Improving approximation via linear combinations of Bernstein operators

Abstract

In this paper, a new way to improve approximation operators is proposed. Two well-known methods are combined: the linear combination of the classic Bernstein operator and the iteration technique. The main goal is to make the convergence to the function faster and to obtain more accurate numerical results for practical use. The approximation error is analyzed using standard tools, such as the K-functional and the modulus of continuity. It is also proved that the proposed operator converges uniformly to the target function and a  Voronovskaja-type asymptotic formula is derived. Finally, numerical examples based on test functions are presented to show that the new operator compares favorably with existing methods.