A New Modification of the Bernstein Operator Based on Shifted Evaluations

Abstract

This paper introduces a new modification of the classical Bernstein operator by replacing the point-wise evaluations \(\psi(w/n)\) with a linear symmetric combination of neighboring values \(\psi((w-1)/n)\) and \(\psi((w+1)/n)\), weighted by expressions depending on \(x\) and \(w\). The modified operator preserves the classical Bernstein structure while incorporating local smoothness into the evaluation process. We establish uniform convergence using Korovkin’s theorem and derive a Voronovskaja-type asymptotic formula. Numerical comparisons confirm that the proposed operator provides an improved approximation accuracy.