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

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.

Approximation by Szasz-Mirakjan-Durrmeyer operators based on shape parameter $\lambda$

“…For the λ−Szász-Mirakjan operators S m,λ (µ; y), following results are satisfied: Lemma 2. For the operators defined by (6), we obtain the following moments…”

“…Proof. Taking into consideration the linearity and monotonicity properties of the operators (6), it gives…”

“…For the operators defined by (4), they studied some theorems such as Korovkin type convergence, local approximation, Lipschitz type convergence, Voronovskaja and Grüss-Voronovskaja type. Also, we refer some recent works based on shape parameter λ ∈ [−1, 1], see details: [5,6,8,[19][20][21][22][23][24][26][27][28].…”

“…5, we establish a Voronovskaya type asymptotic theorem. Finally, with the aid of Maple software, we present the comparison of the convergence of operators (6) to certain functions with some graphics and error of approximation table.…”

A Novel Generalization of Bézier-like Curves and Surfaces with Shape Parameters

“…They numerically analyzed the theoretical results and gave some computer graphics to understand the importance of this study. With the help of new Bernstein basis functions, Cai et al [30] generalized the q-Bernstein polynomials and gave many approximation properties of q-Bernstein polynomials with shape parameter λ on the symmetric interval [−1, 1]. Hiemstra et al [31] presented an idea to show how to calculate a normalized B-spline-like basis for spline spaces with pieces derived from extended Tchebycheff spaces in an efficient and reliable way.…”