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

Abstract

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.

A Note on Approximation of Blending Type Bernstein–Schurer–Kantorovich Operators with Shape Parameter α

“…Mohiuddine et al [6], Acu et al [7], İçöz and Çekim [8,9], and Kajla and Micláus [10,11] constructed new sequences of linear positive operators to investigate the rapidity of convergence and order of approximation in diferent functional spaces in terms of several generating functions. Some other researchers developed many other useful operators [6,[12][13][14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30] in the same feld. In the recent past, for g ∈ [0, 1], m ∈ N and α ∈ [−1, 1], Chen et al [31] constructed a sequence of new linear positive operators as…”

On One- and Two-Dimensional α–Stancu–Schurer–Kantorovich Operators and Their Approximation Properties

“…In [13], the authors investigated various pointwise and uniform approximation results. Furthermore, many researchers, e.g., Kilicman et al [15], Acar et al [16], Aral et al [17], Cai et al [18,19], Çetin et al [20,21], Mohiuddine et al [22], Aslan et al [23,24], Acu et al [25], Agrawal [26], Nasiruzzaman et al [27], and Ayman-Mursaleen et al [28,29], have intensively studied α-Bernstein operators and their modifications for better approximation results. In [30][31][32] some interesting studies have been carried out.…”