Approximation by Schurer Type <i>λ</i>-Bernstein–Bézier Basis Function Enhanced by Shifted Knots Properties

Abstract

In this article, a novel Schurer form of λ-Bernstein operators augmented by Bézier basis functions is presented by utilizing the features of shifted knots. The shifted knots form of Bernstein operators and the Schurer form of the Bézier basis function are used in this article, then, new operators, the Schurer type λ-Bernstein shifted knots operators are constructed in terms of the Bézier basis function. First, the test functions are calculated and the central moments for these operators are obtained. Then, Korovkin’s type approximation properties are studied by the use of a modulus of continuity of orders one and two. Finally, the convergence theorems for these new operators are obtained by using Peetre’s K-functional and Lipschitz continuous functions. In the end, some direct approximation theorems are also obtained.

Keywords

• Bernstein operators
• Bernstein-Schurer shifted knots operators
• Bézier basis function
• modulus of continuity
• λ-Bernstein operators
• Lipschitz type maximal function