Construction of Stancu-Type Bernstein Operators Based on Bézier Bases with Shape Parameter λ

Abstract

We construct Stancu-type Bernstein operators based on Bézier bases with shape parameter λ ∈ [ - 1 , 1 ] and calculate their moments. The uniform convergence of the operator and global approximation result by means of Ditzian-Totik modulus of smoothness are established. Also, we establish the direct approximation theorem with the help of second order modulus of smoothness, calculate the rate of convergence via Lipschitz-type function, and discuss the Voronovskaja-type approximation theorems. Finally, in the last section, we construct the bivariate case of Stancu-type λ -Bernstein operators and study their approximation behaviors.

Approximation Properties of an Extended Family of the Szász–Mirakjan Beta-Type Operators

“…Proof. The assertions (12) and (13) of Lemma 2 follow easily from those of Lemma 1, so we omit the details involved.…”

“…There are several integral and other modifications, variations, and basic (or q-) extensions of the Szász-Mirakjan-type operators. These include the Bézier, Kantorovich, Durrmeyer, and other types of modifications and extensions of the Szász-Mirakjan operators (see, for details, [3][4][5][6][7][8][9][10][11][12][13][14][15]). In particular, Gupta and Noor [6] introduced an integral modification of the Szász-Mirakjan operators in Equation 1by considering a weight function in terms of the Beta basis functions as given below.…”

Weighted statistical approximation properties of univariate and bivariate λ-Kantorovich operators

“…They established uniform convergence of the operators and global approximation result by means of Ditzian-Totik modulus of smoothness. they also constructed the bivariate case of Stancu-type λ-Bernstein operators and studied their approximation behaviors [9]. This paper is divided into five main sections.…”

Approximation on parametric extension of Baskakov–Durrmeyer operators on weighted spaces

“…where S n,k (x) = n k x k (1x) n-k . Many mathematicians researched in this direction and studied various modifications in several functional spaces using different error optimization techniques, i.e., Acar et al [2][3][4][5][6][7], Acu et al [8,9], Barbosu [10], Agrawal et al [11], Aral [12], Mursaleen et al [13][14][15][16][17], Srivastava et al [18][19][20]; for more details see also the references therein and [21][22][23][24][25][26][27][28][29][30].…”