Abstract
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.
Highlights
Quantum calculus, briefly q-calculus, has many applications in various disciplines such as mechanics, physics, mathematics, and so on
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
Summary
Briefly q-calculus, has many applications in various disciplines such as mechanics, physics, mathematics, and so on. Where the generalized q-Bernstein basis functions um,j,a,b are defined as: um,j,a,b (y; q) = They investigated some approximation properties of polynomials (1) and estimated the order of convergence in terms of the moduli of continuity. 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. Srivastava et al [26] proposed and studied a Stancu variant of λ-Bernstein polynomials and examined the uniform convergence, global approximation result, and Voronovskaya’s approximation theorems. One may obtain further results about the saturation order of approximating polynomials defined in this paper as a future study. With the different values of the a, b, m, q, and λ parameters with some graphs and error estimation tables
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