Approximation by Stancu variant of λ-Bernstein shifted knots operators associated by Bézier basis function

Abstract

The current paper presents the λ-Bernstein operators through the use of newly developed variant of Stancu-type shifted knots polynomials associated by Bézier basis functions. Initially, we design the proposed Stancu generated λ-Bernstein operators by means of Bézier basis functions then investigate the local and global approximation results by using the Ditzian–Totik uniform modulus of smoothness of step weight function. Finally we establish convergence theorem for Lipschitz generated maximal continuous functions and obtain some direct theorems of Peetre’s K-functional. In addition, we establish a quantitative Voronovskaja-type approximation theorem.

Keywords

• Bernstein basis polynomial
• Bézier basis function
• λ-Bernstein-polynomial
• Shifted knots
• Stancu operators
• Ditzian–Totik uniform modulus of smoothness