If the following relation: holds, then one can say a function µ belongs to Lip L (ζ). In the following theorem, we estimate the order of convergence in terms of the usual moduli of continuity. 1], and m > 1. Then, the following inequality verifies: where γ m,a,b (y, q) is the same as in Corollary 1.
In the next theorem, we investigate the order of convergence for the function belonging to the Lipschitz-type class. 1], and m > 1, where γ m,a,b (y, q) is the same as in Corollary 1.

Graphs and Error Estimation Tables
In this section, in order to show the convergence behavior of the polynomials (4), we present some graphs and error of estimation tables for the different values of the a, b, m, q, and λ parameters. Furthermore, we compare the convergence of the polynomials (4) with (1) and (3) to the certain functions. Example 1. Let µ(y) = 1 − sin(3πy) (yellow), λ = 1, a = 0.1, and b = 0.6. In Figure 1, we show the convergence of the polynomials (4) to µ(y) for m = 15, 45, 125 (red, green, purple) and q ∈ (0, 1]. Furthermore, in Figure 2, we illustrate the convergence of the polynomials (4) to µ(y) for λ = −1, and the other parameters are the same as in Figure 1. In Table 1, we estimate the error of the approximation of the polynomials (4) to µ(y) with a = 0.1, b = 0.6, −1 ≤ λ ≤ 1, q ∈ (0, 1], and m = 15, 45, 125, respectively. It is obvious from Table 1 that the absolute difference between the polynomials (4) and µ(y) becomes smaller as the value of m increases. Furthermore, in case λ = 1, the polynomials (4) give better approximation results than in cases λ = 0 and λ = −1.

Conclusions
In this research, we proposed and studied several approximation properties of generalized q-Bernstein polynomials based on Bernstein basis functions with shape parameter λ ∈ [−1, 1]. We discussed a Korovkin-type convergence theorem, as well as the order of convergence concerning the usual modulus of continuity and Lipschitz-type functions. To make our research more intuitive, we considered some graphs and error estimation tables. As a result, the newly constructed polynomials (4) gave better approximation results than the previously studied polynomials (1) and (3) in terms of the values of some selected parameters. Because the shape parameter λ is defined on the symmetric interval [−1, 1], there is more flexibility in constructing curves using this basis function, and at the same time, the corresponding linear polynomials with certain values of the shape parameter λ have better convergence and more flexibility in approximating the related function class.