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.

Keywords

• Stancu-type Bernstein operators
• Bézier bases
• Voronovskaja-type theorems
• modulus of continuity
• rate of convergence
• bivariate operators
• approximation properties