Some Approximation Results on $\lambda-$ Szasz-Mirakjan-Kantorovich Operators

Abstract

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.

On a Family of Parameter-Based Bernstein Type Operators with Shape-Preserving Properties

“…Proof. By combining equations ( 11) and ( 12) with the linearity property (8), it is an evident fact.…”

“…Some of the most famous cases are the Schurer polynomials, Kantorovich polynomials, Stancu polynomials, q-Bernstein polynomials, Durrmeyer polynomials, Favard-Szász-Mirakyan operators, Baskakov operators, and numerous others [3][4][5][6]. Some of the recent advances could be traced in [7][8][9][10][11].…”

Statistical Blending-Type Approximation by a Class of Operators That Includes Shape Parameters λ and α

“…Recently, shape parameters α and λ were used to extend Bernstein operators to α-Bernstein type (see [5,6,8,[11][12][13][14][15][16][17][18]) and λ-Bernstein type operators (see [9,[19][20][21][22][23][24][25][26]) in order to approximate functions better, respectively. The convergence of linear positive operators by the shape parameters α and λ in the space of continuous functions of two variables was studied in [27][28][29] and [22,30,31], respectively.…”

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

“…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].…”

“…where s m,j (λ; y) (j = 0, 1, ..∞) given in (5) and λ ∈ [−1, 1]. This work is organized as follows: In Sect.…”