Approximation properties of λ-Bernstein operators

Abstract

Abstract In this paper, we introduce a new type λ-Bernstein operators with parameter λ∈[−1,1] $\lambda\in[-1,1]$, we investigate a Korovkin type approximation theorem, establish a local approximation theorem, give a convergence theorem for the Lipschitz continuous functions, we also obtain a Voronovskaja-type asymptotic formula. Finally, we give some graphs and numerical examples to show the convergence of Bn,λ(f;x) $B_{n,\lambda }(f;x)$ to f(x) $f(x)$, and we see that in some cases the errors are smaller than Bn(f) $B_{n}(f)$ to f.

Keywords

• λ-Bernstein operators
• Bézier basis functions
• Modulus of continuity
• Lipschitz continuous functions
• Voronovskaja asymptotic formula