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.

In this article, we purpose to obtain several approximation properties of Sz\'{a}sz-Mirakjan-Kantorovich operators with shape parameter $\lambda \in\lbrack-1,1]$. We compute some preliminaries results such as moments and central moments for these operators. Next, we derive the Korovkin type convergence theorem, estimate the degree of convergence with respect to the moduli of continuity, for the functions belong to Lipschitz-type class and Peetre's $K$-functional, respectively. Further, we investigate Voronovskaya type asymptotic theorem and give the comparison of the convergence of these newly defined operators to the certain functions with some graphics.

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In this paper, we study several approximation properties of Szasz-Mirakjan-Durrmeyer operators with shape parameter λ∈[−1,1]λ∈[−1,1]. Firstly, we obtain some preliminaries results such as moments and central moments. Next, we estimate the order of convergence in terms of the usual modulus of continuity, for the functions belong to Lipschitz type class and Peetre's K-functional, respectively. Also, we prove a Korovkin type approximation theorem on weighted spaces and derive a Voronovskaya type asymptotic theorem for these operators. Finally, we give the comparison of the convergence of these newly defined operators to the certain functions with some graphics and error of approximation table.

The main concern of this article is to acquire some approximation properties of a new class of Bernstein polynomials based on Bézier basis functions with shape parameter λ ∈ [−1,1] . We prove Korovkin type approximation theorem and estimate the degree of convergence in terms of the modulus of continuity, for the functions belong to Lipschitz type class and Peetre's K -functional, respectively. Additionally, with the help of Maple software, we present the comparison of the convergence of newly defined operators to the certain functions with some graphical illustrations and error estimation tables. Also, we conclude that the error estimation of our newly defined operators in some cases is better than classical Bernstein operators [3], Cai et al. [4] and Izgi [10].