The Bézier variant of Kantorovich type λ-Bernstein operators

Abstract

In this paper, we introduce the Bézier variant of Kantorovich type λ-Bernstein operators with parameter . We establish a global approximation theorem in terms of second order modulus of continuity and a direct approximation theorem by means of the Ditzian–Totik modulus of smoothness. Finally, we combine the Bojanic–Cheng decomposition method with some analysis techniques to derive an asymptotic estimate on the rate of convergence for some absolutely continuous functions.

Approximation by bivariate generalized Bernstein–Schurer operators and associated GBS operators

“…In addition to this, the Stancu type α-Bernstein-Kantorovich, α-Baskakov and their Kantorovich form, and α-Baskakov-Durrmeyer operators were analyzed by Mohiuddine and Özger [32], Aral et al [6,20], and Mohiuddine et al [31], respectively, and for other blending type operators, see [23,27,36]. Some other modifications of Bernstein operators have been studied in [2,15,16,25,33,34,37,42]. Furthermore, Acar and Kajla [3] gave the bivariate α-Bernstein operators and associated generalized Boolean sum operators and then studied the degree of approximation of their operators.…”

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

“…In [16], Cai et al have introduced Bzier variant of Kantorovich type λ-Bernstein operators. A global approximation theorem in terms of second order modulus of continuity and a direct approximation theorem by means of the DitzianTotik modulus of smoothness were established.…”

Approximation properties of λ‐Bernstein‐Kantorovich operators with shifted knots

“…They also showed that this type of generalization improves the rate of convergence than that of the classical Kantorovich operators. Cai considered the Bézier variant of λ ‐Bernstein Kantorovich operators and established approximation theorem by using the usual second order modulus of smoothness and Ditzian‐Totik modulus of smoothness. Cai and Zhou presented the GBS operator of bivariate tensor product of λ ‐Bernstein Kantorovich operators and studied their convergence properties.…”